A new time-scale adaptive denoising method based on wavelet shrinkage

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M ULTISCALE M ODEL.S IMUL.c 2005Society for Industrial and Applied Mathematics Vol.4,No.2,pp.490–530A REVIEW OF IMAGE DENOISING ALGORITHMS,WITH A NEWONE∗A.BUADES†,B.COLL†,AND J.M.MOREL‡Abstract.The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics.In spite of the sophistication of the recently proposed methods,most algorithms have not yet attained a desirable level of applicability.All show an out-standing performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove imagefine structures.The main focus of this paper is,first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and,second,to propose a nonlocal means(NL-means)algorithm ad-dressing the preservation of structure in a digital image.The mathematical analysis is based on the analysis of the“method noise,”defined as the difference between a digital image and its denoised version.The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model.The denoising performance of all considered methods are compared in four ways; mathematical:asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical:the algorithms artifacts and their explanation as a violation of the image model;quantitative experimental:by tables of L2distances of the denoised version to the original image.The most powerful evaluation method seems,however,to be the visualization of the method noise on natural images.The more this method noise looks like a real white noise,the better the method.Key words.image restoration,nonparametric estimation,PDE smoothingfilters,adaptive filters,frequency domainfiltersAMS subject classification.62H35DOI.10.1137/0406160241.Introduction.1.1.Digital images and noise.The need for efficient image restoration meth-ods has grown with the massive production of digital images and movies of all kinds, often taken in poor conditions.No matter how good cameras are,an image improve-ment is always desirable to extend their range of action.A digital image is generally encoded as a matrix of grey-level or color values.In the case of a movie,this matrix has three dimensions,the third one corresponding to time.Each pair(i,u(i)),where u(i)is the value at i,is called a pixel,short for“picture element.”In the case of grey-level images,i is a point on a two-dimensional(2D)grid and u(i)is a real value.In the case of classical color images,u(i)is a triplet of values for the red,green,and blue components.All of what we shall say applies identically to movies,three-dimensional(3D)images,and color or multispectral images.For the sake of simplicity in notation and display of experiments,we shall here be content with rectangular2D grey-level images.∗Received by the editors September30,2004;accepted for publication(in revised form)Janu-ary10,2005;published electronically July18,2005./journals/mms/4-2/61602.html†Universitat de les Illes Balears,Anselm Turmeda,Ctra.Valldemossa Km.7.5,07122Palma de Mallorca,Spain(vdmiabc4@uib.es,tomeu.coll@uib.es).These authors were supported by the Ministerio de Ciencia y Tecnologia under grant TIC2002-02172.During this work,thefirst author had a fellowship of the Govern de les Illes Balears for the realization of his Ph.D.thesis.‡Centre de Math´e matiques et Leurs Applications,ENS Cachan61,Av du Pr´e sident Wilson94235 Cachan,France(morel@cmla.ens-cachan.fr).This author was supported by the Centre National d’Etudes Spatiales(CNES),the Office of Naval Research under grant N00014-97-1-0839,the Direction G´e n´e rale des Armements(DGA),and the Minist`e re de la Recherche et de la Technologie.490ON IMAGE DENOISING ALGORITHMS 491The two main limitations in image accuracy are categorized as blur and noise.Blur is intrinsic to image acquisition systems,as digital images have a finite number of samples and must satisfy the Shannon–Nyquist sampling conditions [31].The second main image perturbation is noise.Each one of the pixel values u (i )is the result of a light intensity measurement,usually made by a charge coupled device (CCD)matrix coupled with a light focusing system.Each captor of the CCD is roughly a square in which the number of incoming photons is being counted for a fixed period corresponding to the obturation time.When the light source is constant,the number of photons received by each pixel fluctuates around its average in accordance with the central limit theorem.In other terms,one can expect fluctuations of order √n for n incoming photons.In addition,each captor,if not adequately cooled,receives heat spurious photons.The resulting perturbation is usually called “obscurity noise.”In a first rough approximation one can writev (i )=u (i )+n (i ),where i ∈I ,v (i )is the observed value,u (i )would be the “true”value at pixel i ,namely the one which would be observed by averaging the photon counting on a long period of time,and n (i )is the noise perturbation.As indicated,the amount of noise is signal-dependent;that is,n (i )is larger when u (i )is larger.In noise models,the normalized values of n (i )and n (j )at different pixels are assumed to be independent random variables,and one talks about “white noise.”1.2.Signal and noise ratios.A good quality photograph (for visual inspec-tion)has about 256grey-level values,where 0represents black and 255represents white.Measuring the amount of noise by its standard deviation,σ(n ),one can define the signal noise ratio (SNR)asSNR =σ(u )σ(n ),where σ(u )denotes the empirical standard deviation of u ,σ(u )= 1|I | i ∈I(u (i )−u )212,and u =1|I | i ∈I u (i )is the average grey-level value.The standard deviation of the noise can also be obtained as an empirical measurement or formally computed whenthe noise model and parameters are known.A good quality image has a standard deviation of about 60.The best way to test the effect of noise on a standard digital image is to add a Gaussian white noise,in which case n (i )are independently and identically distributed (i.i.d.)Gaussian real variables.When σ(n )=3,no visible alteration is usually ob-served.Thus,a 603 20SNR is nearly invisible.Surprisingly enough,one can add white noise up to a 21ratio and still see everything in a picture!This fact is il-lustrated in Figure 1and constitutes a major enigma of human vision.It justifies the many attempts to define convincing denoising algorithms.As we shall see,the results have been rather deceptive.Denoising algorithms see no difference between small details and noise,and therefore they remove them.In many cases,they create new distortions,and the researchers are so used to them that they have created a492 A.BUADES,B.COLL,AND J.M.MORELFig.1.A digital image with standard deviation55,the same with noise added(standard deviation3),the SNR therefore being equal to18,and the same with SNR slightly larger than2. In this second image,no alteration is visible.In the third,a conspicuous noise with standard deviation25has been added,but,surprisingly enough,all details of the original image still are visible.taxonomy of denoising artifacts:“ringing,”“blur,”“staircase effect,”“checkerboard effect,”“wavelet outliers,”etc.This fact is not quite a surprise.Indeed,to the best of our knowledge,all denoising algorithms are based on•a noise model;•a generic image smoothness model,local or global.In experimental settings,the noise model is perfectly precise.So the weak point of the algorithms is the inadequacy of the image model.All of the methods assume that the noise is oscillatory and that the image is smooth or piecewise smooth.So they try to separate the smooth or patchy part(the image)from the oscillatory one.Actually, manyfine structures in images are as oscillatory as noise is;conversely,white noise has low frequencies and therefore smooth components.Thus a separation method based on smoothness arguments only is hazardous.1.3.The“method noise.”All denoising methods depend on afiltering pa-rameter h.This parameter measures the degree offiltering applied to the image.For most methods,the parameter h depends on an estimation of the noise varianceσ2. One can define the result of a denoising method D h as a decomposition of any image v as(1.1)v=D h v+n(D h,v),where1.D h v is more smooth than v,2.n(D h,v)is the noise guessed by the method.Now it is not enough to smooth v to ensure that n(D h,v)will look like a noise. The more recent methods are actually not content with a smoothing but try to recover lost information in n(D h,v)[19,25].So the focus is on n(D h,v).Definition 1.1(method noise).Let u be a(not necessarily noisy)image and D h a denoising operator depending on h.Then we define the method noise of u as the image difference(1.2)n(D h,u)=u−D h(u).This method noise should be as similar to a white noise as possible.In addition, since we would like the original image u not to be altered by denoising methods,theON IMAGE DENOISING ALGORITHMS 493method noise should be as small as possible for the functions with the right regularity.According to the preceding discussion,four criteria can and will be taken into account in the comparison of denoising methods:•A display of typical artifacts in denoised images.