Preservation of interpolation features by fibring

concrete panels. The panels will all be pretensioned in the trans-verse direction during fabrication and post-tensioned together in the longitudinal direction after place-ment. The advantage of using pre-stressed panels is a significant in-crease in the durability of the pavement, with a significant re-duction in required pavement thickness. For example, an 8 in. thick precast, prestressed pave-ment can be designed for the same design life as a 14 in. thick contin-uously reinforced concrete pave-ment by simply adjusting the pre-stress level in the pavement. This adjustment will not only result in significant material cost savings but will also allow for more flexi-bility when pavements are con-structed in areas with overhead clearance restrictions, such as un-der bridges.The proposed concept consists of three different types of panels, as shown in Figure 1. The base panels (Figure 1a) are the “filler”panels between the joint panels and central stressing panel(s). The central stressing panel (Figure 1b) is a panel similar to the base pan-el, with the addition of pockets cast into the panel. These pockets will allow the post-tensioning strands to be stressed at the center of the slab, rather than at the an-chorage, which will be cast into the joint panels. The joint panels (Figure 1c) will contain an expan-sion joint detail (Figure 2), similar to that of bridge expansion joints, which will absorb the significant expansion and contraction move-ments of the pavement with daily and seasonal temperature cycles.A typical panel assembly is shown in Figure 3. The slab length (between expansion joints) will be varied by an increase in the num-ber of base panels between the joint panels and central stressing panels. After all of the panels are set in place, the post-tensioning strands will be inserted into the ducts via the central stressing pockets and threaded through all of the panels to self-locking,spring-loaded post-tensioning an-chors cast into the joint panels.The use of self-locking anchorswill allow the strands to simply bepushed into the anchors fromsome point along the pavement,most likely from small pocketscast into the joint panels.After the post-tensioningstrands are tensioned from thecentral stressing pockets, thepockets will be filled with a fast-setting concrete, which will havesufficient strength by the timetraffic is allowed back onto thepavement. The strands will thenbe grouted in the ducts via inlets/vents located at the expansionjoints and at the stressing pockets.The intermediate joints betweenthe individual panels will then besealed with a low-viscosity, liquidsealant. If needed, the pavementcan then be diamond-ground tosmooth out any major irregulari-ties, and any major voids beneaththe pavement can be filled bystandard grout injection or expan-sive polyurethane foam.To obtain a smooth riding sur-face over the assembled pave-ment, continuous shear keys willbe cast into the panel edges, asshown in Figure 1, to ensure exactvertical alignment of the panels asthey are set in place. Additionally,the panels will be placed over athin, 1 to 2 in. thick, asphalt level-ing course, which should providea smooth, flat surface on whichthe panels can be placed to mini-mize the amount of voids beneaththe panels. A single layer of poly-ethylene sheeting will also beplaced over the asphalt levelingcourse to reduce the friction be-tween the leveling course and theprecast panels.Through the feasibility studydescribed above, the researchersdeveloped a feasible concept for aprecast concrete pavement. Thisconcept should meet the require-ments for both expedited con-struction and increased durability,which will result in both tremen-dous savings in user costs and anincreased design life.With respect to expedited con-struction, the proposed concepthas many features that will allowfor construction to take place dur-ing overnight or weekend opera-tions. First, the asphalt levelingcourse can be placed well in ad-vance of the precast panels. Thiswill allow for the entire asphaltleveling course to be placed at onetime, rather than just prior to theplacement of the precast slabs.Traffic on the leveling courseshould not have a detrimental ef-fect as long as the panels areplaced within a reasonable amountof time after the leveling course.Second, neither the stressingpockets nor the post-tensioningducts must be filled or groutedprior to exposure to traffic. Thepockets can simply be temporarilycovered and the strands can begrouted during a subsequent con-struction operation. Finally, tem-Figure 2. Expansion joint detail to be cast into the joint panels.6"2"1/4" Ø Stainlessporary precast ramps can simply be placed at the end of the slab to provide a transition for traffic onto and off the new pavement. These ramps can then be reused during subsequent operations.User delay costs can be sub-stantially reduced by limiting con-struction to an overnight or week-end timeframe. As an example, the computer program QUEWZ was used to compute and compare user delay costs for precast pave-ment construction and for conven-tional pavement construction. For conventional pavement construc-tion, wherein traffic is diverted through the construction zone for 24 hours per day until construc-tion is complete, the user delay costs were computed as approxi-mately $383,000 per day. On the other hand, precast pavement con-struction, wherein traffic is only diverted from 8 p.m. to 6 a.m. dai-ly, results in user delay costs of only $1,800 per day. Although it may not be possible to place as much precast pavement as con-ventional pavement during one day, the savings in user costs far outweigh any additional construc-tion time.In addition to expedited con-struction, precast pavement also offers enhanced durability. First, the panels will be cast in a con-trolled environment at a precast yard. This will allow for flexibili-ty with the concrete mix, making the use of lightweight, high per-formance, and other concretes possible. Second, because pre-stressing will be incorporated, cracking in the pavement can be prevented. This will reduce, if not eliminate, spalls and punchouts during the design life of the pave-ment. Prevention of cracking will also protect the post-tensioning strands in the pavement. The cast-in-place prestressed pavement constructed in 1985 on Interstate35 in McLennan County, Texas, isa testament to the increased dura-bility of prestressed pavements. Finally, because the precast pan-els will generally be thinner than conventional pavements, and be-cause there will be a great deal of control over the temperature gra-dient in the precast panels during casting, “built-in curl” will be sig-nificantly reduced, if not eliminat-ed. This will greatly reduce tem-perature curling stresses in the pavement.The Researchers Recommend...The proposed concept appears to be a feasible method for expe-diting construction of portland cement concrete (PCC) pave-ments. However, the true feasibil-ity of this concept will be realized only through actual implementa-tion. Therefore, a staged imple-mentation strategy is recommend-ed for testing these concepts and slowly introducing this new con-struction technique into current practices.Staged implementation will begin with small pilot projects aimed at refining the proposed concepts and streamlining the construction process. The pilot projects should be constructed on pavements that can be closed dur-Figure 3. Typical panel assembly.ing construction with a very mini-mal impact on traffic, such as cer-tain frontage roads or rest area roads. Any necessary laboratory testing should be completed prior to the construction of the pilot projects to ensure the viability of certain aspects, such as the spring-loaded anchors and strand place-ment procedures.The pilot projects will be fol-lowed by rural implementation, wherein the construction process will be further streamlined under simulated time constraints. As with the pilot projects, rural imple-mentation should be undertaken on pavements that will not have a very significant impact on traffic if problems occur during construc-tion. Rural implementation should take place, however, on a road that will experience significant traffic loading, such as a rural interstate.Finally, after rural implementa-tion, urban implementation will present the most challenges to pre-cast pavement construction. Urban implementation should take place on an urban intersection or major arterial where road closure must be limited to overnight or weekend operations. By the time urban im-plementation is undertaken, how-ever, the construction process should be fully streamlined to ac-commodate strict time constraints.Implementation will ultimately determine the feasibility of the precast concrete pavement con-cepts presented in this report. In the end, a simple concept that is easily adaptable to existing tech-niques yet not restricted by current practices will ensure the viability of precast concrete pavements.DisclaimerFor More Details …Research Supervisor: B. Frank McCullough, Ph.D., P.E., phone: (512) 232-3141,email:************************.eduTxDOT Project Director:Gary Graham, P.E., phone: (512) 467-5926,email:*****************The research is documented in the following report:Report 1517-1, The Feasibility of Using Precast Concrete Panels to Expedite HighwayPavement Construction, Draft January 2001To obtain copies of the report, contact: CTR Library, Center for Transportation Research,phone:(512)232-3138,email:*************.This research was performed in cooperation with the Texas Department of Transportation and the U. S. Department of Transportation, Federal Highway Administration. The contents of this report reflect the views of the authors, who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the official view or policies of the FHWA or TXDOT.This report does not constitute a standard, specification, or regulation, nor is it intended forconstruction, bidding, or permit purposes. Trade names were used solely for information and not for product endorsement. The engineer in charge was Dr. B. Frank McCullough, P.E. (Texas No. 19914).。

