New Method Based on Curvelet Transform for Image Denoising

WAVELET TRANSFORM MOMENTS FOR FEATURE EXTRACTIONFROM TEMPORAL SIGNALSIgnacio Rodriguez CarreñoDepartment of Electrical and Electronic Engineering, Public University of Navarre, Arrosadia, Pamplona, SpainEmail: irodriguez@unavarra.esMarko VuskovicDepartment of Computer Science, San Diego State University, San Diego, California, USAEmail: marko@Keywords: Pattern recognition, EMG, feature extraction, wavelets, moments, support vector machinesAbstract: A new feature extraction method based on five moments applied to three wavelet transform sequences has been proposed and used in classification of prehensile surface EMG patterns. The new method has essentially ex-tended the Englehart's discrete wavelet transform and wavelet packet transform by introducing more efficientfeature reduction method that also offered better generalization. The approaches were empirically evaluated onthe same set of signals recorded from two real subjects, and by using the same classifier, which was the Vapnik'ssupport vector machine.1 INTRODUCTIONThe electromyographic signal (EMG), measured at the surface of the skin, provides valuable information about the neuromuscular activity of a muscle and this has been essential to its application in clinical diagno-sis, and as a source for controlling assistive devices, and schemes for functional electrical stimulation. Its application to control prosthetic limbs has also pre-sented a great challenge, due to the complexity of the EMG signals.An important requirement in this area is to accu-rately classify different EMG patterns for controlling a prosthetic device. For this reason, effective feature extraction is a crucial step to improve the accuracy of pattern classification, therefore many signal represen-tations have been suggested.Various temporal and spectral approaches have been applied to extract features from these signals. A comparison of some effective temporal and spectral approaches is given in (Du & Vuskovic 2004), where the authors have applied moments to short time Fou-rier transform (STFT), and short time Thompson transform (STTT) on prehensile EMG patterns.The wavelet transform-based feature extraction techniques have also been successfully applied with promising results in EMG pattern recognition by Englehart and others (1998).The discrete wavelet transform (DWT) and its generalization, the wavelet packet transform (WPT), were elaborated in (Englehart 1989a). These tech-niques have shown better performance than the oth-ers in this area because of its multilevel decomposi-tion with variable trade-off in time and frequency resolution. The WPT generates a full decomposition tree in the transform space in which different wavelet bases can be considered to represent the signal. The techniques were applied to feature extraction from surface EMG signals.However, these techniques produce a large amount of coefficients, since the transform space has very large dimension. This fact suggests the system-atic application of feature selection or projection methods and dimensionality reduction techniques to enable the methodology for real time applications.Englehart applied feature selection and feature pro-jection that yielded better classification results and improved time efficiency. Specifically, the principal component analysis (PCA) was used due to its ability to model linear dependencies and to reject irrelevant information in the feature set (Englehart etal. 1999).This paper continues the work described above by taking a different approach to feature reduction. Extending the idea of spectral moments suggested in (Du & Vuskovic 2004) the sequences of wavelet coefficients are further subjected to the calculation of their temporal moments. The main goal of this work is to propose and empirically compare two different novel feature extraction approaches based on simple two-scale DWT and WPT with the two best Engle-hart’s approaches using the DWT and the WPT in combination with principal component analysis (PCA).In this new approach, the first five raw moments were applied to DWT transformed prehensile EMG sequences, which has proven to be very advanta-geous in the classification stage. The methods em-ployed a simple DWT or WPT with only three trans-form sequences, instead of the full DWT or WPT used by Englehart. This has eliminated the tedious feature reduction procedures and PCA.The evaluation of the three approaches was car-ried out on the same set of data, and with an identical classifier based on Vapnik's support vector machines (SVM) with a linear kernel.2 PREHENSILE EMGSThe research presented here was motivated by the need for classification of prehensile electromiographic signals (EMG) for control of a multifunctional pros-thetic hand (Vuskovic etal. 1995). Since the hand-preshaping phase in an average object grasp