•A formal computation of the method noise on smooth images,evaluating how small it is in accordance with image local smoothness.•A comparative display of the method noise of each method on real images with σ=2.5.We mentioned that a noise standard deviation smaller than 3is subliminal,and it is expected that most digitization methods allow themselves this kind of noise.•A classical comparison receipt based on noise simulation:it consists of taking a good quality image,adding Gaussian white noise with known σ,and then computing the best image recovered from the noisy one by each method.A table of L 2distances from the restored to the original can be established.The L 2distance does not provide a good quality assessment.However,it reflects well the relative performances of algorithms.On top of this,in two cases,a proof of asymptotic recovery of the image can be obtained by statistical arguments.1.4.Which methods to compare.We had to make a selection of the denoising methods we wished to compare.Here a difficulty arises,as most original methods have caused an abundant literature proposing many improvements.So we tried to get the best available version,while keeping the simple and genuine character of the original method:no hybrid method.So we shall analyze the following:1.the Gaussian smoothing model (Gabor quoted in Lindenbaum,Fischer,andBruckstein [17]),where the smoothness of u is measured by the Dirichlet integral |Du |2;2.the anisotropic filtering model (Perona and Malik [27],Alvarez,Lions,andMorel [1]);3.the Rudin–Osher–Fatemi total variation model [30]and two recently proposediterated total variation refinements [35,25];4.the Yaroslavsky neighborhood filters [41,40]and an elegant variant,theSUSAN filter (Smith and Brady [33]);5.the Wiener local empirical filter as implemented by Yaroslavsky [40];6.the translation invariant wavelet thresholding [8],a simple and performingvariant of the wavelet thresholding [10];7.DUDE,the discrete universal denoiser [24],and the UINTA,unsupervisedinformation-theoretic,adaptive filtering [3],two very recent new approaches;8.the nonlocal means (NL-means)algorithm,which we introduce here.This last algorithm is given by a simple closed formula.Let u be defined in a bounded domain Ω⊂R 2;thenNL (u )(x )=1C (x )e −(G a ∗|u (x +.)−u (y +.)|2)(0)2u (y )d y ,where x ∈Ω,G a is a Gaussian kernel of standard deviation a ,h acts as a filtering parameter,and C (x )= e −(G a ∗|u (x +.)−u (z +.)|2)(0)h 2d z is the normalizing factor.In orderto make clear the previous definition,we recall that(G a ∗|u (x +.)−u (y +.)|2)(0)= R 2G a (t )|u (x +t )−u (y +t )|2d t .494 A.BUADES,B.COLL,AND J.M.MORELThis amounts to saying that NL(u)(x),the denoised value at x,is a mean of the values of all pixels whose Gaussian neighborhood looks like the neighborhood of x.1.5.What is left.We do not draw into comparison the hybrid methods,in particular the total variation+wavelets[7,12,18].Such methods are significant improvements of the simple methods but are impossible to draw into a benchmark: their efficiency depends a lot upon the choice of wavelet dictionaries and the kind of image.Second,we do not draw into the comparison the method introduced recently by Meyer[22],whose aim it is to decompose the image into a BV part and a texture part(the so called u+v methods),and even into three terms,namely u+v+w, where u is the BV part,v is the“texture”part belonging to the dual space of BV, denoted by G,and w belongs to the Besov space˙B∞−1,∞,a space characterized by the fact that the wavelet coefficients have a uniform bound.G is proposed by Meyer as the right space to model oscillatory patterns such as textures.The main focus of this method is not yet denoising.Because of the different and more ambitious scopes of the Meyer method[2,36,26],which makes it parameter-and implementation-dependent, we could not draw it into the st but not least,let us mention the bandlet[15]and curvelet[34]transforms for image analysis.These methods also are separation methods between the geometric part and the oscillatory part of the image and intend tofind an accurate and compressed version of the geometric part. Incidentally,they may be considered as denoising methods in geometric images,as the oscillatory part then contains part of the noise.Those methods are closely related to the total variation method and to the wavelet thresholding,and we shall be content with those simpler representatives.1.6.Plan of the paper.Section2computes formally the method noise for the best elementary local smoothing methods,namely Gaussian smoothing,anisotropic smoothing(mean curvature motion),total variation minimization,and the neighbor-hoodfilters.For all of them we prove or recall the asymptotic expansion of thefilter at smooth points of the image and therefore obtain a formal expression of the method noise.This expression permits us to characterize places where thefilter performs well and where it fails.In section3,we treat the Wiener-like methods,which proceed by a soft or hard threshold on frequency or space-frequency coefficients.We examine in turn the Wiener–Fourierfilter,the Yaroslavsky local adaptive discrete cosine trans-form(DCT)-basedfilters,and the wavelet threshold method.Of course,the Gaussian smoothing belongs to both classes offilters.We also describe the universal denoiser DUDE,but we cannot draw it into the comparison,as its direct application to grey-level images is unpractical so far.(We discuss its feasibility.)Finally,we examine the UINTA algorithms whose principles stand close to the NL-means algorithm.In section5,we introduce the NL-meansfilter.This method is not easily classified in the preceding terminology,since it can work adaptively in a local or nonlocal way.We first give a proof that this algorithm is asymptotically consistent(it gives back the conditional expectation of each pixel value given an observed neighborhood)under the assumption that the image is a fairly general stationary random process.The works of Efros and Leung[13]and Levina[16]have shown that this assumption is sound for images having enough samples in each texture patch.In section6,we com-pare all algorithms from several points of view,do a performance classification,and explain why the NL-means algorithm shares the consistency properties of most of the aforementioned algorithms.ON IMAGE DENOISING ALGORITHMS4952.Local smoothingfilters.The original image u is defined in a bounded domainΩ⊂R2and denoted by u(x)for x=(x,y)∈R2.This continuous image is usually interpreted as the Shannon interpolation of a discrete grid of samples[31]and is therefore analytic.The distance between two consecutive samples will be denoted byε.The noise itself is a discrete phenomenon on the sampling grid.According to the usual screen and printing visualization practice,we do not interpolate the noise samples n i as a band limited function but rather as a piecewise constant function, constant on each pixel i and equal to n i.We write|x|=(x2+y2)12and x1.x2=x1x2+y1y2as the norm and scalar productand denote the derivatives of u by u x=∂u∂x ,u y=∂u∂y,and u xy=∂2u∂x∂y.The gradientof u is written as Du=(u x,u y)and the Laplacian of u asΔu=u xx+u yy.2.1.Gaussian smoothing.By Riesz’s theorem,image isotropic linearfiltering boils down to a convolution of the image by a linear radial kernel.The smoothing requirement is usually expressed by the positivity of the kernel.The paradigm of suchkernels is,of course,the Gaussian x→G h(x)=1(4πh2)e−|x|24h2.In that case,G h hasstandard deviation h,and the following theorem is easily seen.Theorem2.1(Gabor1960).The image method noise of the convolution with a