ESCAPING NASH INFLATIONIN-KOO CHO,NOAH WILLIAMS,AND THOMAS J.SARGENTA BSTRACT.Mean dynamics describe the convergence to self-confirming equilibria of self-referential systems under discounted least squares learning.Escape dynamics recurrentlypropel away from a self-confirming equilibrium.In a model with a unique self-confirmingequilibrium,the escape dynamics make the government discover too strong a version ofthe natural rate hypothesis.The escape route dynamics cause recurrent outcomes close tothe Ramsey(commitment)inflation rate in a model with an adaptive government.Key Words:Self-confirming equilibrium,mean dynamics,escape route,large deviation,natural rate of unemployment,adaptation,experimenta-tion trap.‘If an unlikely event occurs,it is very likely to occur in the most likely way.’Michael Harrison1.I NTRODUCTIONBuilding on work by Sims(1988)and Chung(1990),Sargent(1999)showed how a government adaptivelyfitting an approximating Phillips curve model recurrently sets inflation near the optimal time-inconsistent ouctome,although later inflation creeps back to the time-consistent suboptimal outcome of Kydland and Prescott(1977).The good outcomes emerge when the government temporarily learns the natural rate hypothe-sis.The temporary escapes from the time-consistent outcome symptomize a remarkable type of escape dynamics that promote experimentation and that are induced by unusual shock patterns that interact with the government’s adaptive algorithm and its imperfect model.The escapes lead to dramatic changes in the government’s inflation policy as it temporarily overcomes its inflationary bias.Some simulated time paths of inflation for different specifications of the model are shown in Figure1.Inflation starts and remains near the high time-consistent value for a while,is rapidly cut to zero,but then gradually2IN-KOO CHO,NOAH WILLIAMS,AND THOMAS J.SARGENTF IGURE ofthe model.approaches the time-consistent high value again.This paper explains the dynamic forces that drive these outcomes.Escape dynamics from self-confirming equilibria can occur in a variety of models with large agents who use adaptive algorithms to estimate approximating models.1For con-creteness,this paper focuses on the Phillips curve model studied by Sargent(1999).The model has the following features:(1)the monetary authority controls the inflation rate, apart from a random disturbance;(2)the true data generating mechanism embodies a version of the natural rate hypothesis in an expectational Phillips curve;(3)as in Kydland and Prescott(1977),a purposeful government dislikes inflation and unemployment and a private sector forecasts inflation optimally;but(4)the monetary policy makers don’t know the true data generating mechanism and instead use a goodfitting approximating model.The fundamentals in the economy arefixed,including the true data generating mechanism,preferences,and agents’methods for constructing behavior rules.Changes in the government’s beliefs about the Phillips curve,and how it approximates the natural rate hypothesis,drive the inflation rate.Inspired by econometric work about approximat-ing models by Sims(1972)and White(1982),we endow the monetary authority,not with the correct model,but with an approximating model that it nevertheless estimates with good econometric procedures.We use the concept of a self-confirming equilibrium,a natural equilibrium concept for behavior induced by an approximating model.2In a self-confirming equilibrium,beliefs are correct about events that occur with positive probability in equilibrium.The approxi-mating model is‘wrong’only in its description of events that occur with zero probability in equilibrium.Among the objects determined by a self-confirming equilibrium are theESCAPING NASH INFLATION3 parameters of the government’s approximating model.While the self-confirming equi-librium concept differs formally from a Nash(or time consistent)equilibrium,3it turns out that the self-confirming equilibrium outcomes are the time-consistent ones.Thus,the suboptimal time consistent outcome continues to be our benchmark.Like a Nash equilibrium,a self-confirming equilibrium restricts population objects (mathematical expectations,not sample means).We add adaptation by requiring the government to estimate its model from historical data in real time.We form an adap-tive model by having the monetary authority adjust its behavior rule in light of the latest model estimates.Thus,we attribute‘anticipated utility’behavior(see Kreps(1998))to the monetary authority.Following Sims(1988),we study a‘constant gain’estimation al-gorithm that discounts past observations.Called a‘tracking algorithm’,it is useful when parameter drift is suspected(see e.g.Marcet and Nicolini(1997)).Results from the literature on least squares learning(e.g.,Marcet and Sargent(1989a), Woodford(1990),Evans and Honkapohja(1998))apply and take us part way,but only part way,to our goal of characterizing the dynamics of the adaptive system.That litera-ture shows how the limiting behavior of systems with least squares learning is described by an ordinary differential equation called the‘mean dynamics’.They describe the(un-conditionally)average path of the government’s beliefs,in a sense that we shall describe precisely.For our model,the mean dynamics converge to the self-confirming equilibrium and the time consistent outcome.Thus,the mean dynamics do not account for the recur-rent stabilizations in the simulations of Sims(1988),Chung(1990),and Sargent(1999). We show that these stabilizations are governed by another deterministic component of the dynamics,described by another ODE,the‘escape’dynamics.They point away from the self-confirming equilibrium and toward the Ramsey(or optimal-under-commitment) equilibrium outcome.So two sorts of dynamics dominate the behavior of the adaptive system.(1)The mean dynamics come from an unconditional moment condition,the least squaresnormal equations.These dynamics drive the system toward a self-confirmingequilibrium.4(2)The escape route dynamics propel the system away from a self-confirming equilib-rium.They emerge from the same least squares moment conditions,but they areconditioned on a particular“most likely”unusual event,defined in terms of the disturbance sequence.This most likely unusual event is endogenous.The escape route dynamics have a compelling behavioral interpretation.Within the confines of its approximate model,learning the natural rate hypothesis requires that the government generate a sufficiently wide range of inflation experiments.To learn even an imperfect version of the natural rate hypothesis,the government must experiment more than it does within a self-confirming equilibrium.The government is caught in an experimentation trap.The adaptive algorithm occasionally puts enough movement into the government’s beliefs to produce informative experiments.4IN-KOO CHO,NOAH WILLIAMS,AND THOMAS J.SARGENT1.1.Related literature.Evans and Honkapohja(1993)investigated a model with mul-tiple self-confirming equilibria having different rates of inflation.When agents learn through a recursive least squares algorithm,outcomes converge to a self-confirming equi-librium that is stable under the learning algorithm.When agents use afixed gain algo-rithm,Evans and Honkapohja(1993)demonstrated that the outcome oscillates among different locally stable self-confirming equilibria.They suggested that such a model can explain widefluctuations of market outcomes in response to small shocks.In models like Evans and Honkapohja(1993)and Kasa(1999),the time spent in a neighborhood of a locally stable equilibrium and the escape path from its basin of at-traction are determined by a large deviation property of the recursive algorithm.As the stochastic perturbation disappears,the outcome stays in a neighborhood of a particular locally stable self-confirming equilibrium(exponentially)longer than the others.This observation provided Kandori,Mailath,and Rob(1993)and Young(1993)with a wayto select a unique equilibrium in evolutionary models with multiple locally stable Nash equilibria.An important difference from the preceding literature is that our model has a unique self-confirming equilibrium.Despite that,the dynamics of the model resemble those for models with multiple equilibria such as Evans and Honkapohja(1993).With multiple locally stable equilibria,outcomes escape from the basin of attraction of a locally stable outcome to the neighborhood of another locally stable equilibrium.The fact that our model has a globally unique stable equilibrium creates an additional challenge for us, namely,to characterize the most likely direction of the escape from a neighborhood of the unique self-confirming equilibrium.As we shall see,the most likely direction entails the government’s learning a good,but not self-confirming,approximation to the natural rate hypothesis.anization.Section2describes the model in detail.Section3defines a self-confirming equilibrium.Section4describes a minimal modification of a self-confirming equilibrium formed by giving the government an adaptive algorithm for its beliefs.Section5uses re-sults from the theory of large deviations to characterize convergence to and escape froma self-confirming equilibrium.Section6shows that numerical simulations of escape dy-namics,like those in Sargent(1999),are well described by the numerically calculated theoretical escape paths.For the purpose of giving intuition about the escape dynamics, Section7specializes the shocks to be binomial,then adduces a transformed measure of the shocks that tells how particular endogenously determined unusual shock sequences drive the escape dynamics.Section8concludes.The remainder of this introduction de-scribes the formal structure of the model andfindings of the paper.1.3.Overview.The government’s beliefs about the economy are described by a vector of regression coefficients.It chooses a decision rule that makes the stochastic process for the economy be.But for the stochastic process,the bestfitting model ofthe economy has coefficients.A self-confirming equilibrium is afixed point of .The orthogonality conditions pinning down the bestfitting model can be expressed (1.1)ESCAPING NASH INFLATION5 We shall show thatwhereA self-confirming equilibrium is a set of population regression coefficients.We form an adaptive model by slightly modifying a self-confirming equilibrium.Rather than usingpopulation moments tofit its regression model,the government uses discounted leastsquares estimates from historical samples.We study how the resulting adaptive systemconverges to or diverges from a self-confirming equilibrium.Each period the govern-ment uses the most recent data to update a least squares estimate of its model co-efficients,then sets its policy according to.This is what Kreps(1998)calls an anticipated utility model.The literature on least squares learning in self-referential sys-tems(see Marcet and Sargent(1989a),Marcet and Sargent(1989b),Woodford(1990),andEvans and Honkapohja(2000))give conditions under which the limiting behavior of thegovernment’s beliefs are nearly deterministic and approximated by the following ordi-nary differential equation(ODE)is governed by the uniqueness and stability of the stationary points of the ODE.Our model has a unique self-confirming equilibrium.It supports the high inflationtime-consistent outcome of Kydland and Prescott(1977).The ODE(1.3),(1.4),is veryinformative about the behavior of our adaptive model.It is globally stable about theself-confirming equilibrium,and describes how the adaptive system is gradually drawnto the self-confirming equilibrium.But to understand how the sample paths recurrentlyvisit the better low-inflation outcome,we need more than the ODE(1.3,1.4).Until our work,such‘escape dynamics’had not been characterized analytically.Thispaper shows that they are governed by the ODE6IN-KOO CHO,NOAH WILLIAMS,AND THOMAS J.SARGENTrate hypothesis.Thus,like the mean dynamics,the escape dynamics are deterministic. We verify that these deterministic dynamics do a good job of describing the simulations. As Sims(1988)and Sargent(1999)emphasize,the evolution of beliefs during an es-cape is economically interesting because then the government discovers a good enough approximate version of the natural rate hypothesis to cause it to pursue superior policy that is supported by beliefs that are‘wrong’in the sense that they are not a self-confirming equilibrium.Nevertheless,in another sense those beliefs are more‘correct’than those in a self-confirming equilibrium because they inspire the government to leave the‘experi-mentation trap’that confines it within a self-confirming equilibrium.2.S ETUPTime is discrete and indexed by.Let be an i.i.d.sequence of random vectors with mean zero and covariance matrix.Let,respectively,be the unemployment rate,the rate of inflation,the public’s expected rate of inflation,and the systematic part of inflation determined by government policy.The government sets ,the public sets,then nature chooses shocks that determine and.The economy is described by the following version of a model of Kydland and Prescott(1977):(2.8)(2.9)(2.10)(2.11)where(2.12)Equation(2.8)is a natural rate Phillips curve;(2.9)says that the government sets infla-tion up to a random term;(2.10)imposes rational expectations for the public;(2.11)is the government’s decision rule for setting the systematic part of inflation.The de-cision rule is a function of the government’s beliefs about the economy,which are parameterized by a vector.For some purposes below we consider the simpler model in which the government only estimates a static regression of unemployment on inflation and a constant(i.e. ).We call this the static model.Since there is no temporal dependence in(2.8),(2.9),all of the temporal dependence in the model comes through the government’s beliefs.Under the static model specification,the government’s control rule can be calculated explicitly, allowing some of our characterizations to be sharper.2.1.The government’s beliefs and control problem.The government’s model of the economy is a linear Phillips curve(2.13)where the government treats as a mean zero,serially uncorrelated random term beyond its control.We shall eventually restrict,but temporarily regard it as arbitrary.TheESCAPING NASH INFLATION7 government’s decision rule(2.11)solves the problem:(2.14)where denotes the expectations operator induced by(2.13)and the minimization is subject to(2.13)and(2.9).We call problem(2.14)the Phelps problem.Versions of it were studied by Phelps(1967), Kydland and Prescott(1977),Barro and Gordon(1983),and Sargent(1999).We identify three salient outcomes associated with different hypothetical government’s beliefs: Belief1.If,then the Phelps problem tells the government to set for all.This is the Nash outcome of Sargent(1999),i.e.,the time-consistent outcome of Kydland and Prescott(1977).Belief2.If,for any,the government setsfor all.This is the Ramsey outcome,i.e.,the optimal time-inconsistent outcome of Kydland and Prescott(1977).Belief3.If the coefficients on current and lagged’s sum to zero,then asfrom below,the Phelps problem eventually sends arbitrarily close to.Under the actual probability distribution generated by(2.8),(2.9),(2.10),the value of the government’s objective function(2.14)is larger under the outcome than under outcome.Under Belief1,the government perceives a trade-off between in-flation and unemployment and sets inflation above zero to exploit that trade-off.Under Belief2,the government perceives no trade-off,sets inflation at zero,and accepts what-ever unemployment emerges.Under Belief3,the government thinks that although there is a short-term trade-off between inflation and unemployment when,there is no ‘long-term’trade-off.Through the workings of an‘induction hypothesis’that opens an apparent avenue by which the government can manipulate the current position of the Phillips curve(see Cho and Matsui(1995)and Sargent(1999)),the Phelps problem tells the government eventually to set inflation close to zero when is close to.In a common-knowledge model in which(2.13)is dropped and replaced by the as-sumption that the government knows the model,the outcome emerges as what Stokey(1989)and Sargent(1999)call the Nash outcome,and emerges as the Ram-sey outcome.In the common-knowledge model,these varying outcomes reflect different timing protocols and characterize a time-consistency problem analyzed by Kydland and Prescott.The mapping from government beliefs to outcomes is interesting only when the gov-ernment’s beliefs might be free.Our equilibrium concept,a self-confirming equilibrium, restricts those beliefs,and thereby narrows the outcomes relative to those enumerated above.However,the mapping from beliefs to outcomes play a role during escapes from self-confirming equilibria.8IN-KOO CHO,NOAH WILLIAMS,AND THOMAS J.SARGENT3.S ELF-CONFIRMING EQUILIBRIUM3.1.Restrictions on government’s beliefs.Define and(3.15)Let denote the history of the joint shock process up to.Evidently,from(2.8),(2.9),(2.10),(2.11),and therefore the process are both functions of:(3.16)Definition3.1.A self-confirming equilibrium is a that satisfies(3.17)The expectation in(3.17)is taken with respect to the probability distribution generated by(2.8),(2.9),(2.10),(2.11).Notice that is the time value of the object set to zero by the following least squares orthogonality condition:(3.18)Equations(3.18)are the orthogonality conditions that make in(2.13)a least-squares regression.Condition(3.17)thus renders the government’s beliefs consistent with the data.Condition(3.17)can be interpreted as asserting that is afixed point in a mapping from the government’s beliefs about the Phillips curve to the actual Phillips curve.Thus, let(3.19)and.Then notice that(3.20)(3.22)Given a government model in the form of a perceived regression coefficient vector and the associated government best response function,is the actual least squares regression coefficient induced by.Thus,maps government model to a bestfitting model.Equation(3.22)shows that(3.17)asserts that,so thatESCAPING NASH INFLATION9 the government’s model is the bestfitting model.See Marcet and Sargent(1989a)for a discussion of the operator in a related class of models.Elementary calculations show that there is a unique self-confirming equilibrium.It cor-responds to the beliefs(1)mentioned above.These beliefs support the Nash equilibrium outcome in the sense of Stokey(1989)and Sargent(1999).4.A DAPTATION4.1.Discounted least squares updating of.We modify the model now to consist of (2.8),(2.9),(2.10)as before,but replace(2.11)with(4.23)where remains the best-response function generated by the Phelps problem,and is the government’s time estimate of the empirical Phillips curve.The government estimates by the following recursive least squares algorithm:(4.24)(4.25)where is a gain parameter that determines the weight placed on current observations relative to the past.In this paper we consider the case in which the gain is constant.We want to study the behavior of system formed by(2.8),(2.9),(2.10),(4.23),(4.24)and(4.25).4.2.Mean dynamics.Wefind thefirst important component of dynamics by adapting the stochastic approximation methods used by Woodford(1990),Marcet and Sargent (1989a),and Evans and Honkapohja(2000).We call this component the mean dynamics because it governs the(unconditionally)expected evolution of the government’s beliefs. While previous applications of stochastic approximation results in economics have gener-ally considered recursive least squares with decreasing gain,we consider the case where the gain is constant.5Broadly similar results obtain in the constant and decreasing gain cases,but there are important differences in the asymptotics and the sense of convergence that we discuss below.To present convergence proofs,it helps to group together the components of the gov-ernment’s beliefs into a single vector.Define(4.26)Then the updating equations(4.24),(4.25)can be written(4.27)Now break the“update part”into its expected and random components.Defineis the mean of defined as(4.28)5See Evans and Honkapohja(2000)for extensive discussion of constant gain algorithms.10IN-KOO CHO,NOAH WILLIAMS,AND THOMAS J.SARGENTwhere(4.29)Then we can write the composite dynamics as(4.30))over time.As in the decreasing gain case,we can show that the asymptotic behavior of(4.30)is governed by an ODE,but the estimates converge in a weaker sense.Specifically,decreas-ing gain algorithms typically converge with probability one along a sequence of iterations as,but constant gain algorithms converge weakly(or in distribution)as across sequences of iterations,each of which is indexed by the gain.Note that we can rewrite(4.30)as(4.31)This equation resembles afinite-difference approximation of a derivative with time step ,but is perturbed by a noise term.The convergence argument defines a continuous time scale as,and interpolates between the discrete iterations to get a continuous process.Then by letting,the approximation error in thefinite difference goes to zero,and a weak law of large numbers insures that the noise term becomes negligible. We are left with the ODE:(4.33)We need the following set of assumptions.For reference,we also list the original num-ber in Kushner and Yin(1997).To emphasize the asymptotics,we include the superscript on the parameters denoting the gain setting.Assumptions A.A8.5.0:The random sequence is tight.6A8.5.1:For each compact set is uniformly integrable.7A8.5.3:For each compact set the sequence6A random sequence is tight if7A random sequence is uniformly integrable ifA8.5.4a:The ODE that is asymptotically stable.8A8.1.6:The functionthat is the self-confirming equilibrium,the estimate sequence converges weakly to the self-confirming equilibrium.Therefore,with high probability,as and we would expect the government’s beliefs to be near their self-confirming values,and the economy to be near the Nash outcome.However,in the next section we shall see that the beliefs can recur-rently escape the self-confirming equilibrium.Although the impact of noise terms goes to zero with the gain,for a given,“rare”sequences of shocks can have a large impact on the estimates and the economy.5.E SCAPEIn this section we determine the most likely rare events and how they push the gov-ernment’s beliefs away from a self-confirming equilibrium.To this end,wefirst present some general results from the theory of large deviations,a general method for analyzing small probability events.We then present results from Williams(2000),who applies these general results analytically to characterize the escape dynamics.5.1.Escape dynamics as a control problem.Throughout,we will only be interested in characterizing the escape problem for the Phillips curve coefficients.This motivates the following definition.Definition5.1.An escape path is a sequence of estimates that leave a set containing the limit pointfor someFollowing a convention in the large deviation literature,we set the initial point of an escape path to be the stable point,let be the set of all escape paths.For each,define8A point as and for each there exists an such that if for allDefinition5.2.Let be the(first)exit time associated with escape path. An absolutely continuous trajectory is a dominant escape path ifwill occur along with very high probability,if an escape ever occurs.To analyze the escape dynamics,we adapt the general results of Dupuis and Kushner (1989),which are themselves extensions of the theory of Freidlin and Wentzell(1984) for stochastic approximation models.After presenting some general results,we apply results of Williams(2000),who obtains explicit solutions of the escape dynamics that can be used to interpret the simulations calculated earlier by Sims(1988),Chung(1990),and Sargent(1999).Given the recursive formula(4.30),define the-functional as(5.35),and with the evolution of following the mean dynamics conditional on .(We let for trajectories that are not absolutely continuous.)In the context of continuous time diffusions,Freidlin and Wentzell(1984)characterized the dominant escape path as a solution of a variational problem.Their results have been extended to discrete time stochastic approximation models by Dupuis and Kushner(1985)and Dupuis and Kushner(1989).We adapt these results in the following theorem,whose main object is the solution of the following variational problem:(5.38)for someThe minimized value(1)Suppose that the shocks are i.i.d.and unbounded but that there exists a algebraand constants such that for all anda.s.Then we have:for some(2)If the shocks are i.i.d.and bounded,andbe the terminal point of the dominant escape path.Then for any and:.The next three parts establish stronger results under the assumption that the errors are bounded.Part(2)shows that under bounded errors,the asymptotic inequality in part(1)becomes an asymptotic equality. Part(3)shows that for small the time it takes beliefs to escape the self-confirming equi-librium becomes close to.It is known(see Benveniste,Metivier,and Priouret(1990)for example)that the asymptotic distribution of Markov processes can be characterized by the Poisson equa-tion,so it is natural that it appears here.This analysis then leads to a representation of the-functional as a quadratic form in,with a normalizing matrix that depends on the solution of the Poisson equation associated with.In general the solution of the Poisson equation can itself be a difficult problem,as it involves solving a functional equation.However in the important linear-quadratic-Gaussian case(which includes our model),the problem can be solved in the space of quadratic functions,and therefore the Poisson equation reduces to a matrix Lyapunov equation.This provides a tremendous simplification,as there are efficient numerical methods for solving Lyapunov equations. We summarize these arguments in the following theorem and remark.Theorem5.4.Suppose that Assumptions A hold,that follows a stationary functional au-toregression with a unique stationary distribution and Lipschitz continuous mean and variance functions,and that the function is Lipschitz continuous in.Then there is a matrix-valued function such that the dominant escape path and rate function can be determined by solving the following variational problem:(5.39)subject to(5.41)(5.42)for someProof.See Williams(2000).Remark5.5.In our model,follows a linear autoregression,the are i.i.d.normal,and is a quadratic function of.Then is a fourth moment matrix that can be calculated explicitly by solving matrix Lyapunov equations described in Appendix C.This theorem provides a clearer interpretation and analysis of the variational problem. The escape dynamics perturb the mean dynamics by a forcing sequence.Then is a quadratic cost function that measures the magnitude of the perturbations during the episode of an escape.In particular,we can think of(5.39)as a least squares problem, where plays the role of a covariance matrix.If we had then the beliefs adhere to the mean dynamics,and the cost would be zero.For the beliefs to escape from.Tofind the dominant escape path,we solve the control problem in(5.39).We form the Hamiltonian with co-states for the evolution of:It is easy to verify that the Hamiltonian is convex,so thefirst order conditions are nec-essary and sufficient.Taking thefirst order conditions,we see that the dominant escape path is characterized by the following set of differential equations:The path that achieves the minimum is the dominant escape path.This path characterizes the evolution of the parameters on the most likely path away from the stable point.The minimized value.There is a unique self-confirming equilibrium,depicted in Figure2.It has.To solve the problem numerically,it helps to recast the boundary value problem as an initial value problem.In the ODE system(5.43)and boundaries(5.42),the only com-ponents left undetermined are the initial conditions for the co-states.We can solve the problem by minimizing over these initial conditions,and determine the escape times and。