takes about 500 ms, it is important to accomplish the feature extraction and classification in less than 400 ms, pref-erably in 200 ms. Such a difficult task requires very strong feature extractor and classifier.The mioelectric control of multifingered hand prostheses was studied in several papers, for example (Nishikava 1991), (Uchida 1992), (Farry 1996), and (Huang 1999). Most of the ideas in these efforts were inspired by Hudgins (Hudgins etal. 1991). In this work the concept of preshaping of multifunctional grasps was based on the recognition of a particular finger joint movement. In an earlier work done at San Diego State University, the approach was rather dif-ferent, based on grasp types, instead of hand configu-rations in joint space. Once a grasp type is recognized from the recorded EMGs, it can be then synergisti-cally mapped into the desired joint configuration (Vuskovic 1995) for any hand, with any number of degrees of freedom. We have considered four basic grasp types according to the Schlesinger classification (Schlesinger 1919): cylindrical grasp (C), spherical grasp (S), lateral grasp (L) and precision grasp (P), see figure 1.3 EXPERIMENTAL SETUPFour-channel surface EMG signals from two healthy subjects were recorded at 1000 Hz sampling fre-quency. The recording was done while the subject has repeatedly performed the four grasp motions. There were 216 grasp recordings evenly distributed across the four grasps types: 60 (subject 1) + 4 (sub-ject 2) for cylindrical grasp, 30+10 for precision grasp, 30+10 for lateral grasp and 60 + 12 for spheri-cal grasp. Three different EMG sequence lengths were used: 200 ms, 300 ms and 400 ms. The 200 and 300 ms sequences were obtained by truncating the recordings of 400 ms sequences. (The sequences of300 ms were not presented in this paper.)Figure 1: Four grasp types.4 DISCRETE WA VELET TRANS-FORMThe DWT is a transformation of the original tem-poral signal into a wavelet basis space. The time-frequency wavelet representation is performed byrepeatedly filtering the signal with a pair of filtersthat cut the frequency domain in the middle.Specifically, the DWT decomposes a signal into an approximation signal and a detail signal. The ap-proximation signal is subsequently divided into new approximation and detail signals. This process is carried out iteratively producing a set of approxima-tion signals at different detail levels (scales) and a final gross approximation of the signal.The detail Dj and the approximation Aj at level j can be obtained by filtering the signal with an L -sample high pass filter g , and an L -sample low pass filter h . Both approximation and detail signals are downsampled by a factor of two. This can be expressed as follows:1110[][][][2],L j j j k A n A n h k A n k −−−===∑Η− (1) 1110[][][][2],L j j j k D n D n g k A n k −−−===∑G − (2) where 0, n = 0,1,…N-1 is the original temporalsequence, while H and G represent the convolu-tion/down sampling operators. Sequences g[n] andh[n] are associated with wavelet function []A n ()ψt andthe scaling function through inner products:()ϕt[]((2),g n t t n =ψ− (3)[]((2).h n t t n =ϕ− (4)Operators H and G can be applied repeatedly in al-teration, for example: 0AA A =H H , DD=0A =G G , AD =0A G H ,DA =0The A and D sequences obtained as the result ofDWT are still massive in terms of the number ofsamples, which contributes to large dimensionality offeature space. Besides, the sequences have a highnoise component inherited from the original EMGsignal.A =H G , etc.A feature extraction approach based on DWTapplied by Englehart (1998, 1998a) consists of fourdifferentiated phases:1. Perform full DWT decomposition of the EMGsignals, until scale j = log2 (N), with the Coifletwavelet of order 4 (C4);2. Square the DWT coefficients;3. Apply PCA for dimensionality reduction tech-niques;4. Determine the optimal number of features perchannel based on the target classifier.An optimization phase is needed before selecting the adequate number of PCA features in order to maxi-mize the performance of the target classifier. The optimum number of features was 100 DWT coeffi-cients per channel of the EMG signal used in this work.5 WA VELET PACKET TRANS-FORMThe WPT is a generalization of DWT. It generates a full wavelet basis decomposition tree. In each scale,not only the approximation signal as in DWT, butalso the detail signals are filtered to obtain another two low and high frequency signals. Many different representations of a signal can be obtained by select-ing different wavelet packet basis. In this regardWPT is superior to DWT, as the chosen basis can be optimized with respect to frequency or time resolu-tion. Englehart (1999) generated a feature extraction method based on the WPT for EMG signals. In this method a previous phase must be applied to the set of training signals. The underlying idea is to select theWPT