Gaussian kernel G h isu−G h∗u=−h2Δu+o(h2).A similar result is actually valid for any positive radial kernel with bounded variance,so one can keep the Gaussian example without loss of generality.The preceding estimate is valid if h is small enough.On the other hand,the noise reduction properties depend upon the fact that the neighborhood involved in the smoothing is large enough,so that the noise gets reduced by averaging.So in the following we assume that h=kε,where k stands for the number of samples of the function u and noise n in an interval of length h.The spatial ratio k must be much larger than1to ensure a noise reduction.The effect of a Gaussian smoothing on the noise can be evaluated at a referencepixel i=0.At this pixel,G h∗n(0)=i∈IP iG h(x)n(x)d x=i∈Iε2G h(i)n i,where we recall that n(x)is being interpolated as a piecewise constant function,the P i square pixels centered in i have sizeε2,and G h(i)denotes the mean value of the function G h on the pixel i.Denoting by V ar(X)the variance of a random variable X,the additivity of variances of independent centered random variables yieldsV ar(G h∗n(0))=i ε4G h(i)2σ2 σ2ε2G h(x)2d x=ε2σ28πh2.So we have proved the following theorem.Theorem2.2.Let n(x)be a piecewise constant white noise,with n(x)=n i on each square pixel i.Assume that the n i are i.i.d.with zero mean and varianceσ2. Then the“noise residue”after a Gaussian convolution of n by G h satisfiesV ar(G h∗n(0)) ε2σ2 8πh2.496 A.BUADES,B.COLL,AND J.M.MORELIn other terms,the standard deviation of the noise,which can be interpreted as thenoise amplitude,is multiplied by εh √8π.Theorems 2.1and 2.2traduce the delicate equilibrium between noise reductionand image destruction by any linear smoothing.Denoising does not alter the image at points where it is smooth at a scale h much larger than the sampling scale ε.The first theorem tells us that the method noise of the Gaussian denoising method is zero in harmonic parts of the image.A Gaussian convolution is optimal on harmonic functions and performs instead poorly on singular parts of u ,namely edges or texture,where the Laplacian of the image is large.See Figure 3.2.2.Anisotropic filters and curvature motion.The anisotropic filter (AF)attempts to avoid the blurring effect of the Gaussian by convolving the image u at x only in the direction orthogonal to Du (x ).The idea of such a filter goes back to Perona and Malik [27]and actually again to Gabor (quoted in Lindenbaum,Fischer,and Bruckstein [17]).SetA F h u (x )=G h (t )u x +t Du (x )⊥|Du (x )|dt for x such that Du (x )=0and where (x,y )⊥=(−y,x )and G h (t )=1√2πh e −t 22h 2is theone-dimensional (1D)Gauss function with variance h 2.At points where Du (x )=0an isotropic Gaussian mean is usually applied,and the result of Theorem 2.1holds at those points.If one assumes that the original image u is twice continuously dif-ferentiable (C 2)at x ,the following theorem is easily shown by a second-order Taylor expansion.Theorem 2.3.The image method noise of an anisotropic filter A F h isu (x )−A F h u (x ) −12h 2D 2u Du ⊥|Du |,Du ⊥|Du | =−12h 2|Du |curv (u )(x ),where the relation holds when Du (x )=0.By curv (u )(x ),we denote the curvature,i.e.,the signed inverse of the radius of curvature of the level line passing by x .When Du (x )=0,this means thatcurv (u )=u xx u 2y −2u xy u x u y +u yy u 2x(u 2x +u 2y )32.This method noise is zero wherever u behaves locally like a one-variable function,u (x,y )=f (ax +by +c ).In such a case,the level line of u is locally the straight line with equation ax +by +c =0,and the gradient of f may instead be very large.In other terms,with anisotropic filtering,an edge can be maintained.On the other hand,we have to evaluate the Gaussian noise reduction.This is easily done by a 1D adaptation of Theorem 2.2.Notice that the noise on a grid is not isotropic;so the Gaussian average when Du is parallel to one coordinate axis is made roughly on √2more samples than the Gaussian average in the diagonal direction.Theorem 2.4.By anisotropic Gaussian smoothing,when εis small enough with respect to h ,the noise residue satisfiesVar (A F h (n ))≤ε√2πhσ2.ON IMAGE DENOISING ALGORITHMS 497In other terms,the standard deviation of the noise n is multiplied by a factor at mostequal to (ε√2πh)1/2,this maximal value being attained in the diagonals.Proof .Let L be the line x +t Du⊥(x )|Du (x )|passing by x ,parameterized by t ∈R ,and denote by P i ,i ∈I ,the pixels which meet L ,n (i )the noise value,constant on pixel P i ,and εi the length of the intersection of L ∩P i .Denote by g (i )the averageof G h (x +t Du⊥(x )|Du (x )|)on L ∩P i .Then one has A F h n (x )i εi n (i )g (i ).The n (i )are i.i.d.with standard variation σ,and thereforeV ar (A F h (n ))= i ε2iσ2g (i )2≤σ2max(εi ) iεi g (i )2.This yieldsVar (A F h (n ))≤√2εσ2 G h (t )2dt =ε√2πhσ2.There are many versions of A F h ,all yielding an asymptotic estimate equivalent to the one in Theorem 2.3:the famous median filter [14],an inf-sup filter on segments centered at x [5],and the clever numerical implementation of the mean curvature equation in [21].So all of those filters have in common the good preservation of edges,but they perform poorly on flat regions and are worse there than a Gaussian blur.This fact derives from the comparison of the noise reduction estimates of Theorems2.1and 2.4and is experimentally patent in Figure3.2.3.Total variation.The total variation minimization was introduced by Ru-din and Osher [29]and Rudin,Osher,and Fatemi [30].The original image u is supposed to have a simple geometric description,namely a set of connected sets,the objects,along with their smooth contours,or edges.The image is smooth inside the objects but with jumps across the boundaries.The functional space modeling these properties is BV (Ω),the space of integrable functions with finite total variation T V Ω(u )= |Du |,where Du is assumed to be a Radon measure.Given a noisy image v (x ),the above-mentioned authors proposed to recover the original image u (x )as the solution of the constrained minimization problemarg min uT V Ω(u ),(2.1)subject to the noise constraintsΩ(u (x )−v (x ))d x =0and Ω|u (x )−v (x )|2d x =σ2.The solution u must be as regular as possible in the sense of the total variation,while the difference v (x )−u (x )is treated as an error,with a prescribed energy.The constraints prescribe the right mean and variance to u −v but do not ensure that it is similar to a noise (see a thorough discussion in [22]).The preceding problem is naturally linked to the unconstrained problem arg min u T V Ω(u )+λΩ|v (x )−u (x )|2d x (2.2)498 A.BUADES,B.COLL,AND J.M.MORELfor a given Lagrange multiplierλ.The above functional is strictly convex and lower semicontinuous with respect to the weak-star topology of BV.Therefore the minimum exists,is unique,and is computable(see,e.g.,[6]).The parameterλcontrols the tradeoffbetween the regularity andfidelity terms.Asλgets smaller the weight of the regularity term increases.Thereforeλis related to the degree offiltering of the solution of the minimization problem.Let us denote by TVFλ(v)the solution of problem(2.2)for a given value ofλ.The Euler–Lagrange equation associated with the minimization problem is given by(u(x)−v(x))−12λcurv(u)(x)=0(see[29]).Thus,we have the following theorem.Theorem2.5.The image method noise of the total variation minimization(2.2) isu(x)−TVFλ(u)(x)=−12λcurv(TVFλ(u))(x).As in the anisotropic case,straight edges are maintained because of their small curvature.However,details and texture can be oversmoothed ifλis too small,as is shown in Figure3.2.4.Iterated total variation refinement.In the original total variation model the removed noise,v(x)−u(x),is treated as an error and is no longer studied.In practice,some structures and texture are present in this error.Several recent works have tried to avoid this effect[35,25].2.4.1.The Tadmor–Nezzar–Vese approach.In[35],the authors have pro-posed to use the Rudin–Osher–Fatemi model iteratively.They decompose the noisy image,v=u0+n0,by the total variation model.So taking u0to contain only ge-ometric information,they decompose by the very same model n0=u1+n1,where u1is assumed to be again a geometric part and n1contains less geometric informa-tion than n0.Iterating this process,one obtains u=u0+u1+u2+···+u k as a refined geometric part and n k as the noise residue.This strategy is in some sense close to the matching pursuit methods[20].Of course,the weight parameter in the Rudin–Osher–Fatemi model has to grow at each iteration,and the authors propose a geometric seriesλ,2λ,...,2kλ.In that way,the extraction of the geometric part n k becomes twice more taxing at each step.Then the new algorithm is as follows:1.Starting with an initial scaleλ=λ0,v=u0+n0,[u0,n0]=arg minv=u+n|Du|+λ0|v(x)−u(x)|2d x.2.Proceed with successive applications of the dyadic refinement n j=u j+1+n j+1,[u j+1,n j+1]=arg minn j=u+n|Du|+λ02j+1|n j(x)−u(x)|2d x.3.After k steps,we get the following hierarchical decomposition of v:v=u0+n0=u0+u1+n1=.....