Eurographics Symposium on Rendering(2005)Kavita Bala,Philip Dutré(Editors)Multiresolution Reflectance FilteringPing Tan1†Stephen Lin2Long Quan1Baining Guo2Heung-Yeung Shum2Hong Kong University of Science and Technology1Microsoft Research,Asia2AbstractPhysically-based reflectance models typically represent light scattering as a function of surface geometry at the pixel level.With changes in viewing resolution,the geometry imaged within a pixel can undergo significant varia-tions that result in changing reflectance characteristics.To address these transformations,we present a multires-olution reflectance framework based on microfacet normal distributions within a pixel over different scales.Since these distributions must be efficiently determined with respect to resolution,they are recorded at multiple reso-lution levels in mipmaps.The main contribution of this work is a real-time mipmapfiltering technique for these distribution-based parameters that not only provides smooth reflectance transitions in scale,but also minimizes aliasing.With this multiresolution reflectance technique,our system can rapidly and accurately incorporatefine reflectance detail that is customarily disregarded in multiresolution rendering methods.Categories and Subject Descriptors(according to ACM CCS):I.3.5[Computer Graphics]:Physically based modeling1.IntroductionVarious computer graphics techniques have been presented for rendering objects at different resolutions.For example, object geometry at different scales has been addressed by work in multiresolution meshes(e.g.,[Hop96]),and color textures can be rendered at different levels of detail us-ing mipmaps[Wil83].These multiresolution representations serve the purpose of reducing run-time computation by ap-proximating small-level details in a manner that maintains accurate appearance.Although geometry and textures are often modelled at multiple scales,the reflectance properties of an object are customarily represented by afixed bidirectional reflectance distribution function(BRDF).Reflectance characteristics such as shading and highlights,however,clearly depend upon the surface geometry imaged within a pixel,and typi-cally the geometric characteristics at the pixel level change with scale,as shown in Fig.1.As the viewer zooms out from a surface,fine-level geometric elements known as mesostructure reduce in scale into the pixel domain.Since changes in resolution can alter geometric characteristics at †Correspondence email:ptan@t.hk the pixel level as illustrated in Fig.2,reflectance should also change accordingly.To address this problem,we propose a technique for real-time rendering of multiresolution reflectance with respect to the geometric structure within each pixel.In represent-ing pixel-level geometry,most physically-based reflectance models consider a surface to be composed of planar micro-facets,and their surface normal directions are represented by a Gaussian distribution[CT81,APS00].As seen in Fig.2,a single Gaussian is often inadequate for modeling the more complex microfacet normal distributions of low resolution pixels which view a broader surface area,so a Gaussian mix-ture model(GMM)may be employed to more accurately convey multi-resolution variations in pixel geometry.Con-ventional microfacet-based reflectance models can be di-rectly extended to compute reflectance from GMMs,and in our implementation,we utilize a combination of extended Lambertian and Cook-Torrance models[CT81]for the dif-fuse and specular components of reflectance,respectively. The principal challenge of employing a multiresolution reflectance model is in how to efficiently and accurately de-termine the reflectance parameters at different scales.Since mesostructure geometry can vary over a surface,these re-flectance parameters can be spatially variant as well.In ob-taining GMM parameters at multiple resolutions,it is im-c The Eurographics Association2005.(a)(b)Figure1:Changing reflectance characteristics of a desert at different resolutions.(a)Rendering results with our method;(b)Real photographs captured at similar resolutions. practical to assemble and process the microfacets of each pixel at runtime,so we precompute and store them in mipmaps.However,since reflectance is a non-linear func-tion of distribution-based GMM parameters,these GMM mipmaps cannot be trilinearly interpolated like color tex-tures.Moreover,since reflectance is very sensitive to geo-metric information,filtering of GMM parameters must be carefully considered for aliasing to be minimized.The primary contribution of our work is a method for constructing andfiltering reflectance mipmaps that represent pixel geometry information.To promote accurate mipmap interpolation,we propose a Bayesian algorithm for prefilter-ing normal distributions such that a correspondence of distri-bution features is obtained.For corresponding features,we present a rapidfiltering technique that capitalizes on hard-ware mipmapping functions to perform the non-linearfil-tering needed to optimally interpolate the distribution-based quantities.Though the formulation of the prefiltering algo-rithm is somewhat involved,the runtime mipmap interpo-lation and reflectance computation are simple and straight-forward to implement.With this method,accurate scale-dependent reflectance effects with smooth transitions are rendered at high frame rates with negligible aliasing.2.Previous WorkNumerous reflectance models have been proposed in com-puter graphics,and they have generally been designed for rendering BRDFs at afixed resolution.Although some empirical models,such as spherical harmonics[W AT92], have theflexibility to represent multiresolution reflectance, we take a physically-based approach and focus on the re-flectance effects of changing pixel geometry through multi-ple resolutions.To account for the effect of pixel geometry on reflectance, Becker and Max[BM93]tabulate at rendering time the distribution of unoccluded normals from the projected dis-placement map in each pixel.For a substantial reductionfacet normal distributionpixel areaFigure2:Normal distributions at coarser resolution(top), andfiner resolution(bottom).Distributions within a pixel change with respect to rendering resolution,which causes reflectance variations over different scales.in run-time computation,mipmaps of geometric attributes have been employed.For a mipmap of bump maps,Toksvig [Tok04]presents a method for adjusting surface roughness values over multiple scales based on a simple consistency measure of surface normal directions.In rendering com-plex volumetric structures at multiple scales,Neyret[Ney98] mipmapped intra-pixel geometry that was expressed in terms of an ellipsoid model of normal distributions.At coarse resolutions,the geometry within a pixel can be-come so complicated that a more general model is needed for accurate representation.More important,the subsampling of normal distributions that effectively occurs with simpli-fied models can lead to significant aliasing that is magnified in rendering due to the sensitivity of reflectance to surface normals,as illustrated in Fig.4(a).A high sampling rate is utilized in a method by Fournier[Fou92],which mod-els reflectance as a set of seven Phong peaks per texel.In the mipmapfiltering process,Phong peaks from interpolated texels are scaled and aggregated such that up to56peaks are used in rendering a pixel,which entails a non-trivial ex-pense in computation.Similar to Fournier’s work,we repre-sent pixel geometry in terms of a GMM of normal direc-tions.However,efficientfiltering of these highly-sampled distribution-based parameters poses a challenging problem that was not addressed in[Fou92].In this work,we success-fully interpolate these reflectance mipmaps by using a pro-posed prefiltering technique for GMM mipmap construction and by non-linearfiltering of the reflectance parameters in a manner that exploits linear hardwarefilters.With this ap-proach,resolution dependencies of reflectance are modeled accurately and efficiently.3.Multiresolution Reflectance ModelMicrofacet normal distributions are typically represented by a single Gaussian lobe,which is suitable for micro-scale surface normal variations referred to as surface rough-ness[TS67].At coarser resolutions where surface geometry perturbs the distribution of normals in a pixel,the single-Gaussian model becomes less valid,as illustrated in Fig.2.c The Eurographics Association2005.For a multiresolution representation,a more general descrip-tor is needed,so we employ a Gaussian mixture model (GMM)of normal distributions.GMMs have long been used for modeling general distributions,and the use of two Gaus-sians for representing normal distributions has been sug-gested previously[CT81].Since a single Gaussian describes a micro-scale normal distribution at a single point,the distri-bution for a pixel that encompasses multiple such points can be expected to have a mixture of Gaussians formG(n)=N∑i=1αi g(n;µi,σi),where N is the number of Gaussians,g(·)is a Gaussian with mean normal vectorµi=(µi(x),µi(y))and univariate stan-dard deviationσi,andαi are mixture weights.Considering surface geometry as a heightfield z=f(x,y),a unit normal n can be represented as a2D vector in x-y space.For a GMM distribution of microfacet normals,the Lam-bertian and Cook-Torrance models can be directly extended by summing the reflectance values for each individual Gaus-sian.For the Cook-Torrance component,this sum can be ex-pressed asI CT=ρs F(u)N∑i=1αi g(h,µi,σi)µi·vwhereρs is the illumination color,l and v are lighting and viewing directions defined in the local coordinate frame, and F(u)is the simplified Fresnel function of the material [Sch94]with respect to u=max{v·h,0}where h denotes the bisector of l and v.Similarly,the Lambertian component can be formulated for a sum of N Gaussians,whose mean normal vectorsµi are used to represent surface normals:I Lamb=ρd(1−F(u))N∑i=1αi[µi·l].whereρd is the average albedo of the microfacets in a pixel as determined from a color mipmap.A scale dependent reflectance function that employs GMMs can then be ex-pressed as a sum of these two components:I=I CT+I Lamb.4.Reflectance MipmapsFor efficient rendering of the scale-dependent reflectance function,the GMM parameters need to be rapidly deter-mined.To facilitate this computation,we precompute a re-flectance mipmap for which the GMM parameters of each texel are computed according to normal samples drawn from thefinest-resolution surface heightfield.Infiltering a reflectance mipmap,a GMMˆΘ={ˆg i:= (ˆαi,ˆµi,ˆσi);1≤i≤N}needs to be accurately and efficiently interpolated from the K GMMs{Θk;1≤k≤K}={g ki:= (αki,µki,σki);1≤i≤N,1≤k≤K}of the neighboring K texels.Although standard graphics hardware provides an ef-ficient trilinear interpolation function,the distribution-based GMM parameters are non-linearly related to reflectance,and therefore cannot be accuratelyfiltered by trilinear interpola-tion.As noted by[Fou92],K N-modal distributions may be(a)(b)Figure3:Interpolation of unaligned(a)and aligned(b) GMMs.Gaussian components are numbered to show the correspondence.Interpolation takes place between Gaus-sian lobes with same index.While alignment maintains proper geometric characteristics after interpolation,mis-alignment results in lost detail and essentially a reduction in sampling rate.interpolated by an NK-modal distribution,and all NK com-ponents should ideally be rendered and summed for accu-rate computation of reflectance.However,this accumulation of distribution components dramatically degrades rendering efficiency.To address this problem,we propose a GMM preprocess-ing andfiltering technique that accurately and efficiently computes interpolated GMMs that have the same number of Gaussian components as the GMMs of mipmap texels. For each Gaussian componentˆg i of thefiltered GMM,we interpolate corresponding Gaussians g ki,1≤k≤K,from the GMMs of the K neighboring texels.To elevate accuracy, we present an algorithm that computes mipmap GMMs in a manner that yields similar means for corresponding Gaus-sians g ki among neighboring texels.For efficientlyfiltering this precomputed reflectance mipmap at rendering time,we describe a technique for optimal non-linear interpolation of corresponding GMM components that takes advantage of trilinearfilters in standard graphics hardware.4.1.GMM AlignmentInterpolating a set of K Gaussians by a single Gaussian can be accurate only when the centers of these Gaussians are closely located.To minimize interpolation error,it is there-fore necessary for neighboring texels to have aligned GMMs such that corresponding Gaussian components have simi-lar means.As illustrated in Fig.3,if corresponding Gaus-sians have means that are distant from one another,thefil-tered GMM tends to be averaged towards