basis that best classifies all classes of signals. For this purpose, Englehart proposed a modified ver-sion of the local discriminant basis (LDB) algorithm(Englehart 1998a, 2001), to maximize the discrimi-nation ability of the WPT by using a class separabil-ity cost function (Saito & Coifman 1995). Once thebest basis for classification is defined (for differentchannels and different signal lengths), the followingsteps must be performed:1. Perform the full WPT decomposition untilscale j = log 2 (N), with the Symlet wavelet oforder 5 (S5);2. Square the WPT coefficients;3. Average energy maps within each subband;4. Select the WPT coefficients from a basis cho-sen previously for each channel and for differ-ent signal lengths;5. Extract the optimal number of features basedon the target classifier;6. Apply PCA transform to the feature space fordimensionality reduction (removing the eigen-vectors whose eigenvalues are zero);7. Extract the optimum number of features per channel for the target classifier;The optimal number of features for Englehart’s WPT based approach and for the support vector ma-chine as the target classifier (see section 7) was found to be three features per channel, per signal length.6 DWT AND WPT MOMENTS 1The new approach for feature extraction presented here is based on DWT and WPT, and on the calcula-tion of their temporal moments. The approach was first proposed in (Rodriguez & Vuskovic 2005) as an extension of the idea of spectral moments (Du & Vuskovic 2004).Specifically, we used two different wavelets suc-cessfully applied by Englehart on surface EMG sig-nals: C4 and S5In order to reduce the dimensionality and to smooth out the noise, we applied six moments to transformed signals (DWT and WPT):10[],0,1,2,...,5,j mN m n jnM S n m N −=⎛⎞==⎜⎟⎜⎟⎝⎠∑ (5) where represents sequences A, D, AA, DD, AD and DA used in algorithms described below, while N []S n j is number of samples at the corresponding level of decomposition.The new approach based on DWT consists of the following steps:1. Perform two-scale decomposition of the inputsignal;2. Compute moments for three transform se-quences (D, AA, AD);3. Apply logarithm transform to each feature,log(0.1+f);4. Normalize all features using mean value andstandard deviation computed for each feature across all samples.The choice of sequences D, AA and AD was made empirically; it has given the best results in av-1DWT and WPT moments should not be confused with wavelet vanishing moments.erage for the given set of data. Similar choice was made for WPT algorithm.The WPT-based method has the following steps:1. Perform two-scale decomposition of the inputsignal;2. Select basis obtained from previous applicationof the best basis Coifman algorithm; 3. Compute moments for three transform se-quences (A, DA, DD);4. Apply logarithm transform to each feature,log(0.1+f);5. Normalize all features using mean value andstandard deviation computed for each feature across all samples.The optimal basis selection in this method was based on a single channel. The same basis thus ob-tained was subsequently used for single and multiple channels, and for different sequence lengths.Log transformation was applied to moments as it effectively reduces the skewness and the kurtosis of data, consequently resulting in an estimated probabil-ity density that appears more like normal distribution (Vuskovic atal. 1995). The nonlinear transformation of features has significantly improved the classifier performance.7 THE SVM CLASSIFIERThe support vector machines ( Christianini & Shaw-Taylor 2000) are a family of learning algorithms based on the work of Vapnik (1998), which have recently gained a considerable interest in pattern rec-ognition community. The success of SVM comes from their good generalization ability, robustness in high dimensional feature spaces and good computa-tional efficiency.In this work, a standard SVM classifier with lin-ear kernel has been used for dichotomic (binary) clas-sification (Gunn 1997). The multiclass SVM can also be considered, but this is out of the scope of this pa-per.The previous work on the classification of pre-hensile EMG patterns (Vuskovic 1996) has shown that the most difficult is to discriminate cylindrical from spherical grasps (C/S), and then lateral from precision grasps (L/P). Therefore the SVM is applied to these pairs of grasp types and the feature extraction methods were evaluated accordingly.The classification tests were performed with leave-one-out method, where one sample was re-moved from the data set and the rest of the samples were used to train the SVM. The procedure was re-peated for each sample in the data set, and the aver-age hit rate was computed afterwards.8 COMPUTATIONAL COMPLEX-ITYApplication of WPT and calculation of J scales, 2log ≤J N , where N is the length of the original temporal signal, results in JN coefficients. Conse-quently, the computational cost of the full-scale WPT is in the order of 2 (Englehart 2001). Similarly, the computational complexity of full-scale DWT is half the computational complexity of the WPT, i.e.