=u0+u1+···+u k+n k.ON IMAGE DENOISING ALGORITHMS 499The denoised image is given by the partial sum k j =0u j ,and n k is the noise residue.This is a multilayered decomposition of v which lies in an intermediate scale of spaces,in between BV and L 2.Some theoretical results on the convergence of this expansion are presented in [35].2.4.2.The Osher et al.approach.The second algorithm due to Osher et al.[25]also consists of an iteration of the original model.The new algorithm is as follows:1.First,solve the original total variation model u 1=arg min u ∈BV|∇u (x )|d x +λ(v (x )−u (x ))2d x to obtain the decomposition v =u 1+n 1.2.Perform a correction step to obtain u 2=arg min u ∈BV|∇u (x )|d x +λ(v (x )+n 1(x )−u (x ))2d x ,where n 1is the noise estimated by the first step.The correction step adds this first estimate of the noise to the original image and raises the decomposition v +n 1=u 2+n 2.3.Iterate:compute u k +1as a minimizer of the modified total variation mini-mization,u k +1=arg min u ∈BV|∇u (x )|d x +λ (v (x )+n k (x )−u (x ))2d x ,wherev +n k =u k +1+n k +1.Some results are presented in [25]which clarify the nature of the above sequence:•{u k }k converges monotonically in L 2to v ,the noisy image,as k →∞.•{u k }k approaches the noise-free image monotonically in the Bregman distanceassociated with the BV seminorm,at least until u ¯k −u ≤σ2,where u isthe original image and σis the standard deviation of the added noise.These two results indicate how to stop the sequence and choose u ¯k .It is enoughto proceed iteratively until the result gets noisier or the distance u ¯k −u 2gets smallerthan σ2.The new solution has more details preserved,as Figure 3shows.The above iterated denoising strategy being quite general,one can make the computations for a linear denoising operator T as well.In that case,this strategyT (v +n 1)=T (v )+T (n 1)amounts to saying that the first estimated noise n 1is filtered again and its smooth components are added back to the original,which is in fact the Tadmor–Nezzar–Vese strategy.2.5.Neighborhood filters.The previous filters are based on a notion of spatial neighborhood or proximity.Neighborhood filters instead take into account grey-level values to define neighboring pixels.In the simplest and more extreme case,the de-noised value at pixel i is an average of values at pixels which have a grey-level value close to u (i ).The grey-level neighborhood is thereforeB (i,h )={j ∈I |u (i )−h <u (j )<u (i )+h }.。

怎样适应新模式英文作文英文：Adapting to a new English writing style can be both challenging and rewarding. Here are some strategies I've found helpful:1. Immerse Yourself: Surround yourself with English as much as possible. Watch English movies, listen to English music, and read English books. This helps you get a feel for the language and its nuances.2. Practice Writing Regularly: The more you write, the better you'll get. Start with simple sentences and gradually move on to more complex structures. Don't be afraid to make mistakes; it's all part of the learning process.3. Learn from Examples: Reading well-written English essays can give you a sense of how to structure your ownwriting. Pay attention to the use of language, sentence structure, and vocabulary.4. Expand Your Vocabulary: The more words you know, the more options you have when expressing yourself. Try tolearn a few new words every day and use them in your writing.5. Get Feedback: Ask a teacher or native speaker to review your writing and provide feedback. This can help you identify areas for improvement and learn from your mistakes.6. Be Patient: Learning a new writing style takes time, so don't get discouraged if you don't see immediate results. Keep practicing, and you'll get there eventually.中文：适应新的英语写作风格既具挑战性又有回报。

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MingXin，LAI XueJia，XIAO GuoZhen，QIN Lei6.A chaos-based image encryption algorithm using alternate stru cture ZHANG YiWei，WANG YuMin，SHEN XuBang7.Impossible differential cryptanalysis of advanced encryption sta ndard CHEN Jie，HU YuPu，ZHANG YueYu8.Classification and counting on multi-continued fractions and its application to multi-sequences DAI ZongDuo，FENG XiuTao9.A trinomial type of σ-LFSR oriented toward software implemen tation ZENG Guang，HE KaiCheng，HAN WenBao10.Identity-based signature scheme based on quadratic residues CHAI ZhenChuan，CAO ZhenFu，DONG XiaoLei11.Modular approach to the design and analysis of password-ba sed security protocols FENG DengGuo，CHEN WeiDong12.Design of secure operating systems with high security levels QING SiHan，SHEN ChangXiang13.A formal model for access control with supporting spatial co ntext ZHANG Hong，HE YePing，SHI ZhiGuo14.Universally composable anonymous Hash certification model ZHANG Fan，MA JianFeng，SangJae MOON15.Trusted dynamic level scheduling based on Bayes trust model WANG Wei，ZENG GuoSun16.Log-scaling magnitude modulated watermarking scheme LING HeFei，YUAN WuGang，ZOU FuHao，LU ZhengDing17.A digital authentication watermarking scheme for JPEG image s with superior localization and security YU Miao，HE HongJie，ZHA NG JiaShu18.Blind reconnaissance of the pseudo-random sequence in DS/ SS signal with negative SNR HUANG XianGao，HUANG Wei，WANG Chao，L(U) ZeJun，HU YanHua1.Analysis of security protocols based on challenge-response LU O JunZhou，YANG Ming2.Notes on automata theory based on quantum logic QIU Dao Wen3.Optimality analysis of one-step OOSM filtering algorithms in t arget tracking ZHOU WenHui，LI Lin，CHEN GuoHai，YU AnXi4.A general approach to attribute reduction in rough set theory ZHANG WenXiuiu，QIU GuoFang，WU WeiZhi5.Multiscale stochastic hierarchical image segmentation by spectr al clustering LI XiaoBin，TIAN Zheng6.Energy-based adaptive orthogonal FRIT and its application in i mage denoising LIU YunXia，PENG YuHua，QU HuaiJing，YiN Yong7.Remote sensing image fusion based on Bayesian linear estimat ion GE ZhiRong，WANG Bin，ZHANG LiMing8.Fiber soliton-form 3R regenerator and its performance analysis ZHU Bo，YANG XiangLin9.Study on relationships of electromagnetic band structures and left/right handed structures GAO Chu，CHEN ZhiNing，WANG YunY i，YANG Ning10.Study on joint Bayesian model selection and parameter estim ation method of GTD model SHI ZhiGuang，ZHOU JianXiong，ZHAO HongZhong，FU Qiang。

提升你的视角英语作文Broadening Your Perspective: The Power of Diverse Views。

In today's interconnected world, the ability to broaden one's perspective is more crucial than ever. With the rapid advancements in technology and the ease of global communication, we are constantly exposed to a plethora of ideas, cultures, and viewpoints. Embracing this diversity can enrich our lives, foster creativity, and lead to a deeper understanding of the world around us. This essayaims to explore the importance of broadening our perspectives and the benefits it brings to both individuals and society.Understanding Different Cultures。

One of the most significant ways to broaden our perspective is by learning about different cultures. Each culture has its own unique history, traditions, and values that shape the way its people think and behave. Byimmersing ourselves in different cultures, we can gain a greater appreciation for diversity and develop empathy towards others.Traveling to foreign countries, trying new cuisines, and learning a new language are all excellent ways to experience different cultures firsthand. However, even if traveling is not an option, we can still broaden our horizons by reading books, watching documentaries, or attending cultural events that showcase different cultures.Embracing Different Viewpoints。