a single lobe, effectively resulting in a downsampled representation that causes strong aliasing.To facilitate alignment of GMM com-ponents among neighboring mipmap texels,we propose a modification to the Expectation-Maximization(EM)algo-rithm[DLR77,Bil97]that solves for the texel GMMs in a manner that favors close alignment of neighboring GMM components.With the traditional EM algorithm,the parametersΘof an N-Gaussian GMM are computed according to normal sam-ples X={n j;1≤j≤M}drawn from the original high-resolution heightfield,such thatΘ=argmax P(X|Θ)=argmaxΠM j=1 ΣN i=1αi g(n j;µi,σi) .c The Eurographics Association2005.To globally align GMM components of neighboring texels,we introduce the following prior constraint on GMM param-eters Θwith respect to the GMM parameters of neighboring texels N (Θ)={Θk ;1≤k ≤K }:P (Θ|N (Θ))=ΠK k =1P (Θ|Θk )=ΠK k =1ΠN i =1exp (−αki ||µki −µi ||2)P (Θ|Θk )is at its largest when the corresponding means µi ,µki are identical,and maximizing it will favor neighbor-ing GMMs with similar means for corresponding Gaussian components.Moreover,the penalty for discrepancies among Gaussian means ||µki −µi ||2is scaled by αki ,which allows a larger discrepancy for less important Gaussians with a smaller mixture weight.In some cases,the computed weight of a Gaussian component may be zero,αki =0,when the Gaussian does not contribute significantly to the GMM and does not correspond well with the neighboring GMMs.As such,the GMM parameters can be well defined by the fol-lowing maximum a posterior (MAP)formula:Θ=argmax P (Θ|X )=argmax P (X |Θ)P (Θ|N (Θ))=argmax [log (P (X |Θ))+log (P (Θ|N (Θ)))]This posterior can be maximized by the Expectation-Maximization algorithm [Bil97].For each texel in the mipmap,we alternately compute the expectation of sample n j drawn from the i -th Gaussian componentEz i j =αi g (n j ;µi ,σi )/N∑k =1αk g (n j ;µk ,σk ),and the maximization of the GMM parametersαi =∑M j =1Ez i j M σ2i =M ∑j =1Ez i j ||n j −µi ||2M ·d µi ≈∑M j =1Ez i j n j +c ∑Kk =1αki µki ∑M j =1Ez i j +c ∑Kk =1αki until the GMM parameters converge.Here,αki ,µki are theGMM parameters of neighboring texel k ,d =2is the dimen-sion of the unit normal vector,and c is a constant that mod-ulates smoothness among corresponding Gaussians,which we set to 1for all experiments.To accelerate convergence,normal samples are clustered into N groups and each group is used to compute the initial values of g i =(αi ,µi ,σi );1≤i ≤N .In a mipmap structure,each texel has 13neighbors:eight at the same level,one from the immediate coarser level,and four at the finer level.Be-cause of this network relationship,all the GMM parameters should ideally be estimated simultaneously,but this would result in a complex and expensive optimization.Instead,we compute the GMM of each texel one by one in scanline or-der,moving from coarser to finer levels,during which only those texels whose GMM parameters have been computed are counted as valid neighbors.Generally,this mipmap pre-computation step converges quickly,in about 5minutes for a 512×512height field with a 2.8GHz Pentium IV CPU.4.2.Interpolation of Aligned GMM ComponentsWith GMMs aligned by the described EM algorithm and stored in the reflectance mipmap,filtering isper-(a)(b)Figure 4:A closeup section of the carved silver disk in Fig.6for three consecutive frames.(a)Rendered using the Toksvig method with a 16-bit floating-point normal mipmap;(b)Rendered with our method.Our results are more consis-tent between frames,while the Toksvig method has signif-icant animation aliasing.In comparison to a ground truth image that traces 16rays/pixel with our detailed reflectance model,the normalized Euclidean color error per pixel is 0.060for our method and 0.165for the Toksvig method.formed among corresponding GMM components.For the i -th component of a GMM,an interpolated Gaus-sian g (n ,ˆµi ,ˆσi )is computed according to samples drawn from the K corresponding Gaussian distribu-tions g (n ;µki ,σki ),k =1,2,···,K .An optimal interpolationcan be expressed in closed-form as ˆµi =1K ∑Kk =1µki and ˆσi 2=1K∑K k =1σki 2+||µki ||2 −ˆ||µi ||2.Although the Gaussian mean ˆµi can be directly computed by trilinear in-terpolation,the variance ˆσ2cannot be interpolated linearly.However,the above closed-form solutions can be expressed using the the hardware trilinear interpolation function T (p )for parameter p ,such that optimal non-linear filtering of ˆσcan nevertheless be efficiently computed in hardware asˆµi =T (µi )ˆσi 2=T (σ2i +||µi ||2)−||ˆµi ||2.Taking the GMM mixture coefficients αi into account,theabove interpolation formulas can be extended toˆαi =T (αi )ˆµi =T (αi µi )/T (αi )ˆσi 2=T (αi (σ2i +||µi ||2))/T (αi )−||ˆµi ||2as derived in the Appendix.To facilitate this computation,the i -th Gaussian is stored in a floating-point texture map as(αi ,αi µi (x ),αi µi (y ),αi (σ2i +||µi ||2)),and hardware trilinear interpolation can directly be used to filter it.Since graph-ics hardware performs trilinear interpolations in real time,the distribution-based reflectance mipmaps are interpolated rapidly enough for real-time rendering.Our current imple-mentation utilizes mixtures of four Gaussians,where each Gaussian component is stored as a texture.5.ResultsWe implemented our algorithm on a PC with a 2.8GHz Pen-tium IV CPU and an nVIDIA 6800GT graphics card.ThecThe Eurographics Association 2005.(a)(b)(c)(d)(e)Figure 5:Interpolation of independently computed GMMs in (a),and aligned GMMs from our method in (b).In (a)and (b),each black square represents a texel,and the dash square denotes a pixel area that overlaps parts of four tex-els.Gaussian components are numbered to show the corre-spondence,and interpolation takes place between compo-nents with the same index.Interpolation of (a)generates (c),which is far from the ground truth normal distribution (d).Interpolation of (b)generates a distribution (e)closer to (d).hardware-accelerated rendering is implemented in a single pass using Pixel Shader 3.0and Vertex Shader 3.0with Di-rectX.In total,241instructions are used in the Pixel Shader among which 4are texture instructions and 45instructions are used for the Vertex Shader.Toksvig [Tok04]proposed a simple method that utilizes a single Gaussian with variant deviation to model microfacet normal distributions at different resolutions.However,sub-sampling a complex normal distribution by a single Gaus-sian can lead to significant aliasing.In Fig.4,we compare aliasing of the method of Toksvig [Tok04]with that of our method for three consecutive frames from a camera zooming out.For a clearer contrast between the two methods,we refer the reader to the accompanying video,in which our method renders this example at 120fps while our implementation of the Toksvig method renders at 125fps.In Fig.5,the importance of GMM alignment is illus-trated for the metal disk of Fig.6.GMM interpolation results for aligned GMMs and unaligned,independently computed GMMs are displayed as microfacet normal histograms in x-y space.If GMM parameters are computed independently at each texel,the clarity of surface details is decreased and aliasing is introduced,as displayed in Fig.6.The need for multiresolution reflectance is exemplified by both the desert in Fig.1and the armor in Fig.8,which con-sists of many small hemispherical metal scales.At coarser resolutions,the normal distribution within each pixel broad-ens,which effectively increases the surface roughness.Inthe(a)(b)(c)Figure 6:GMM alignment for filtering.(a)With unaligned GMM mipmap;(b)Ground truth by multiple ray tracing;(c)With aligned GMM mipmap,formed by our method for normal distribution prefiltering.Interpolation of unaligned GMMs results in significant aliasing and loss ofdetail.Figure 7:Comparison of our result (top)and a result us-ing Fournier’s method (bottom).In comparison to multi-ple ray tracing,the color error in terms of normalized Eu-clidean color distance is 0.041for our method and 0.123forFournier’s method.figure,we compare our rendering results,results with a fixed BRDF represented by our reflectance model at highest reso-lution,and a ground truth image computed by multiple ray tracing (16/pixel).Since a fixed BRDF does not account for changes in pixel geometry at different scales,an incorrect amount of shine appears at coarse resolutions.Our frame rates for the desert and armor examples are 45fps and 107fps respectively at a 500×500screen size,while multiple ray tracing renders at about 0.01fps for the armor.A comparison to Fournier’s method with four Phong peaks is exhibited in Fig.7,which displays a carved disk viewed from an oblique angle.A technical difficulty in using a combination of Phong peaks is its sensitivity in data fitting,which leads to some rendering artifacts.This example high-lights the ability of our method to utilize additional hard-ware mipmap features,such as anisotropic filtering in this case,which is made possible by its use of hardware trilinear interpolation functions.Additionally,our method renders an order of magnitude faster than a hardware implementation of Fournier’s method,at 120fps vs.13fps for this disk,even though our method utilizes a more sophisticated reflectance model.For further comparisons,we refer the reader to the accompanying video.cThe Eurographics Association 2005.(a)(b)(c)(d)Figure8:Golden armor rendered at different resolutions.From a higher resolution(a)to a lower resolution(b),our method captures changes in reflectance properties,as seen in comparison to a ground truth image(c)rendered by multiple ray tracing. If reflectance variation is ignored and the armor is rendered with afixed reflectance,overly shiny armor results at a coarse resolution(d).At the coarse resolution,the normalized Euclidean color error per pixel in comparison to multiple ray tracing is 0.058for our method and0.175for thefixed BRDF.6.ConclusionWe have presented a real-time technique for multiresolu-tion rendering of geometry-based reflectance.To handle distribution-based reflectance parameters needed for model-ing multiresolution pixel geometry,we presented a method for mipmap precomputation that computes GMM parame-ters in a manner that yields accuratefiltering and minimal aliasing.Although reflectance is represented by distribution-based quantities,we show how linear hardwarefilters can be exploited to attain real-time rendering. AcknowledgementThis work is supported by the Hong Kong RGC grant HKUST6182/04E and6188/02E.References[APS00]A SHIKHMIN M.,P REMONZE S.,S HIRLEY P.:A microfacet-based brdf generator.In Proc.SIGGRAPH’00 (2000),pp.65–74.[Bil97]B ILMES J.:A Gentle Tutorial on the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models.Tech.Rep.ICSI-TR-97-021,Univ.of California-Berkeley,1997.[BM93]B ECKER B.G.,M AX N.L.:Smooth transitions between bump rendering algorithms.In Proc.SIGGRAPH’93(1993), pp.183–190.[CT81]C OOK R.L.,T ORRANCE K.E.:A reflectance model for computer graphics.In Proc.SIGGRAPH’81(1981),pp.307–316.[DLR77]D EMPSTER A.P.,L AIRD N.M.,R UBIN D. B.: Maximum-likelihood from incomplete data via the em algorithm.Journal of the Royal Statistical Society39,1(1977),1–38. [Fou92]F OURNIER A.:Normal distribution functions and multi-ple surfaces.In Graphics Interface Workshop on Local Illumina-tion(1992),pp.45–52.[Hop96]H OPPE H.:Progressive meshes.In Proc.SIGGRAPH ’96(1996),pp.99–108.[Ney98]N EYRET F.:Modeling,animating,and rendering com-plex scenes using volumetric textures.IEEE Trans.on Visualiza-tion and Computer Graphics4,1(1998),55–70.[Sch94]S CHLICK C.:A survey of shading and reflectance puter Graphics Forum13,2(1994),121–132.[Tok04]T OKSVIG M.:Mipmapping Normal Maps.Tech.Rep.TB-01256-0001,nVIDIA Corporation,2004.[TS67]T ORRANCE K.E.,S PARROW E.M.:Theory for off-specular reflection from roughened surfaces.J.Optical Societyof America57(1967),1105–1114.[WAT92]W ESTIN S.H.,A RVO J.R.,T ORRANCE K.E.:Pre-dicting reflectance functions from complex surfaces.In Proc.SIGGRAPH’92(1992),pp.255–264.[Wil83]W ILLIAMS L.:Pyramidal parametrics.In Proc.SIG-GRAPH’83(1983),pp.1–11.AppendixTo linearly interpolate GMMs{Θk;1≤k≤K}according to texelweights{w k;1≤k≤K},we totally draw N samples from theseGMMs,among which N k samples come from GMMΘk.Here,N=∑K k=1N k and N k/N=w k.The optimal interpolationˆg i isfitto samples drawn from corresponding Gaussians{g ki;1≤k≤K}.The number of samples drawn from g ki is N kαki.The mean and vari-ance of these samples areµki andσ2ki respectively.Therefore,ˆαi=1NK∑k=1N kαki=K∑k=1N kNαki=K∑k=1w kαki=T(αi)ˆµi=∑K k=1(N kαki)µki∑K k=1N kαki=∑K k=1N k Nαkiµki∑K k=1N kαki=∑K k=1w k(αkiµki)∑K k=1w kαki=T(αiµi)T(αi) ||ˆµi||2+ˆσi2=∑K k=1(N kαki)(||µki||2+σ2ki)∑k=1N kαki=∑K k=1N k Nαki(||µki||2+σ2ki)∑K k=1N k Nαki =∑K k=1w k αki(||µki||2+σ2ki)∑K k=1w kαki=T αi(||µi||2+σ2i)T(αi)c The Eurographics Association2005.。