()≤O JN (log )O N N (2log 2O N N ). Since our new ap-decomposition, we can enumerate all the approaches with respect to their computational complexity in the increasing order: DWT(new) < WPT (new) < DWT (Englehart) < WPT (Englehart). The complexities are summarized in table 1.Table 1: Computational complexityNew approachEnglehartDWT WPT DWT WPTO(N)O(2N)O(N logN/2)O(N logN)9 EXPERIMENTAL EV ALUATIONIn this section we discuss the methodology for the experimental evaluation of DWT and WPT ap-proaches.9.1 Cluster VisualizationIn order to compare the effectiveness of a feature extraction method there is needed some method to compare the discrimination of clusters in feature space, either by 2D or 3D scatter plots, or by some distance measure between clusters. Both methods are normally based on the transformation of the feature space through PCA or Fisher-Rao transform, which both use the inverse of the cluster covariance matri-ces. Unfortunately the dimensionality of the feature space is often larger than the number of samples, which makes the methods inapplicable due to the singularity or ill-conditioning of the covariance ma-trices. However, the support vector machines offered new possibilities. SVM maximize the margin be-tween clusters and the separation hyperplane in the original or kernel-induced feature space without a need to use covariance matrices.We use in this work a projection of the original feature space onto the line perpendicular to the maximal-margin separation hyperplane: (6) ,T p X w =where X is N ×d sample (feature) matrix, w is unit, d -dimensional normal to the separation hyperplane, and p is N -vector of projected samples. In order to get a 2D plot of samples another projection vector is needed: (7) .T q X u =The d -dimensional projection vector u doesn't have to be orthonormal to w , but has to be unique in some way. Therefore we used the direction of the minimal variance of both clusters, which is nearly laying in the separation hyperplane. The vector coin-cides with the eigenvector that corresponds to the smallest non-zero eigenvalue of the pooled covari-ance matrix:11221212(1)(1)(,),2N S N S S pool S S N N −+−==+− (8) where N i (N 1+N 2 = N ) and S i are sizes and covari-ance matrices of the two clusters. An example of cluster diagrams, plot(p,q), is shown in figure 3, which will be discussed later.9.2 Hotelling DistanceA useful quantitative measure of cluster discrimina-tion in multidimensional space is Hotelling distance between cluster means (T 2 statistic). The T 2 can be computed for projected clusters:211212121212()()(,),T N N T c c C c N N C pool C C −=−+=,c − (9), where i and C c i are sample means and sample co-variance matrices of projected clusters respectively. In order to establish the significance of the distance under some confidence level, the T 2 distance needs to be compared with the corresponding critical value2c T . The critical value can be obtained if we assume that the quantity21212(1(2N N r T N N r+−−+−)) has F-distribution with degrees of freedom r and 121f N N r =+−−, where r = 2 in case of 2D pro-jections (Seber 1984). The above is true under the assumption that clusters have normal distributions with nearly equal sizes and covariance matrices. If this is not the case, a stronger statistic has to be used. In this work we used statistic suggested in (Yao 1965), where the cluster distance was computed as:()1212112212()//()T T c c C N C N c c −=−+−T . (10)The degrees of freedom for the F-distribution were estimated from the data (Seber 1984) (not pre-sented here due to limited space). The test works for unequal clusters that can have any bell-shaped distri-bution. The T 2 values are shown in tables 2 and 3, and in the scatter diagrams in figure 3. The critical values c were all below 11. The value of cluster distances as a quantitative measure of cluster dis-crimination is that they can be easily and quickly computed.29.3 Number of MomentsOnce the classification pairs are determined, the next step is to determine the optimal number of DWT and WPT moments, which will be used for feature reduction. This was done experimentally by extensive application of feature extractions and classiffications to different EMG signal lengths and different number of channels.Figure 2: Hotelling distances versus number of momentsfor WPT: (a) 200 ms, single channel, (b) 200ms, four channels, (c) 400 ms, single channel, (d) 400 ms, four channels (C/S grasps – lower bars, L/P grasps upper barBased on the bar graphs the selection of five mo-ments (M 0, M 1,…,M 4) was a clear choice.10 THE RESULTSThe comparison of four different approaches: the five-moment DWT and WPT as proposed in