初中生写有关人工智能一类的英语作文全文共6篇示例，供读者参考篇1Artificial Intelligence: The Future is Here!Hi everyone! I'm a 7th grader and I've been really interested in artificial intelligence (AI) lately. It seems like something straight out of a sci-fi movie, but it's actually becoming a reality right before our eyes. Let me tell you what I've learned about this fascinating and mind-boggling technology.First off, what exactly is AI? Basically, it refers to computer systems that can perform tasks that normally require human intelligence, like learning, reasoning, problem-solving, and even creativity. These systems use complex algorithms and massive amounts of data to "learn" and make decisions, just like our brains do. Crazy, right?One of the most well-known examples of AI is virtual assistants like Siri, Alexa, and Google Assistant. These helpful little robots can understand our voice commands, look up information for us, set reminders, play music, and even crackjokes sometimes (though their sense of humor could use some work!). But AI goes way beyond virtual assistants.Self-driving cars are another incredible application of AI. These vehicles use sensors, cameras, and advanced software to navigate roads, avoid obstacles, and make driving decisions without any human input. Companies like Tesla, Waymo, and Uber are racing to perfect this technology and make our roads safer. Imagine never having to worry about distracted or drunk drivers again!AI is also transforming fields like healthcare and scientific research. Smart diagnostic systems can analyze medical images and data to detect diseases earlier and more accurately than human doctors. And AI algorithms can sift through massive datasets and spot patterns that lead to new scientific discoveries, from better drugs to cleaner energy solutions.Personally, I can't wait to see what AI has in store for the future. Maybe one day, we'll have robot tutors that can customize lessons just for us based on how we learn best. Or AI assistants that can help with our homework and answer any question we have. Heck, AI might even be able to compose essays for us (though I doubt it could make them as entertaining as this one!).Speaking of the future, some scientists are working on artificial general intelligence (AGI) – AI systems that can match or exceed human intelligence across all domains. We're still probably decades away from AGI, but if we ever achieve that level of AI, it could lead to a technological singularity where progress happens at an unimaginable pace. Whole industries and ways of life could be transformed overnight.As exciting as AGI sounds, it's also a little scary to think about. What if superintelligent AI systems become uncontrollable or decide humans are a threat? Will we become obsolete and get taken over by our own creations, like in the Terminator movies? I really hope the AI researchers are taking safety seriously and putting safeguards in place.Well, those are just some of my thoughts on this wild and rapidly evolving field of AI. Whether you find it thrilling or terrifying, there's no denying that it's going to have a huge impact on all of our lives in the years ahead. We might as well buckle up and enjoy the ride into our AI-powered future!篇2The Fascinating World of Artificial IntelligenceHey there! My name is Alex, and I'm a 13-year-old student who's super interested in technology, especially artificial intelligence (AI). AI is like really smart computer programs that can do amazing things like understand human language, recognize images and speech, and even beat human masters at complex games like chess and Go.I first learned about AI a couple of years ago when I saw a video of this crazy robot that could walk around and do backflips and stuff. I thought that was so cool! Then I started reading about how AI can be used for all sorts of helpful tasks like assisting doctors in diagnosing diseases, controlling self-driving cars, and providing suggestions for movies or products you might like based on your interests.At first, some of the technical details about AI went over my head. Like how AI systems use things called neural networks that are inspired by the human brain to process data in a way that mimics how we learn and make decisions. But the more I read, the more fascinated I became.One type of AI I find particularly interesting is called machine learning. Basically, instead of being programmed with tons of rules like traditional software, machine learning systems can study data and examples to figure things out on their own. It'slike how we learn language and skills as babies by observing patterns rather than following strict rules. With enough data to train on, machine learning can allow AI to do amazing things like understand natural human speech, translate between languages, recognize faces, objects and even emotions in images and video.Speaking of recognizing images, another awesome AI capability is computer vision. By analyzing digital images and videos, computer vision algorithms can automatically identify people, objects, text, scenery and activities. They can even track movement of things over time. It's thanks to computer vision that AI can power so much modern facial recognition for security and photo tagging on social media. Self-driving cars also rely heavily on computer vision to detect other vehicles, pedestrians, traffic signals and road conditions.While those are some of the current major applications, the possibilities for AI seem almost limitless going forward. I could see AI being used to help solve challenging problems like climate change by analyzing environmental data and testing potential solutions through simulation. AI tutors and personalized learning tools could transform education by adapting to each student's unique needs and learning style. AI might even help uscommunicate with animals by interpreting their vocalizations and behaviors!Those are valid concerns, but I don't think we should be afraid of AI overall. We just need to make sure it's developed responsibly and its applications are guided by ethics around protecting people's privacy, preventing harm, and respecting human rights. With the proper care, AI can be an amazing tool to help solve humanity's greatest challenges.Personally, I'd love to have a career in the field of AI once I'm older. It would be so rewarding to help advance this incredible technology in ways that improve people's lives. Maybe I could work on creating AI assistants to help people with disabilities, or AI systems to diagnose diseases earlier through analyzing medical scans and data. Or who knows, perhaps I could even contribute towards the development of artificial general intelligence (AGI) - an AI that can think, learn and reason just as flexibly as the human mind!Even if I don't directly work in AI, I know it's a field that will increasingly intersect with almost every career and industry in the future. So it's definitely something all students like me should learn about so we can make the most of AI's potential. Atthe very least, we need to understand AI well enough to not be replaced by it, ha!In all seriousness though, I don't think we should view AI as a threat to human jobs or humanity itself. Instead, we should see it as an amazing tool that can collaborate with us and empower us to achieve so much more. I mean, we've already used inventions like the printing press, steam engine, and computers to massively expand human knowledge, productivity and reach. AI will take that even further by amplifying our intelligence in incredible new ways.AI may seem like something from science fiction, but the foundations for it are very real thanks to decades of work by computer scientists, mathematicians, cognitive scientists and others. I'm so excited to see where the latest advancements in machine learning, neural networks and other AI capabilities lead. From smarter digital assistants to new scientific and medical breakthroughs, I really think AI will help create a better world and push humanity forward.Those are just my thoughts as a kid fascinated by AI and its vast potential! I'm sure there's still so much about this field that I have to learn. But I'm looking forward to it and can't wait to see what the future of artificial intelligence has in store. Hopefullyyou found my perspective interesting, even if it's not the most advanced take on the topic. Let me know if you have any other questions - I'm always eager to learn more!