extrapolation方法和interpolation方法Extrapolation and interpolation are both methods used in mathematics and statistics to estimate values within or beyond a given set of data points. However, they differ in the way they make these estimates.Extrapolation:Extrapolation is the process of estimating values outside the range of observed data. It involves extending a curve or line beyond the given data points to make predictions or estimates. This method assumes that the pattern observed in the existing data will continue to hold for values beyond the observed range. However, extrapolation can be unreliable if the underlying pattern changes or if the data is subject to significant variability.Interpolation:Interpolation is the process of estimating values within the range of observed data. It involves constructing a curve or line that passes through the given data points to estimate values at specific points. Interpolation assumes that there is a smooth and continuous relationship between the observed data points. This method is useful for estimating values when there is a need for finer granularity within the observed range.In summary, extrapolation estimates values outside the range of observed data, while interpolation estimates values within the observed range. Both methods involve making assumptions about the underlying pattern in the data, but extrapolation carries a higher risk of being less accurate due to potential changes or variability in the data.。

船舶英语floating launch飘浮下⽔gravity launch重⼒下⽔grease launch油脂下⽔steel ball method of launch钢珠下⽔mechaniz launch机械化下⽔end launch纵向下⽔side launch横向下⽔incline launch倾斜下⽔jump launch跳跃式下⽔dipp launch跌⼊式下⽔two-point support launch两⽀点下⽔up to slipway上排marine slipway下⽔重量launch weight下⽔重量launch grease下⽔油脂pressure bear layer乘压层transitional layer过渡层slid layer滑动层weight of up to slipway上排重量lift by the stern尾浮lift尾浮fore poppet pressure前⽀架压⼒zone of pivot pressure action前⽀架压⼒作⽤区way end pressure滑道末端压⼒dropp⾸落bow drop⾸落dipp⾸沉tipp尾落launch poppet下⽔⽀架fore poppet前⽀架after poppet后⽀架launch beam下⽔横梁trigger⽌滑器check arrangement制动设施sub-assembl部件装配section assembl分段装配unit assembl分段装配berth assembl船台装配unit erection at berth船台装配sectional method of hull construction分段建造法block method of hull construction总段建造法horizontal method of hull construction⽔平建造法pyramid method of hull construction塔式建造法island method of hull construction岛式建造法upright method of hull section construction正装法upsidedown method of hull section construction倒装法sideward method of hull section construction侧装法join ship sections afloat⽔上合拢mould bed胎架jig胎架fixed mould bed固定胎架adjustable mould bed可调胎架positioner摇摆胎架tilt jig回转胎架tilt device回转装置specialized mould bed专⽤胎架bed mould胎板fabrication platform装焊平台electromagnetic platform电磁平台position strip定位马shore定位马electromagnetic strip电磁马jip for fitt-up T-sectionsT型材装焊机temporary stiffen bar临时加强材assembly frame假舱壁sub-assembly船体部件panel板列section分段assembly分段unit分段basic section基准分段join section嵌补分段flat section平⾯分段curved section曲⾯分段three-dimensional unit⽴体分段two-dimensional unit半⽴体分段block总段complete cross section总段section assembly flow line分段装配流⽔线lay-out for assembl装配划线sequence of erection and weld装配焊接顺序plate alignment拼板position定位frame r肋⾻框架reference line for assembl装配定位线seperate method of assembl分离装配法cell method of assembl放射式装配法interpolate method of assembl插⼊装配法centerlineof berth船台中⼼线sight column望⼴柱positioner for unit erection at berth船台装配定位器frame station line on berth船台肋⾻线light target光靶pre-set焊接反变形manual loft⼿⼯放样scale loft⽐例放样full scale loft实尺放样longitudinal shrink scale loft纵向缩尺放样fair光顺fairness光顺性deviation from offsets修正值coincide in three projection planes三⾯吻合lay off of hull lines型线放样mathematical loft数学放样mathematical lines数学型线mathematical fair of lines型线数学光顺sectional curved method剖⾯线法least square method最⼩⼆乘法linear programm method线形规划法variational method变分法curved surface method曲⾯法point by point method of modification逐点修改法local linear interpolation直尺卡样法elastic local linear interpolation卡尺回弹法bend-modify relaxation method弯调松弛法interpolation point正插点inversed interpolation point反插点line of intersect points⾓点线line of tangential points切点线line of centers圆⼼线localized fair局部光顺circular spline圆弧光顺spline function样条函数fitt拟合fitt to curved surface曲⾯拟合parametric curve with sle flecnode单拐参数曲线criterion of fairness光顺判别准则inflexion preservation保凹凸recurrent force回弹⼒coincident recurrent force正回弹⼒incoincident recurrent force返回弹⼒recurrent magnitude回弹量bend direction of curve弯向chang of curvature弯势excess flecnode多余拐点bad point坏点bad point interval坏点区discriminat function定改函数mathematical development of shell plate外板数学展开equation of frame肋⾻⽅程equation of longitudinal seam纵缝⽅程equation of transverse seam横缝⽅程differential geodesic line method微分测地线法tangential development method切⾯展开法thickness modification in plate development板厚修正shipbuild information data process造船信息数据处理系统mathematical loft system for pip管路数放系统graphic data process system图⽰数据处理系统line fair program型线光顺程序seam arrangement program板缝排列程序shell plate development program外板展开程序mathematical seam arrangement板缝数学排列nest program套料程序structural process program结构处理程序nest mathematical process套料数学处理structural mathematical process结构数学处理tape out-put program纸带输出程序draw process program绘图处理程序numerical controlled draw 数控绘图data treatment and memory program数据处理和存储程序station No站号frame No肋⾻号contour line外廓线shell pate development外板展开dutum line准线mean normal method准线展开法center spot squar method⼗字线展开法geodesic line测地线geodesic line development method测地线展开法frame section surface肋⾻平⾯normal plane of shell plate外板法⾯trible of buttock lines纵剖线族tribe of body lines横剖线族tribe of water lines⽔线族camber table梁拱表frame interpolation肋⾻插值frame body plan肋⾻型线图returned table of offsets完⼯型值表finished table of offsets完⼯型值表draft mark吃⽔标志paint line⽔线标志structural member loft结构放样edge line接⼝线upper edge line上⼝线lower edge line下⼝线seam arrangement板缝排列longitudinal seam纵向接缝transverse seam横向接缝curved surface development曲⾯展开structural member development构件展开curvature of frame肋⾻弯度jig for plate development外板展开器selected mould sectionprojected frame line on selected mould section改型肋⾻型线diagonal check methods对⾓线检查法sketch for mark off号料草图template draw样板图scale draw transparency投影底图glass negative投影底⽚profile draw仿形图bridge line连割线template样板mould样板mock-up样箱cradle mould样箱batten样条mould bar洋棒multi-shape template格⼦样板adjustable mould可调样板triangular template三⾓加⼯样板beam mould梁拱样板frame mould肋⾻洋板section mould加⼯洋板template for secondary mark装配划线样板angular template for assembl装配夹⾓样板adjustable angular template可调⾓度样板template for mark-off号料样板loft office放样间loft floor样台scale loft table⽐例放样台manual mark-off⼿⼯号料mark-off from templates样板号料mark-off from mould bars样棒号料mark-off from moulds样箱号料mark-off from sketches草图号料secondary mark-off⼆次号料nest套料mark signs号料符号fram line构架线automatic mark-off⾃动号料projection mark投影号料projection tower投影塔projection room投影室projection mark table投影号料台positive projection正⽚投影negative projection负⽚投影image distortion影像畸变absolute distortion绝对畸变relative distortion相对畸变pincushion distortion枕形畸变barrel distortion桶形畸变photosensitive mark感光号料facsimile mark电传真号料facsimile mark machine电传真号料机electro-print mark电印号料electro-print mark apparatus电印号料装置synchronized scann projection同步扫描投影photo-conductive powder光电粉末electrostatic latent image静电潜像NCshapes mark machine数控型材号料pneumatic mark punch风动号料冲头shipbuild technolgy造船⼯艺two-part construction两段造船法method of build ship in two sections两段造船法tandem shipbuild method串联造船法technology of hull construction船体建造⼯艺loft放样mark off号料hull steel fabrication⾦属船体加⼯assembly and erection船体装配weld焊接launch下⽔outfitt and machinery installation船舶设备和系统安装pre-erection outfitt预安装outfitt and machinery installation on berth船台安装quay outfitt and installation码头安装dock outfitt and installation船坞安装unit outfitt单元安装unit pre-outfitt单元预装bollard test系泊实验sea trial航⾏实验construction start date建造开⼯⽇time on build berth船台周期shipbuild period造船周期launched displacement of ships下⽔量ship output船舶产量ship delivery交船operation stage⼯艺阶段operation items⼯艺项⽬operational path⼯艺路线technological process⼯艺过程technological instruction⼯艺规程method plann of ship construction原则⼯艺standard technological procedure典型⼯艺operation items detail chart⼯艺项⽬明细表master schedule总⼯艺进度表"positional flow" method流⽔定位法"specialized gang" method流⽔定员法edge preparation边缘加⼯individuals of hull structure船体零件symbols for hull steel fabrication船体加⼯符号groove preparation坡⼝加⼯edge plann刨边edge mill铣边edge grind磨边carbon arc air goug碳弧⽓刨plate-edge planer刨边机plate-edge miller铣边机shear剪切roll shear滚剪blank冲切scarf削斜flame cutt⽕焰切割plasma cutt等离⼦切割arc air cutt电弧⽓割gas cutt⽓割oxy cutt氧⽭切割oxy-flux cutt氧熔剂切割laser cutt激光切割under water cutt⽔下切割stack cutt迭板切割shear squence剪切顺序drag后拖量shear machine剪床guillotine龙门剪床roll shear machine滚剪机disk shear圆盘剪circular shear圆盘剪nibbl machine振动剪床shear and punch machine联合冲剪机semi-automatic gas cutt machine半⾃动⽓割机gantry type semi-automatic gas cutt machine半⾃动桥式⽓割机semi-automatic gas cutt machine for mark circular shape半⾃动割圆机semi-automatic multitorch gas cutt machine半⾃动多炬⽓割机全位置仿形⽓割机all position copy gas cutt machine全位置仿形⽓割机automatic gas cutt machine⾃动⽓割机profile gas cutt machine靠模⽓割机copy gas cutt machine靠模⽓割机photo-electric trac gas cutt machine光电跟踪⽓割机photo-electric linefollow gas cutt machine光电跟线⽓割机NCgas cutt machine数控⽓割机NCuniversal gas cutt machine数控通⽤⽓割机cutt torch割炬cutt nozzle割嘴preheat⽓割预热roll bend滚弯press bend压弯knuckle折⾓angle bend less than 90折锐⾓angle bend greater than 90折钝⾓flang折边die bend强制弯曲free bend⾃由弯曲concave bend of frame⾻材内弯convex bend of frame⾻材外弯length wise wavws around edge皱折minimum bend radius最⼩弯曲半径pre-bend预弯surplus straight edge剩余直边spr back弯曲回弹rubber mold form橡⽪模成型spinn旋压flow form旋压bulg胀形shot blast form喷丸成型stretch metal in center portion of sheet放中tight metal in center portion of sheet收中stretch metal around edges of sheet放边tight metal around edges of sheet收边hammer solidly around edges of sheet拨缘channell压筋rectification of distortion矫形rectification of distortion by flame⽕⼯矫形rectification of distortion by flame heat and water cool⽔⽕矫形hot form⼤⽕成形line heat form⽔⽕成形heat line加热线heated-side cool method正⾯⽔冷法back-side cool method背⾯⽔冷法width of heat affect zone热影响区宽度length of heat line焰道长度heat speed热源速度flame power⽕焰功率angular distortion⾓变形explosion form爆炸成形unconfined system explosion form⽆模爆炸成形three-roll bend machine三辊弯板机shipyard roll press造船⽤滚压两⽤机shipyard combined plate bend and flang machine造船⽤滚压两⽤机four-roll bend machine四辊弯板机press压⼒机hydraulic press with adjustable head for ship use造船⽤压头可移式液压机three-row plunger type press三排柱塞式弯板机multi-plunger hydraulic press多柱塞式弯板机NChydraulic press数控成形液压机sail-shaped plate帆形板saddle-shaped plate鞍形板screw-shaped plate螺形板corrugated plate波形板channelled plate槽形板cylindrical shaped plate桶形板conical shaped plate锥形板spherical shaped plate球形板pure bend纯弯板concentrated force bend集中⼒弯曲stretch bend拉弯torsional deflection旁弯变形cold frame bend machine肋⾻冷弯机frame roll machine肋⾻滚弯机frame stretch bend machine肋⾻拉弯机cold frame bend machine with-adjustable die可调模具式肋⾻冷弯机cold frame bend machine with multiplungers多压头式肋⾻冷弯机three cylinder frame bend machine三缸肋⾻冷弯机frame pure-bend machine纯弯曲肋⾻冷弯机frame bend machine of medium frequency induction heat中频肋⾻热弯机NCcold frame bend machine数控肋⾻弯曲机concave bend frame内弯肋⾻convex bend frame 外弯肋⾻s-bend frameS-形肋⾻steel pretreatment刚材预处理steel levell and straighten钢材矫正levell of piates钢板矫平mangle辊式矫平机plate straightener辊式矫平机straighten of shapes型钢矫直shapes straighten machine型钢矫直机steel preheat钢材预热rust removal of steel钢材除锈abrasive blast喷射除锈abrasive磨料shot blast抛丸除锈shot blast machine抛丸除锈机shot blast room抛丸室abrasive impeller抛丸器automatic primer spray⾃动喷涂底漆chemical clean化学除锈acid clean of steel酸洗除锈pickl酸洗处理dip pickl浸渍酸洗spray pickl喷淋酸洗neutraliz中和处理passivat钝化处理phosphat磷化处理pickl bath酸洗池pickl waste treatment废酸液处理acid pickl inhibitor酸洗缓蚀剂vacu-blast clean真空喷射除锈arc rust removal电⽕花除锈mechanical rust removal机械除锈secondary rust removal⼆次除锈shot peen喷丸除锈sand blast喷沙除锈wet sand blast湿喷沙除锈spray pickl in circulation循环喷射酸洗acid paste pickl化学膏除锈hydroblast⾼压⽔除锈flame clean⽕焰除锈remote control descal machine遥控除锈机adjust needle cleaner针束除锈器stripp film clean可剥性薄膜除锈section shot-blast分段抛丸除锈section shot-peen分段喷丸除锈section pickl分段酸洗除锈shot-peen room喷丸间brush paint刷涂spray pant喷涂airless spray paint⽆空⽓喷涂electrostatic spray静电喷涂agent for electrostaic spray静电喷涂助剂electrocoat电泳涂漆flow line of pickl酸洗磷化电泳涂漆流⽔线phosphat and electrocoat酸洗磷化电泳涂漆流⽔线abrasive磨料abrasive blast喷射除锈abrasive impeller抛丸器absolute distortion绝对畸变acceptance document验收移交书acceptance of shaft轴系安装验收acid clean of steel酸洗除锈acid paste pickl化学膏除锈acid pickl inhibitor酸洗缓蚀剂actual throat thickness焊缝厚度additional test复验adjustable angular template可调⾓度样板adjustable block机械墩adjustable mould可调样板adjustable mould bed可调胎架adjust needle cleaner针束除锈器after poppet后⽀架agent for electrostaic spray静电喷涂助剂ahead and astern reach measurement惯性试验air tight test of pipeline管路⽓密实验airless spray paint⽆空⽓喷涂alignment找正alignment by flanges直接找正alignment of bottom cas and bear seat下⽓缸和轴承座找正all position copy gas cutt machine全位置仿形⽓割机all position weld全位置焊接anchor test in trial锚设备效⽤实验angle bend greater than 90折钝⾓angle bend less than 90折锐⾓angular distortion⾓变形angular template for assembl装配夹⾓样板anneal weld退⽕焊approved certificate for product产品合格证arc air cutt电弧⽓割arc blow磁偏吹arc clean action阴极破碎arc length弧长arc rust removal电⽕花除锈arc strike引弧arc weld电弧焊arcvoltage电弧电压argon arc weld氩弧焊assembl error装配误差assembl on board上船组装assembly分段assembly and erection船体装配assembly frame假舱壁automatic gas cutt machine⾃动⽓割机automatic primer spray⾃动喷涂底漆automatic weld⾃动焊automatic weld in vertical position垂直⾃动焊auxiliary boiler test辅锅炉试验average linear load平均线负荷back chipp清根back goug清根back run打底焊back settl回淤back weld打底焊backfire回⽕back weld封底焊bad point坏点bad point interval坏点区band test弯曲试验bank slope驳岸barrel distortion桶形畸变base line基准线base metal母材basic covered electrode碱性焊条basic section基准分段basin港池batten样条beam mould梁拱样板beams gate迭梁式坞门bear metal cracks轴承龟裂bed mould胎板bend direction of curve弯向bend of pipewith core有芯弯管bend prior to weld先弯后焊berth船台berth assembl船台装配berth force挤靠⼒bevel angle坡⼝⾓度bilge block舭墩blade rust叶⾯锈斑blade tip deflection螺旋桨叶稍上翘blank冲切block总段block load搁顿负荷block method of hull construction总段建造法block sequence分段多层焊blocks墩blocks with sand box沙箱墩blowhole⽓孔blow吹车boat tank跳船boiler burner test锅炉燃烧器试验boiler water fill上⽔（锅炉）bollard test系泊实验bonded flux陶质焊剂bonded slag粘渣bond of plastic pipe塑料管粘接bond of propeller to shaft螺旋桨胶合联接bor for shaft line轴系镗孔bow drop⾸落box包⾓焊braz钎焊braz filler metal钎料braz flux钎剂break test for anchor chain锚链拉断实验breakwater防波堤bridge approach引桥bridge gauge桥规bridge line连割线brush paint刷涂build berth造船台build dock造船坞bulg胀形burn loss烧损burn of bear metal⽩合⾦烧熔butt joint对接接头butt weld对接焊缝butter预堆边焊caisson坞门camber table梁拱表cambered way弧形滑道canalled dock运河式船坞cantilever end load悬臂端负荷carbon arc air goug碳弧⽓刨carbon equivalent炭当量cargo handl test起重设备负荷试验cast of propeller in metal patten螺旋桨⾦属型铸造cell method of assembl放射式装配法center block for shaft alignment轴系中⼼定位块center line strip on berth船台中⼼线板center spot squar method⼗字线展开法center校中center of mov parts运动件校中center of shaft by indicators指针法轴系校中center of shaft by light光学法轴系校中center of shaft by ruler and gauge直尺和厚薄规法轴系校中center of shaft轴系校中center of shaft by calculation method计算法轴系校中center of shaft by direct connection平轴法轴系校中center of shaft by load method轴承负荷法轴系校中center or rotor转⼦校中centerline of berth船台中⼼线centrifugal cast of propeller螺旋桨离⼼铸造ceramic flux陶质焊剂chain intermittent fillet weld对称间断焊缝chang of curvature弯势channelled plate槽形板channell压筋check of axial clearance轴向间隙确定check of dynamic balance of rotor转⼦校动平衡check of play between piston and cylinder cover校⽓缸余隙check of propeller pitch螺旋桨螺距校正check of reference center line中⼼线测量check of the tappet clearance校⽓门间隙check arrangement制动设施check of bear clearance with lead压铅法check of propeller pitch螺旋桨螺距检验check of surface in contact by mark compound涂⾊油法chemical analysis化学分析chemical clean化学除锈circular shear圆盘剪circular spline圆弧光顺circumferential weld环焊缝CO2 weld⼆氧化碳⽓体保护焊coat药⽪coincide in three projection planes三⾯吻合coincident recurrent force正回弹⼒cold bend of pipe管⼦冷弯cold crack冷裂缝cold frame bend machine肋⾻冷弯机cold frame bend machine with multiplungers多压头式肋⾻冷弯机complete cross section总段complete engine set installation整机安装complete fusion焊透compressed air test⽓压实验concave bend frame内弯肋⾻concave bend of frame⾻材内弯concentrated force bend集中⼒弯曲configuration of bent pipe管⼦弯势conical shaped plate锥形板connection weld联系焊缝construction start date建造开⼯⽇consumable guide electroslag weld熔嘴电渣焊consumables焊接材料continuous weld连续焊缝continuous weld连续焊contour line外廓线convex bend frame 外弯肋⾻convex bend of frame⾻材外弯copper back铜垫copper shoe铜滑块copy gas cutt machine靠模⽓割机core wire焊芯core wire diameter焊条直径coreless bend of pipe⽆⼼弯管corner joint⾓接接头corrugated plate波形板course keep test航向稳定性试验covered berth室内船台covered electrode焊条cover药⽪crack sensitivity焊接裂缝敏感性crack test抗裂试验crack test with 500x2000mm test pieces⼤板对接抗裂试验cradle mould样箱crater弧坑crater crack弧坑裂缝cribb井字墩criterion of acceptance of radiographic examination of welded seam焊缝质量射线检查评定标准criterion of fairness光顺判别准则criterion of ultrasonic examination of welded seam焊缝超声波探伤标准cruciform crack test⼗字接头抗裂试验cruciform joint⼗字接头curvature of frame肋⾻弯度curved section曲⾯分段curved surface development曲⾯展开curved surface method曲⾯法cutt nozzle割嘴cutt torch割炬cylindrical shaped plate桶形板data treatment and memory program数据处理和存储程序dead center gauge⽌点规defective items after inspection废品通知单deflection of main journal主轴颈下沉量degree of mechanization in weld焊接机械化程度deposited metal熔敷⾦属deposition rate熔敷系数depth of fusion熔深design element of weld焊缝结构要素design throat thickness焊缝计算厚度design water lever设计⽔位destructive examination破坏性检验detectaphone侦听起detection sensitivity探伤仪灵敏度determination of center line and axis找中determination of center line or axis找中determination of shaft linep轴系找中deviation from offsets修正值diagonal check methods对⾓线检查法dial gauge for measur of crank spread曲轴量表die bend强制弯曲differential geodesic line method微分测地线法diffusible hydrogen content扩散氢含量dip pickl浸渍酸洗dip transfer短路过渡dipp⾸沉dipp launch跌⼊式下⽔discriminat function定改函数disk shear圆盘剪dock船坞dock bottom坞底dock entrance坞⼝dock head坞⾸dock outfitt and installation船坞安装dock sill坞坎dock wall坞壁dockgate坞门dockgate channel门槽dock weight进坞重量dolphin pier靠船墩double bevel grooveK型坡⼝draft mark吃⽔标志drag后拖量draw process program绘图处理程序drop test for anchor锚投落试验dropp⾸落dry dock⼲坞dry extension of electrode焊丝⼲伸长dutum line准线dynamic characteristic电源动特性dynamic test of propeller螺旋桨浆动平衡试验dynamometer测功器eccentricity偏⼼度edge grind磨边edge joint端接接头edge line接⼝线edge mill铣边edge plann刨边edge preparation边缘加⼯elastic local linear interpolation卡尺回弹法electric dynamometer电⼒测功器electrocoat电泳涂漆electrode diameter焊条直径electrode extension焊丝伸出长度electrode for vertical down weld下⾏焊条electrode negative正接electrode positive反接electromagnetic platform电磁平台electromagnetic strip电磁马electronbeam weld电⼦束焊electronic indicator电⼦⽰功器electroslag weld电渣焊electroslag weld with plate electrode板极电渣焊electroslag weld with wire electrode丝极电渣焊electrostatic latent image静电潜像electrostatic spray静电喷涂elevation⽔位emergency drain test应急疏⽔实验enclosed weld强迫成形焊end launch纵向下⽔equation of frame肋⾻⽅程equation of longitudinal seam纵缝⽅程equation of transverse seam横缝⽅程equilibrium tide平潮evaluation test of power plant on board动⼒装置实船鉴定试验examination of opened up parts拆验examination of ream紧配螺栓配合检查excess flecnode多余拐点excess of water过⽔excess weld metal加强⾼expander胀管器explosion expand of pipe爆炸胀管explosion form爆炸成形explosive weld爆炸焊接exposed berth露天船台external characteristic电源外特性fabrication platform装焊平台facsimile mark电传真号料facsimile mark machine电传真号料机fair光顺fairness光顺性fatigue test疲劳试验filler metal填充⾦属filler wire焊丝fillet weld⾓焊缝fillet weld in flat position船形焊fillet weld Tee bend testT型接头弯曲试验fillet weld⾓焊final cutt of pipe割管finished table of offsets完⼯型值表firecracker weld躺焊first product inspection⾸件检验FISCO type crack test加压固定对接抗裂试验fishbone crack test鱼⾻型抗裂试验fitt拟合fitt of compreg层压胶⽊镶嵌fitt of lignumvitae铁梨⽊镶嵌fitt of rubber seal r in stern tube尾轴管衬套橡胶嵌套fitt to curved surface曲⾯拟合fixed foot bridge固定引桥fixed mould bed固定胎架flame bend of pipe⽕焰弯管flame clean⽕焰除锈flame cutt⽕焰切割flame power⽕焰功率flangeless joint⽆法兰联接flang折边flap gate卧倒式坞门flash back回⽕flat position weld平焊flat section平⾯分段flatten test⾦属管压扁试验float caisson浮箱式坞门float launch飘浮下⽔float spar fender浮式护⽊flood dock注⽔式船坞flood test灌⽔法flow form旋压flow line of pickl酸洗磷化电泳涂漆流⽔线flow rate of shield gas保护⽓体流量flux焊剂flux back焊剂垫fog test⽓雾法fore poppet前⽀架fore poppet pressure前⽀架压⼒formation of weld寒风成形frame bend machine of medium frequency induction heat中频肋⾻热弯机frame body plan肋⾻型线图frame interpolation肋⾻插值frame lines肋⾻检验线frame mould肋⾻洋板frame No肋⾻号frame r肋⾻框架frame roll machine肋⾻滚弯机frame section surface肋⾻平⾯frame station line on berth船台肋⾻线frame stretch bend machine肋⾻拉弯机fram line构架线free bend⾃由弯曲free runn空车运转freeboard measurement of life boat救⽣艇⼲舷测定full penetration焊透full scale loft实尺放样fume of weld焊接烟尘furnace brickwork炉墙砌筑furnace explosion dur light up冷爆fused flux熔炼焊剂gallery廊道gantry龙门架gap两轴曲折gas cutt⽓割gas inflation liferaft test⽓胀式救⽣筏试验gas metal arc weld熔化级⽓体保护焊gas mixture metal arc weld混合⽓体保护焊gas pore⽓孔gas tightness⽓密性gas tungsten arc weld⾮熔化级⽓体保护焊gas weld⽓焊gate chamber门库gate channel for repair修理门槽gate pier门墩generat set portion of bollard test发电机组系泊试验generat set portion of trial test发电机组航⾏试验geodesic line测地线geodesic line development method测地线展开法glass negative投影底⽚globular transfer⼤滴过渡graphic data process system图⽰数据处理系统gravity launch重⼒下⽔gravity weld重⼒焊grease launch油脂下⽔groove坡⼝groove angle坡⼝⾓度groove depth坡⼝深度groove preparation坡⼝加⼯groove radius坡⼝半径guided bend of pipe靠模弯管guillotine龙门剪床hammer solidly around edges of sheet拨缘handl test of life boat救⽣艇和⼯作艇降落试验hard solder钎料hardness test硬度试验hardness test of weld zone焊接接头硬度试验heat affected zone热影响区heat input线能量heat line加热线heat speed热源速度hell test倾斜试验high frequency bend of pipe⾼频弯管hogg拱曲horizontal build berth⽔平船台horizontal method of hull construction⽔平建造法horizontal position weld横焊hot bend of pipe管⼦热弯hot crack热裂缝hot form⼤⽕成形hot wire submerged arc weld预热焊丝埋弧焊hull steel fabrication⾦属船体加⼯hydraulic bend of pipe液压弯管hydraulic dynamometer⽔⼒测功器hydraulic press with adjustable head for ship use造船⽤压头可移式液压机hydraulic subassembly for crankshaft曲轴液压套合hydraulic test of boiler锅炉⽔压实验hydraulic test of pipeline管路液压实验hydroblast⾼压⽔除锈hydrogen content in welds焊缝氢含量idl惰转image distortion影像畸变impact force撞击⼒impact test冲击实验imperfection in bond轴承脱壳incline launch倾斜下⽔inclined build berth倾斜船台inclined Tee joint斜接接头incoincident recurrent force返回弹⼒incomplete fusion未焊透indicator⽰功器individuals of hull structure船体零件inflexion preservation保凹凸insert pipe嵌补管inspection勘验inspection of levelness⽔平度测量inspection by mould bar样条法检验inspection by sketch method草图法检验inspection by templet样板法检验inspection dur hull steel fabrication船体零部件加⼯检验inspection of abutt edges of block总段对接检验inspection of block总段装配检验inspection of center line of build berth船台中⼼线测量inspection of draft mark and loaded water linesinspection of frame spac肋⾻间距测量inspection of groove坡⼝测量inspection of hull deformation after launch全船校正检查inspection of hull form after completion船体完⼯测量inspection of hull lines线型检验inspection of hull loft船体放样检验inspection of install of propeller螺旋桨安装检查inspection of mould bed胎架检验inspection of mount of bear seats轴承座安装检验inspection of panel平⾯分段检验inspection of permanent ballast before launch下⽔压载检查inspection of preassembled frame r肋⾻框架装配检验inspection of ship structure全船结构检查inspection of squar accuracy⾓尺线检查inspection of three dimensional unit⽴体分段检验inspection of thrust block推⼒块厚度检查inspection of vertical surface垂直度测量inspection on development of structural members船体构件展开检验inspection on shell plate development外板展开检验installation of boiler锅炉安装installation of desel engine柴油机安装installation of main engine主机安装installation of marine power plant船舶动⼒装置安装installation of pip管系安装installation of pip unit管系单元安装installation of shaft轴系安装installation of turbine汽轮机安装interlayer temperature层间温度intermittent weld间断焊缝interpass temperature层间温度interpolate method of assembl插⼊装配法interpolation point正插点inversed interpolation point反插点iron powder electrode铁粉焊条iron wedge楔铁island method of hull construction岛式建造法items for acceptance of vessel验收项⽬jetty突堤码头jig胎架jig for plate development外板展开器join lines for hull assembly船体装配对合线join section嵌补分段join ship sections afloat⽔上合拢jump launch跳跃式下⽔keel block龙⾻墩knuckle折⾓knuckl line slipway折线滑道lack of fusion未焊透lagg of boiler锅炉绝热层包扎lamellar tear层状撕裂land tie拉桩lap joint搭接接头laser alignment激光找正laser cutt激光切割launched displacement of ships下⽔量launch下⽔launch beam下⽔横梁launch grease下⽔油脂launch poppet下⽔⽀架launch way滑道launch weight下⽔重量layer焊层lay off of hull lines型线放样least square method最⼩⼆乘法leg of fillet weld焊脚length of heat line焰道长度length wise wavws around edge皱折levell找平levell by level gauge液⾯法找⽔平levell of engine bed机座找平levell of foundation机座找平levell of piates钢板矫平lifebuoy test救⽣圈试验lift by the stern尾浮lift尾浮light target光靶line fair program型线光顺程序line heat form⽔⽕成形line of centers圆⼼线line of intersect points⾓点线line of tangential points切点线line outs of ship equipments船舶综合放样linear porosity链状⽓孔linear programm method线形规划法listen侦听listen rod听⾳棒local linear interpolation直尺卡样法localized fair局部光顺localized porosity密集⽓孔loft放样loft floor样台loft office放样间longitudinal bead bend test纵向焊道弯曲试验longitudinal bead notched bend test纵向焊道缺⼝弯曲试验longitudinal seam纵向接缝longitudinal shrink scale loft纵向缩尺放样longitudinal slipway纵向滑道longitudinal weld纵向焊缝low pressure cast of propeller螺旋桨低压铸造lower edge line下⼝线lowest pressure curve of compartments舱室最⼩压⼒曲线magnetic detection磁性探伤main boiler test主锅炉试验main engine portion of bollard test主机系泊试验main engine portion of trial test主机航向实验mangle辊式矫平机manipulation of electrode运条⽅式manual bend of pipe⼿⼯弯管manual emergency anchor test⼈⼒应急起锚试验manual emergency steer test⼈⼒应急操舵试验manual loft⼿⼯放样manual metal arc weld⼿弧焊manual weld⼿⼯焊marginal wharf顺岸码头marine slipway下⽔重量mark for pipes管⼦划线mark for shaft bor轴系孔划线mark off号料mark signs号料符号master schedule总⼯艺进度表match key on propeller shaft推进器轴配件mathematical development of shell plate外板数学展开mathematical fair of lines型线数学光顺mathematical lines数学型线mathematical loft数学放样mathematical loft system for pip管路数放系统mathematical seam arrangement板缝数学排列maximum linear load最⼤线负荷mean normal method准线展开法measurement of curvature of frame肋⾻弯度测量measurement of finished propeller螺旋桨完⼯测量measurement of fit clearance of propeller螺旋桨配合间隙测量measurement of fit clearance of shaft轴系安装间隙测量measurement of fit clearance of sterntube推进器轴安装间隙测量measur error测量误差measur of crank spread曲轴臂距差测量mechanial bend of pipe机械弯管mechanical blocks机械墩mechanical expand of pipe机械胀管mechanical indicator机械⽰功器mechanical rust removal机械除锈mechanized slipway机械化道脂mechanized weld机械化焊接mechaniz launch机械化下⽔medium frequency bend of pipe中频弯管melt rate熔化系数metal transfer熔滴过渡method of build ship in two sections两段造船法method plann of ship construction原则⼯艺MIG weld熔化级⽓体保护焊mill铣切。