篇3The Awesome World of AIHi there! My name is Jamie and I'm a 7th grader at Central Middle School. Today I want to tell you all about artificial intelligence, or AI for short. AI is something that seems like science fiction, but it's very real and growing more important every day. Simply put, AI refers to machines that can think and learn like humans.One type of AI that you've probably heard of is virtual assistants like Siri, Alexa, and Google Assistant. These helpful programs use AI to understand our voices and respond to our questions and commands. Let's say I ask Alexa "What's the weather going to be like this weekend?" Alexa will check the online weather forecasts, process that information, and give me a summary in plain English. Amazing!AI assistants can do all sorts of useful tasks like setting reminders, converting units, playing music, and even telling jokes. My mom uses the AI on her smartphone to make grocery lists,find recipes, get directions, and more. She says AI assistants are like having a super smart personal assistant that never gets tired or takes a day off.But AI can do way more than just be a virtual helper. It's being used in self-driving cars that can sense the road and navigate without a human driver. AI software can analyze medical scans and test results to help doctors diagnose diseases. And AI algorithms are used by websites like Netflix to recommend shows you'll probably enjoy based on your viewing history and preferences.One of the most fascinating areas of AI is machine learning. This is where the AI software can study huge amounts of data to detect patterns and make predictions all by itself, just like how our brains learn over time from experience. For example, an AI could examine millions of past home sales to figure out the biggest factors that influence housing prices. Or it could analyze thousands of security camera videos to get really good at recognizing suspicious behavior.Machine learning is how AI systems are trained to master skills like recognizing spoken words, identifying objects in images, translating between languages, and playing complex games like chess and Go. The more data the AI has to learn from,the smarter and more capable it becomes. This is letting AI take on challenges that were incredibly difficult to program using traditional software rules and logic.There's also the challenge of making AI systems that are robust, unbiased, and aligned with human ethics and values. We need to make sure the AI doesn't learn harmful biases from the data it's trained on, and that it remains under meaningful human control. We wouldn't want an AI that was racist or sexist, or that could be misused by bad people to cause harm.Some people worry that AI will eventually become super intelligent and turn against its human creators. But many AI researchers think we're nowhere close to that level of general AI yet, and that we'll have plenty of warning if it starts happening so we can shape AI positively. I think it's important not to be afraid of new technologies, but to learn about篇4The Brilliant World of AIMy name is Alex and I'm in the 8th grade. I'm really interested in technology, especially artificial intelligence or AI for short. AI is all about creating computer systems that can perform tasks that normally require human intelligence. Things likelearning, problem-solving, decision-making, recognizing speech and images, and so on. AI is becoming super advanced and it's going to change the world in amazing ways!One of the coolest areas of AI is machine learning. This is where computers can learn and improve from data without being explicitly programmed. It's kind of like how we learn - through experiences. With machine learning, computers study huge amounts of data to find patterns and insights. They use algorithms to build models that allow them to make predictions or decisions. The more data they have, the better they get!A common use of machine learning is for things like product recommendations on sites like Amazon and Netflix. Have you ever noticed how Netflix seems to know exactly what movies and shows you'll like? That's machine learning hard at work! The algorithms study your viewing history and preferences to personalize the recommendations just for you.But machine learning can do way more than just product recs. It's being used for all kinds of amazing applications like detecting fraud, improving cyber security, forecasting weather, making medical diagnoses, and even composing music or artwork! The possibilities are mind-blowing.Another fascinating area of AI is natural language processing or NLP. This is what allows computers to understand, interpret and generate human language. Virtual assistants like Siri, Alexa and Google Assistant all use NLP to communicate with us. When you ask Alexa to add an item to your shopping list or to play your favorite music, it comprehends your speech and intent through NLP.NLP is also what powers real-time translation apps and software. You know how on Google Translate you can have whole conversations translated instantly across languages? That's next-level NLP at work! The technology is analyzing the languages, context and even things like idioms and slang to produce smooth, natural translations. It's like real-life universal translators from science fiction!Computer vision is another awesome application of AI that allow machines to identify and process images and videos just like humans can. It combines machine learning with understanding the visual world. Computer vision already helps power face recognition for tagging friends in pics on social media. But it also has way bigger uses like aiding self-driving cars to "see" the road, assisting doctors to diagnose diseases fromscan images, and tons of applications for security and surveillance.Speaking of self-driving cars, they simply wouldn't be possible without AI! Autonomous vehicles rely on multiple AI capabilities like computer vision, sensor data processing, navigation, path planning and decision making. There's no way conventional programming could account for the infinite number of potential scenarios a self-driving car could encounter on the roads. But with advanced AI systems, they can dynamically analyze situations and make smart decisions in real-time while driving.AI is also bringing huge improvements to areas like robotics, manufacturing, logistics and more through machine learning, planning and perception. Robots can be trained using AI to intelligently coordinate and carry out complex physical tasks and processes. It allows systems to constantly adapt and optimize in ways old-school programming could never match.What really excites me most about AI though, is the potential it has to help solve humanitarian issues and push forward scientific breakthroughs. There are already examples of AI being used for good in areas like:Protecting the environment by monitoring deforestation, air and water pollution, wildlife populations etc.Tackling hunger and food insecurity by optimizing crop sustainability and yieldsProviding quality education for all through intelligent tutoring systems and adaptive learningAdvancing healthcare through drug discovery, treatment design, and preventive careMitigating climate change by modeling impacts and solutionsAnd those are just a few examples! With AI's incredible processing power, predictive capabilities and never-ending learning potential, I'm confident it will unlock solutions to our biggest global challenges that we can't even imagine yet.But if we get it right, artificial intelligence will be one of the most transformative forces for good in human history! I can't wait to see how AI continues evolving and changing the world for the better as I get older. Maybe I'll even end up having a career developing these incredible technologies one day. For now though, I'll just keep learning everything I can about AI and spread the word about why it's so brilliant!