Interpolation Using Prelookup :: Blocks (Simulink®) jar:file:///D:/Program%20Files/matlab/help/toolbox/simulink/help.jar!/slr...®SimulinkProvide feedback about this page Interpolation Using PrelookupUse output of Prelookup block to accelerate approximation of N-dimensional functionLibraryLookup TablesDescriptionThe Interpolation Using Prelookup block is intended for use with the Prelookup block. The Prelookup block calculates the index and interval fraction计算索引值和间隔部分 that specifies how its input value relates to the breakpoint data set. You feed将..供给 the resulting index and fraction values into an Interpolation Using Prelookup block to interpolate内插 an n-dimensional table. This combination of blocks performs the equivalent operation模块组合与…有等效的作用 that a single instance of the Lookup Table (n-D) block performs. However, the Prelookup and Interpolation Using Prelookup blocks offer greater flexibility更加灵活that can provide more efficient simulation and code generation更加高效的仿真和代码生成.To use this block, you must define a set of output values as the Table data parameter定义输出值作为列表数据参数. In normal use, these table values correspond to the breakpoint data sets specified in Prelookup blocks列表值与在Prelookup中指定的断点数据相对应. The Interpolation Using Prelookup block generates its output by looking up or estimating table values 通过查表或估算产生输出值 based on the index and interval fraction values (denoted on the block as k a nd f, respectively) fed into the block by each Prelookup block:If the inputs match等于 the values of indices specified in breakpoint data sets, the InterpolationUsing Prelookup block outputs the table value at the intersection of the row, column, 输出行列的交点表值and higher dimension breakpoints.If the inputs do not match不等于 the values of indices specified in breakpoint data sets, theInterpolation Using Prelookup block generates output by interpolating appropriate table values插值适当的表值. If the inputs are beyond the range of breakpoint data sets, the Interpolation UsingPrelookup block can extrapolate its output value外推输出值.The Interpolation Using Prelookup block can perform interpolation on a portion of its table在表的部分进行插值. The Number of sub-table selection dimensions子表选择维数parameter lets you specify that interpolation occur only on a subset of its Table data 仅在表数据的一个子集parameter. For example, if your 3-D table data constitutes a stack of 2-D tables to be interpolated, set the Number of sub-table selection dimensions parameter to 1. The block displays an input port (labeled as sel) used to select and interpolate the 2-D tables.Data Type SupportThe Interpolation Using Prelookup block accepts real signals of any numeric data type任何数字数据类型 supported by Simulink software, except Boolean. The Interpolation Using Prelookup block supports fixed-point data types for signals, table data, and intermediate results.Interpolation Using Prelookup :: Blocks (Simulink®) jar:file:///D:/Program%20Files/matlab/help/toolbox/simulink/help.jar!/slr... For a discussion on the data types supported by Simulink software, see Data Types Supported by Simulink in the Simulink documentation.Parameters and Dialog BoxThe Main pane of the Interpolation Using Prelookup block dialog box appears as follows:Number of table dimensionsThe number of dimensions that the Table data parameter must have表数据必须具有的维数. This determines the number of independent variables for the table and hence the number of inputs to the block表独立变量的数量和输入量的数. Enter an integer between 1 a nd 30 i nto this field.Table dataThe table of output values输出值表. During simulation, the matrix size must match the dimensions defined by the Number of table dimensions parameter. Note that during block diagram editing, you can enter an empty matrix空矩阵 (specified as []) or an undefined workspace variable一个未定义的工作区变量 as the Table data parameter. Use this behavior to postpone specifying acorrectly-dimensioned matrix for the Table data parameter and continue editing the block diagram.For information about how to construct multidimensional arrays如何构建多维数组 in MATLAB software, see M ultidimensional Arrays in the MATLAB online documentation.Click the Edit button to open the Lookup Table Editor (see Lookup Table Editor in theSimulink documentation).Interpolation method插值法None – Flat o r Linear. See Interpolation Methods in the Simulink documentation for moreinformation.Extrapolation method外推法None – Clip o r Linear. See Extrapolation Methods in the Simulink documentation for moreinformation. The Extrapolation method parameter is visible外推方法的参数是可见的only if youselect Linear as the Interpolation method parameter.Note The Interpolation Using Prelookup block does not support Linearextrapolation if its input or output signals specify integer or fixed-point data types整数或定点数据类型.Action for out of range inputSpecifies whether to produce a warning or error message if the input is out of range指定是否将产生一个警告或错误信息，如果输入超出范围. The options are:None — the default, no warning or error messageWarning — display a warning message in the MATLAB Command Window andcontinue the simulationError — halt停止 the simulation and display an error message in the SimulationDiagnostics ViewerCheck index in generated code (Real-Time Workshop license required)Specifies whether Real-Time Workshop software generates code that checks the validity of the indexvalues fed to this block.Valid index input may reach last indexSpecifies how the index and interval fraction inputs to the block (labeled respectively as k a nd f o n theblock) access the last elements of the n-dimensional table specified by the Table data parameter. Ifenabled, the block returns the value of the last element in a particular dimension of its table when kindexes the last table element in the corresponding dimension and f i s 0. If disabled, the block returnsthe value of the last element in a particular dimension of its table when k i ndexes the next-to-last tableelement in the corresponding dimension and f i s 1. Note that index values are zero-based.This parameter is visible only if the Interpolation method specifies Linear a nd theExtrapolation method is None - Clip.Number of sub-table selection dimensionsSpecifies the number of dimensions of the subtable used to compute this block's output. Specify0 (the default) to interpolate the entire table, which disables subtable selection禁用子表选择.Sample timeSpecify the time interval 指定时间间隔between samples. To inherit the sample time, set this parameter to-1. See How to Specify the Sample Time in the Simulink documentation for more information. The Signal Attributes pane of the Interpolation Using Prelookup block dialog box appears as follows:Output minimum输出的最小值Specify the minimum value that the block should output. The default value, [], is equivalent to -Inf.Simulink software uses this value to perform:Parameter range checking (see Checking Parameter Values)Simulation range checking (see Checking Signal Ranges)Automatic scaling自动缩放 of fixed-point data typesOutput maximum输出的最大值Specify the maximum value that the block should output. The default value, [], is equivalent to Inf.Simulink software uses this value to perform:Parameter range checking (see Checking Parameter Values)Simulation range checking (see Checking Signal Ranges)Automatic scaling of fixed-point data typesOutput data type输出数据类型Specify the output data type. You can set it to:A rule that inherits a data type继承数据类型规则, for example, Inherit: Inherit viaback propagation反向传播法The name of a built-in data type内置数据类型, for example, singleThe name of a data type object, for example, a Simulink.NumericType o bjectAn expression that evaluates to a data type, for example, fixdt(1,16,0)Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Output data type parameter.See Specifying Block Output Data Types for more information.Lock scaling against changes by the autoscaling toolSelect to lock scaling of data选择锁定缩放数据.Integer rounding mode数据舍入模式Select the rounding mode for fixed-point operations. For more information, see R ounding in theSimulink Fixed Point User's Guide.Block parameters such as Table data are always rounded to the nearest representable value最接近的可表示值. To control the rounding of a block parameter, enter an expression using a MATLABrounding function into the mask field.Saturate on integer overflow整数溢出饱和Select to have overflows saturate溢出饱和. Otherwise, overflows wrap.When you select this check box, saturation applies to every internal operation on the block, not just the output or result. In general, the code generation process can detect when overflow is not possible, in which case, no saturation code is generated.The Table Attributes pane of the Interpolation Using Prelookup block dialog box appears as follows:Table minimum表最小值Specify the minimum value for table data. The default value, [], is equivalent to -Inf.Table maximum表最大值Specify the maximum value for table data. The default value, [], is equivalent to Inf.Table data typeSpecify the table data type. You can set it to:A rule that inherits a data type, for example, Inherit: Same as outputThe name of a built-in data type, for example, singleThe name of a data type object, for example, a Simulink.NumericType o bjectAn expression that evaluates to a data type, for example, fixdt(1,16,0)Tip Specify a table data type different from the output data type for these cases:Interpolation Using Prelookup :: Blocks (Simulink®) jar:file:///D:/Program%20Files/matlab/help/toolbox/simulink/help.jar!/slr...Lower memory requirement for storing table data that uses a smaller type than theoutput signalSharing of prescaled table data between two Interpolation Using Prelookup blockswith different output data typesSharing of custom storage table data in Real-Time Workshop generated code forblocks with different output data typesThe Intermediate Attributes中间值特性pane of the Interpolation Using Prelookup block dialog box appears as follows:Intermediate results data typeSpecify the intermediate results data type. You can set it to:A rule that inherits a data type, for example, Inherit: Same as outputThe name of a built-in data type, for example, singleThe name of a data type object, for example, a Simulink.NumericType o bjectAn expression that evaluates to a data type, for example, fixdt(1,16,0)Interpolation Using Prelookup :: Blocks (Simulink®) jar:file:///D:/Program%20Files/matlab/help/toolbox/simulink/help.jar!/slr...Click the Show data type assistant button to display the Data Type Assistant,which helps you set the Intermediate results data type parameter.Tip Use this parameter to specify higher precision指明更高的精度for internalcomputations than for table data or output data.CharacteristicsSee AlsoPrelookupProvide feedback about this pageIntegrator Interval Test © 1984-2009 The MathWorks, Inc. •Terms of Use •Patents •Trademarks •Acknowledgments。