篇5The Exciting World of Artificial IntelligenceHi there! My name is Jamie, and I'm a student in middle school. Recently, I've become really interested in a fascinating topic called artificial intelligence, or AI for short. Let me tell you all about it!AI is like having a super-smart robot friend that can help you with all sorts of things. It's a technology that allows machines to think and learn like humans do. Isn't that amazing? These machines, called AI systems, can process information, recognize patterns, make decisions, and even come up with creative ideas –just like our brains do, but way faster and more efficiently!One of the coolest things about AI is that it can learn from experience, just like we do. For example, if you show an AI system a bunch of pictures of dogs, it can study those pictures and learn to recognize dogs in other images or even in real life. The more data and examples you give it, the better it gets at its task. It's like playing a game over and over until you master it, but for an AI, it happens much quicker!AI has already made its way into our daily lives in so many ways. Have you ever used a virtual assistant like Siri or Alexa?Those are AI systems that can understand your voice commands and help you with tasks like setting alarms, getting weather updates, or even cracking jokes. Speaking of jokes, some AI systems are now so advanced that they can write stories, poems, and even funny one-liners!But AI isn't just about fun and games; it's also being used to solve serious problems and make our lives better. For instance, AI can help doctors diagnose diseases more accurately by analyzing medical images and data. It can also help scientists study climate change and find ways to protect our environment. In fact, AI is being used in almost every field imaginable, from finance and transportation to education and entertainment.Personally, I think AI is one of the most exciting technologies of our time. Just imagine having a robot tutor that can explain complex concepts in a way that's easy to understand, or a virtual friend that can play games with you and never gets bored. The possibilities are endless!But what do you think about AI? Do you find it fascinating or a little bit scary? Maybe a mix of both? Either way, I encourage you to learn more about it because it's shaping the world we live in, and who knows, you might even end up working with AI systems in the future!Well, that's all from me for now. I've gotta run and catch up on my favorite AI-generated cartoon series. Until next time, stay curious and keep exploring the amazing world of technology!Word count: 2,012篇6Artificial Intelligence: The Future is HereHave you ever wondered what the future will be like? I think about it a lot. Will we have flying cars and jet packs? Will robots do all our chores and homework for us? The idea of advanced technology has always fascinated me, especially artificial intelligence or AI.AI is basically computer software that can think and learn kind of like a human brain. It can look at data, see patterns, and make decisions without being directly programmed for every situation. AI is used in lots of things we interact with every day like Google searches, Siri and Alexa voice assistants, and even Netflix movie recommendations.But AI is going to be so much more than that. Scientists are working on making AI that can drive cars, diagnose diseases, create art and music, and even tutor students better than humanteachers! Just imagine an AI math tutor that could look at how you are solving problems and give you customized help and practice for the areas you are struggling with most. How cool would that be?Some people are worried that advanced AI could become smarter than humans and take over the world like in the Terminator movies. But most experts say we are still very far away from anything like that. Current AI is extremely good at specific narrow tasks, but it can't reason about the world like a human can. An AI mig。

第 22卷第 4期2023年 4月Vol.22 No.4Apr.2023软件导刊Software Guide基于CEEMDAN-SE-TCN的集群资源预测研究史爱武，张义欣，韩超，黄河（武汉纺织大学计算机与人工智能学院，湖北武汉 430200）摘要：针对服务器集群负载数据的波动性和非线性特点，提出一种基于CEEMDAN-SE-TCN的预测算法。

该算法首先将原始服务器集群数据经过自适应加噪集合经验模态分解（CEEMDAN），有效降低负载序列复杂度。

然后，在得到分解后的相关IMF分量后，利用相关系数法将各IMF分量与原始序列进行比较，去除相关性较弱的分量。

最后，提取各分量相应的特征值输入并加入注意力机制的时间卷积网络（SE-TCN）进行建模预测。

通过Google集群数据集中的CPU负载率序列实测证明，在同等条件下CEEMDAN-SE-TCN模型整体优于其他基准模型，MAPE指标相较于其他模型分别降低7.1%、6.5%、2.5%，证明了该算法的有效性和可行性。

关键词：自适应加噪的集合经验模态分解；相关系数法；注意力机制；时间卷积网络；负载预测DOI：10.11907/rjdk.221466开放科学（资源服务）标识码（OSID）：中图分类号：TP302 文献标识码：A文章编号：1672-7800（2023）004-0043-05Research on Cluster Resource Prediction Based on CEEMDAN-SE-TCNSHI Ai-wu， ZHANG Yi-xin， HAN Chao， HUANG He（School of Computer and Artificial Intelligence， Wuhan Textile University， Wuhan 430200， China）Abstract：Prediction algorithm based on CEEMDAN-SE-TCN is proposed according to the volatility and nonlinearity of server cluster load data. First， the original server cluster data is decomposed into adaptive noisy set empirical mode decomposition （CEEMDAN）， which effec‐tively reduces the complexity of the load sequence. Then， after the decomposed relevant IMF components are obtained， the correlation coeffi‐cient method is used to compare each IMF component with the original sequence to remove the components with weak correlation. Finally， the corresponding eigenvalues of each component are extracted and input into the time convolution network （SE-TCN） with attention mechanism for modeling and prediction. The actual measurement of CPU load rate sequence in Google cluster dataset shows that CEEMDAN-SE-TCN model is better than other benchmark models under the same conditions， and MAPE indicators are reduced by 7.1%， 6.5% and 2.5% respec‐tively compared with other models， which proves the effectiveness and feasibility of this algorithm.Key Words：adaptive denoising ensemble empirical mode decomposition； correlation coefficient method； attention mechanism； time convo‐lution network； load prediction0 引言集群技术是指将多台计算机通过集群软件相互连接，组成一个单一系统模式进行管理，其目的是为了通过较低的成本获取更高性能，增加系统的可扩展性与可靠性。

数字通信中的多抽样率信号处理中英⽂翻译（部分）Multirate Signal Processing Concepts in Digital CommunicationsBojan VrceljIn Partial Fulfillment of the Requirementsfor the Degree ofDoctor of PhilosophyCalifornia Institute of TechnologyPasadena, California2004 (Submitted June 2, 2003)AbstractMultirate systems are building blocks commonly used in digital signal processing (DSP). Their function is to alter the rate of the discrete-time signals, which is achieved by adding or deleting a portion of the signal samples. Multirate systems play a central role in many areas of signal processing, such as filter bank theory and multiresolution theory. They are essential in various standard signal processing techniques such as signal analysis, denoising, compression and so forth. During the last decade, however, they have increasingly found applications in new and emerging areas of signal processing, as well as in several neighboring disciplines such as digital communications.The main contribution of this thesis is aimed towards better understanding of multirate systems and their use in modern communication systems. To this end, we first study a property of linear systems appearing in certain multirate structures. This property is called biorthogonal partnership and represents a terminology introduced recently to address a need for a descriptive term for such class of filters. In the thesis we especially focus on the extensions of this simple idea to the case of vector signals (MIMO biorthogonal partners) and to accommodate for nonintegral decimation ratios (fractional biorthogonal partners).Some of the main results developed here pertain to a better understanding of the biorthogonal partner relationship. These include the conditions for the existence of stable and of finite impulse response (FIR) biorthogonal partners. A major result that we establish states that under some generally mild conditions, MIMO and fractional biorthogonal partners exist. Moreover, when they exist, FIR solutions are not unique. We develop the parameterization of FIR solutions, which makes the search for the best partner in a given application analytically tractable. This proves very useful in the central application of biorthogonal partners, namely, channel equalization in digital communications with signal oversampling at the receiver. Sampling the received signal at a rate higher than that defined by the transmitter provides some flexibility in the design of the equalizer. A good channel equalizer in this context is one that helps neutralize the distortion on the signal introduced by the channel propagation but not at the expense of amplifying the channel noise. This presents the rationale behind the partner design problem which is formulated and solved. Theperformance of such equalizers is then compared to several other equalization methods by computer simulations. These findings point to the conclusion that the communication system performance can be improved at the expense of an increased implementational cost of the receiver.While the multirate DSP in the aforementioned communication systems serves to provide additional degrees of freedom in the design of the receiver, another important class of multirate structures is used at the transmitter side in order to introduce the redundancy in the data stream. This redundancy generally serves to facilitate the equalization process by forcing certain structure on the transmitted signal. If the channel is unknown, this procedure helps to identify it; if the channel is ill-conditioned, additional redundancy helpsVavoid severe noise amplification at the receiver, and so forth. In the second part of the thesis, we focus on this second group of multirate systems, derive some of their properties and introduce certain improvements of the communication systems in question.We first consider the transmission systems that introduce the redundancy in the form of a cyclic prefix. The examples of such systems include the discrete multitone (DMT) and the orthogonal frequency division multiplexing (OFDM) systems. The cyclic prefix insertion helps to effectively divide the channel in a certain number of nonoverlaping frequency bands. We study the problem of signal precoding in such systems that serves to adjust the signal properties in order to fully take advantage of the channel and noise properties across different bands. Our ultimate goal is to improve the overall system performance by minimizing the noise power at the receiver. The special case of our general solution corresponds to the white channel noise and the best precoder under these circumstances simply performs the optimal power allocation.Finally, we study a different class of communication systems with induced signal redundancy, namely, the multiuser systems based on code division multiple access (CDMA). We specifically focus on the special class of CDMA systems called `a mutually orthogonal usercode receiver' (AMOUR). These systems use the transmission redundancy to facilitate the user separation at the receiver regardless of the (different) communication channels. While the method also guarantees the existence of the zero-forcing equalizers irrespective of the channel zero locations, the performance of these equalizers can be further improved by exploiting the inherent flexibility in their design. Weshow how to find the best equalizer from the class of zero-forcing solutions and then increase the size of this class by employing alternative sampling strategies at the receiver. Our method retains the separability properties of AMOUR systems while improving their robustness in the noisy environment.Chapter 1 IntroductionThe theory of multirate digital signal processing (DSP) has traditionally been applied to the contexts of filter banks [61], [13], [50] and wavelets [31], [72]. These play a very important role in signal decomposition, analysis, modeling and reconstruction. Many areas of signal processing would be hard to envision without the use of digital filter banks. This is especially true for audio, video and image compression, digital audio processing, signal denoising, adaptive and statistical signal processing. However, multirate DSP has recently found increasing application in digital communications as well. Multirate building blocks are the crucial ingredient in many modern communication systems, for example, the discrete multitone (DMT), digital subscriber line (DSL) and the orthogonal frequency division multiplexing (OFDM) systems as well as general filter bank precoders, just to name a few. The interested reader is referred to numerous references on these subjects, such as [7]-[9], [17]-[18], [27], [30], [49], [64], [89], etc.This thesis presents a contribution to further understanding of multirate systems and their significance in digital communications. To that end, we introduce some new signal processing concepts and investigate their properties. We also consider some important problems in communications especially those that can be formulated using the multirate methodology. In this introductory chapter, we give a brief overview of the multirate systems and introduce some identities, notations and terminology that will prove useful in the rest of the thesis. Every attempt is made to make the present text as self-contained as possible and the introduction is meant to primarily serve this purpose. While some parts of the thesis, especially those that cover the theory of biorthogonal partners and their extensions provide a rather extensive treatment of the concepts, the material regarding the applications of the multirate theory in communication systems should be viewed as a contribution to a better understanding and by no means the exhaustive treatment of such systems. For a more comprehensive coverage the reader is referred to a range of extensive texts on the subject, for example, [71], [18], [19], [39], [38], [53], etc.1.1 Multirate systems 1.1.1 Basic building blocks The signals of interest in digital signal processing are discrete sequences of real or complex numbers denoted by x(n), y(n), etc. The sequence x(n) is often obtained by sampling a continuous-time signal x c(t). The majority of natural signals (like the audio signal reaching our ears or the optical signal reaching our eyes) are continuous-time. However, in order to facilitate their processing using DSP techniques, they need to be sampled and converted to digital signals. This conversion also includes signal quantization, i.e.,discretization in amplitude, however in practice it is safe to assume that the amplitude of x(n) can be any real or complexSignal processing analysis is often simplified by considering the frequency domain representation of signals and systems. Commonly used alternative representations of x(n) are its z-transform X (z) and the discrete-time Fourier transform X (O'). The z-transform is defined as X(z) = E _.x(n)z-"', and X (e j") is nothing but X(z) evaluated on the unit circle z = e3".Multirate DSP systems are usually composed of three basic building blocks, operating on a discrete-time signal x(n). Those are the linear time invariant (LTI) filter, the decimator and the expander. An LTI filter, like the one shown in Fig.1.1, is characterized by its impulse response h(n), or equivalently by its z-transform (also called the transfer function) H(z). Examples of the M-fold decimator and expander for M = 2 are shown in Fig.1.2. The rate of the signal at the output of an expander is M times higher than the rate at its input, while the converse is true for decimators. That is why the systems containing expanders and decimators are called `multirate' systems. Fig.1.2 demonstrates the behavior of the decimator andthe expander in both the time and the frequency domains.XE(z) = [X (z)]IM XD(z) = [X (z)]iM = X(z M)1 M-1 1 j2 k =M E X(z e n a)k=0for M-fold expander, and (1.1)for M-fold decimator. (1.2)The systems shown in Figs.1.1 and 1.2 operate on scalar signals and thus are called single input-single output (SISO) systems. The extensions to the case of vector signals are ratherstraightforward: the decimation and the expansion are performed on each element separately. The corresponding vector sequence decimators/expanders are denoted within square boxes in block diagrams. In Fig.1.3 this is demonstrated for vector expanders. The LTI systems operating on vector signals are called multiple input-multiple output (MIMO) systems and they are characterized by a (possibly rectangular) matrix transfer function H(z).1.1.2 Some multirate definitions and identitiesThe vector signals are sometimes obtained from the corresponding scalar signals by blocking. Conversely, the scalar signals can be recovered from the vector signals by unblocking. The blocking/unblocking operations can be defined using the delay or the advance chains [61], thus leading to two similar definitions. One way of defining these operations is shown in Fig.1.4, while the other is obtained trivially by switching the delay and the advance operators. Instead of drawing the complete delay/advance chain structure, we often use the simplified block notation as in Fig.1.4. It is usually clear from the context which of the two definitions数字通信中的多抽样率信号处理Bojan Vrcelj博⼠学位论⽂加州技术学会Pasadena, 加州2004 (委托于2003.6.2)摘要多抽样率系统普遍是被运⽤在处理数字信号⽅⾯。