VRayMtl[VRay材质]VRay材质可以得到较好的物理上的正确照明（能源分布）、较快的渲染速度，更方便的反射/折射参数。

在VRay材质中可以运用不同的纹理贴图、控制反射/折射，增加凹凸和置换贴图、强制直接Gl计算，为材质选择不同的BRDF类型。

Basic parameters[基础参数]Diffuse:设置材质的漫反射颜色。

注意：实际的漫反射颜色也取决于反射/折射颜色。

Reflect:反射，设置反射的颜色。

Fresnel reflections:菲涅尔反射，勾选这个复选框后，反射的强度将取决于对象表面的入射角，自然界中有一些材质（如玻璃）的反射就是这种方式。

不过要注意的是这个效果还取决于材质的折射率。

Fresnel loR: 菲舍尔反射率，这个参数在Fresnel reflections选项后面的L（锁定）按钮弹起的时候被激活，可以单独设置菲舍尔反射的反射率。

Hilight glossiness:有光泽的高光，这个是新增加的功能，控制VR材质的高光状态。

默认情况下，L形按钮被按下，Hilight glossiness处于非激活状态。

Refl glossiness:有光泽的反射，这个参数用于设置反射的锐利效果。

值为1意味着是一种完美的镜面反射效果，随着取值的减小，反射效果会越来越模糊。

平滑反射的品质由下面的细分参数来控制。

Subdivs:细分，控制平滑反射的品质。

较小的取值将加快渲染的速度，但是会导致更多的噪波，反之亦然。

Use interpolation:使用插补VRay能够使用一种类似于发光贴图的缓存方案来加快模糊反射的计算速度。

勾选这个复选框表示使用缓存方案。

Max depth: 最大深度，定义反射能完成的最大次数。

注意当场景中具有大量的反射/折射表面的时候，这个参数要设置的足够大才会产生真实效果。

Exit color: 退出颜色，当光线在场景中反射达到最大深度定义的反射次数后就停止反射，此时这个颜色将被返回，并且不再追踪远处的光线。

calibrationCalibration: A Guide to Accurate MeasurementIntroductionCalibration plays a crucial role in ensuring accurate measurements across various fields and industries. Whether it is in laboratories, manufacturing facilities, or even home appliances, calibration is essential for reliable and precise readings. In this document, we will delve into the importance of calibration, its various applications, and the different methods and techniques employed to achieve accurate measurements.What is Calibration?Calibration is the process of comparing measurements or readings obtained from an instrument or device with a known reference or standard. It aims to minimize measurement errors and ensure that the instrument is providing reliable and accurate results. By calibrating an instrument, we can establish its traceability to a recognized national orinternational standard, ensuring that the measurements are accepted universally.The Importance of CalibrationThe significance of calibration cannot be overstated, as it affects various aspects of our lives. Let's explore some of the key reasons why calibration is crucial:1. Accuracy and Reliability: Calibration ensures that instruments provide accurate and reliable measurements. This is particularly critical in scientific research, healthcare, manufacturing processes, and environmental monitoring.2. Compliance: Many industries are subject to regulatory standards and requirements. Calibration helps organizations comply with these standards, ensuring that they meet legal obligations and avoid penalties or legal issues.3. Quality Control: Calibration ensures that products and services meet predetermined quality standards. By calibrating measuring instruments regularly, manufacturing companies can maintain consistent quality, minimize defects, and enhance customer satisfaction.4. Safety: In sectors like healthcare, aviation, and automotive, accurate measurements are vital to ensuring safety. Calibration helps identify and rectify any inaccuracies in instruments, ensuring reliable and safe operations.Methods of CalibrationThere are various methods employed for calibration, depending on the instrument or device being calibrated. Let's explore some common calibration techniques:1. Comparative Calibration: This method involves comparing the measurements of the instrument under calibration with a reference instrument that is known to be accurate. The two instruments are subjected to the same input, and the differences in readings are adjusted to minimize errors.2. Direct or Primary Calibration: In this method, the instrument is calibrated directly against a known and traceable standard. The standard may be a physical artifact, like a weight or length, or it may be an electronic standard, such as a voltage or current.3. Interpolation Calibration: This method is employed when the instrument cannot be calibrated directly against a standard. Instead, measurements are obtained at a few specific points, and the values in between are estimated using mathematical interpolation techniques.4. Recalibration: Over time, instruments may drift or deviate from their original calibration. Recalibration involves adjusting the instrument back to its original calibration standard. Regular recalibration helps maintain accuracy and prolong the life of the instrument.Applications of CalibrationCalibration finds applications across various fields and industries. Some notable examples include:1. Laboratory Testing: Calibration is fundamental in laboratories where accurate and reliable measurements are crucial for research, experiments, and data analysis. Instruments such as balances, pipettes, spectrometers, and thermometers are regularly calibrated to ensure precise results.2. Manufacturing: Calibration is essential in manufacturing processes to maintain consistent quality, avoid defects, and comply with industry standards. Instruments used for measuring dimensions, pressures, temperatures, and electrical parameters are calibrated to ensure accuracy.3. Healthcare: In the healthcare sector, calibration is critical for medical devices such as blood pressure monitors, thermometers, infusion pumps, and diagnostic instruments. Accurate measurements are vital for patient safety and effective treatment.4. Environmental Monitoring: Instruments used for monitoring air quality, water quality, radiation levels, and weather conditions must be calibrated to provide accurate data. This ensures informed decision-making and timely actions for environmental preservation.ConclusionCalibration is a fundamental process that ensures accurate measurements and reliable results in various fields. Whether it is in laboratories, manufacturing facilities, healthcare, or environmental monitoring, the importance of calibration cannot be overstated. By adhering to calibration practicesand using appropriate techniques, we can enhance the accuracy, safety, and quality of measurements, ultimately benefiting society as a whole.。

Lecture 1The Old English Period and the Middle English Period:The Old English PeriodTeaching Process1 The early history of England●The beginning of settlement in Britain took place in about 700 B. C. The settlers wereCeltic-speaking tribes called Britons (or Celts), from whom the island get its name—Britain (the land of Britons). The Britons were a primitive people, a bronze-age people entering an age of iron.●In 55 B. C., Britain was invaded by the Roman general Julius Caesar. The Celts weresubjugated to the role of Rome for nearly 400 years. In 410, the Romans abandoned the island, which marks the end of “Roman Conquest” (55 B. C.—410 A. D.)●After the withdrawal of the Roman troops, England was soon invaded by threeGermanic tribes: the Angles, the Saxons and the Jutes, who came from the Northeast of Europe. The Anglo-Saxon invaders established some small kingdoms in Britain which by the 7th century were combined into a United Kingdom called England (the land of Angles). Its people were called the English. The three dialects spoken by them naturally grew into a single language called Anglo-Saxon, or Old English, a derivative of Low German.●The Anglo-Saxons admired the wisdom and courage of their war leaders. InAnglo-Saxon society, the king was essentially a warrior, whose duty was mainly to lead the tribe to fight against any possible foreign or hostile tribes. The Anglo-Saxons were a heathen people before they accepted Christianity. They believed in the old mythology of Northern Europe. That is why the Northern mythology has left its marks upon the English Language. (e. g. some of the names of the week in English derive from the names of Northern gods: Tuesday—Tiu战神, Wednesday—Woden主神, Thursday—Thor雷神, Friday— Frigg爱情女神). In 597, St. Augustine and his monks landed in Kent. Then England was Christianized part by part in the following century.Monasteries were built all over the country. It was in these monasteries that the earliest English books were written down, because at that time few except monks could read and write. The conversion of The Anglo-Saxons to Christianity changed their intellectual outlook. Christianity offered some values that were more consiste nt and solid than their pagan beliefs. It also enabled them to learn book knowledge in monasteries or schools from the Scriptures and the classical writings of Virgil (a great roman poet) and the ancient Greeks. They travelled long distance to Rome on pilgrimages, and these experiences enriched their understanding of the world, broadened and deepened their mental and spiritual power.●From 787, the English began to be troubled by bands of Danish Vikings (Scandinaviansea robbers). At first they just invaded parts of the country from time to time. Gradually, however, they came to make permanent settlement and ruthlessly destroyed many of the monasteries. The great hero of the time was King Alfred the Great (849-901) of the Kingdom of Wessex in the south, who succeeded in driving off the Danes. It is to him that we owe much of the preservation of Older Anglo-Saxon Literature. Equally important is the fact that he started the Anglo-Saxon Chronicle, a historical register of national events from dim past to his own age. This work was continued by monks long after his death. The Danes were eventually defeated, and the Anglo-Saxons returned to rule in 1042. England did not have peace for long. Soon they were conquered in 1066 by the Norms from France. This is the end of the Anglo-Saxon period.The following is a formula for the nation:Original Angles+ Saxons +Danes +Normans=EnglishBritons Jutes2 The early English Literature●The Britons (Celts) had no written language, so their myths and legends were notrecorded for many centuries. (e.g. The Adventures of King Arthur and His Knights of the Round Table is about a Celtic Chieftain of the 5th or 6th century, but these legends were not written down until they reached eleventh century France and were circulated after the 13th century.) Therefore, English literature actually began with the Anglo-Saxonsettlement in England (See Chang Yaoxin, 2-4).●Beowulf◆ It is a 3182-line alliterative verse, considered the monumental work in English poetry of the Anglo-Saxon period, or the national epic of the English people.◆It is of Germanic heritage, and contains evidently pre-Christian elements. The whole song is essentially pagan in spirit and matter.◆ It is pervaded with paganism. In the noble figure of Beowulf, pagan heroism and fatalism are mingled with Christian qualities. The non-Christian story of monsters and mighty sea beasts is blended with some serious thoughts on the power of a wise God.◆ In the poem, the poet expresses a hope that the evil should be punished, and the righteous will be rewarded.◆The poem probably existed in its oral form as early as the 6th century. It was written down in the 7th or 8th century though the extant manuscript, discovered in 1705, dated back to the 10th century. The whole epic is divided into two parts with an interpolation. There are three episodes related in the career of Beowulf:(1) The fight with the monster, Grendel(2) The fight with Grendel’s mother, a still more frightful she-monster(3) The mortal combat with the Fire Dragon (after being the king of the Geats for 50 years). (For details see Chan Yaoxin, 4-5)◆ The poem is more than a fairy story of heroes combating monsters.▼It recaptures the values, beliefs and longing of the Anglo-Saxon people before they came to England. Though it is inferior to Iliad 《伊里亚特》(written by Homer) in artistic merit, it is nevertheless a dignified noble poem, technically unexcelled in old English poetry.▼It affords us invaluable insight into the heroic ways of life of the Teutonic peoples, as well as into the best qualities of the newer culture they were building in England.▼It clearly mirrors their ideals—valor, the love of glory, honour, and duty, the loyalty of the retainer and the generosity of the lord. It also reflects the typical tone of their literature, with its emphasis on the darker emotions, its grim scenery, and its vigorous portrayal of the sea and of battle.▼The poem unfolds a picture of an early Danish society, of its public life, its customs, rituals and cultural activities. Present-day readers are often charmed by its depiction and primitive beauty.▼Its chief significance lies in the portrayal of a great national hero, who is not simply a courageous warrior, but a wise ruler, and a brave fighter. He is selfless, eager to help others in distress, faithful to his people, and is ready to sacrifice his own life for their welfare.◆ Three important features must be noted while studying the poem:▼The most striking feature is the use of alliteration, which is the repetition of a speech sound in a sequence of nearby words, and is usually applied only to consonants and only when the recurrent sound begins a word or a stressed syllable within a word. This is characteristic of all Old English verse.▼The second feature is the use of metaphors and understatements.▼The third feature of the poem is the mixture of pagan and Christian elements survive: the observing of omens, the attribution of power to Wyrd (i.e. fate), cremation, blood-revenge, and the praise of worldly glory—all woven into the poem. On the other hand, the assumption of God’s domination over the world, of Devil’s agency among men, of existence of Heaven and Hell, of a last judgment, and of the noxiousness of sin—are all Christian.ⅦReflection Questions and AssignmentsAnswer the question “Why does Beowulf tower above Old English L iterature?” in the exercise book.。