this paper, and the DWT and WPT of Englehart (1998a, 1999) have been measured by Hotteling distances and by the classification hit rates applied to two cluster pairs (C/S) and (L/P).The results are presented in tables 2 through 5. The feature extraction was performed for 200 and 400 ms time sequences recorded from a single channel and from four simultaneous channels. Each channel repre-sented one surface EMG electrode attached to the up-per-forehand of the subject. Several different wavelets were used in experiments, but only the two most suc-cessful ones were shown here: the fourth-order Coif-man wavelets (C4) and the fifth-order symlets (S5). The two tables show a roughly good correlation be-tween the Hotelling distances and the classification hit-rates. The small differences can be explained by the fact that the Hotelling distances point the goodness of clustering, while the hit rates stress the generaliza-tion of the trained SVM.An example of four different cluster scatter dia-grams is shown in figure 3.Figure 3: SVM-projected clusters, 200 ms, and four channels, WPT: (a) C/S - new approach, (b) L/P - new approach, (c) C/S - Englehart, (d) L/P - EnglehartThe results suggest clear advantage of our novel method over the Englehart’s approaches mainly due to the moments used for dimensionality reduction, instead of applying PCA. In addition, the application of log transformation on features has helped consid-erably. Our WPT novel method seems to behave bet-ter at classifying the 200 ms sequences. This is due to the WPT basis selection, which better characterizes the frequency structure of the transient signals.Table 2: Hotelling distances (C/S)New approach EnglehartWT WPT Sig. length /chnlsC4 S5 C4 S5 DWT C4WPT S5200/1 75 61 109 97 49 13200/4352 466 424 421 201 73 400/1 92 79 96 79 480 45 400/4 366 570 535 488 295 100Table 3: Hotelling distances (L/P)New approach Englehart DWT WPT Sig.length /chnls C4 S5 C4 S5 DWT C4 WPT S5 200/1 33 65 44 100 362 11 200/4 289 3462 1724 723 756 107 400/1 178 166 233 262 118 60 400/4 560 24680 1472 718 2388 168Table 4: Classification hit rates in % (C/S)New approach EnglehartWT WPT Sig. length /chnlsCO4 SY5 CO4 SY5 DWT C4WPT S5200/1 75.0 76.7 79.2 79.2 60.8 62.5 200/4 90.0 94.2 94.2 95.0 86.7 88.3 400/1 80.8 80.0 80.8 77.5 60.8 59.2 400/4 99.2 96. 7 98.3 97.5 88.3 93.3Table 5: Classification hit rates in % (L/P)New approach Englehart WT WPTSig. length /chnls CO4 SY5 CO4 SY5WT WPT200/1 73.3 81.7 81.7 83.3 56.7 53.3 200/4 91.7 96.7 90.0 98.3 80.0 93.3 400/1 96.7 95.0 93.3 91.7 56.7 63.3 400/4 99.9 99.9 99.9 99.9 88.3 95.011 CONCLUSIONSA new approach of wavelet-based feature extraction from temporal signals has been proposed. The ap-proach extends the Englehart's discrete wavelet trans-form and wavelet packet transform by subjecting the two-scale, three-sequence wavelet coefficients to temporal moment computation. This has helped re-duce significantly the dimensionality of the resulting feature vectors without loosing the essential informa-tion in the original patterns. It was found experimen-tally that first five raw moments represent a good compromise. The new methods are applied to prehen-sile EMG signals of various lengths and various amounts of input signals (surface EMG channels) and compared to the best approaches of Englehart, on the same set of signals. For the comparison are used two quantitative measures: Hotelling statistic and classifi-cation hit rates. The classifier applied to the extracted features was linear support vector machine, which has exceptionally good performance in case of large fea-ture spaces and fewer training samples. The results have shown superior performance of the new ap-proach. A brief complexity analysis also shows that the new approach is more efficient time wise.Although the methodology was demonstrated on EMG signals, we believe the methodology can equally successfully be applied to other temporal sig-nals.REFERENCESChristianini N., Shawe-Taylor, 2000. An Introduction toSupport Vector Machines . Cambridge Univ. Press. Du S. and Vuskovic M., 2004. Temporal vs. Spectral ap-proach to Feature Extraction from Prehensile EMG Sig-nals. In IEEE Int. Conf. on Information Reuse and Inte-gration (IEEE IRI-2004), Las Vegas, Nevada.Englehart K., Hudgins B., Parker P. and Stevenson M., 1998. Time-frequency representation for classification of the transient myoelectric signal. In ICEMBS’98.Proceedings of the 20th Annual International Confer-ence on Engineering in Medicine and Biology Society.ICEMBS Press.Englehart K., 1998a. Signal Representation for Classifica-tion of the Transient Myoelectric Signal. Doctoral The-sis. University of New Brunswick, Fredericton, New Brunswick, Canada.Englehart K., Hudgins B., Parker P. and Stevenson M., 1999. Improving Myoelectric Signal Classification us-ing Wavelet Packets and Principle Component Analy-sis. In ICEMBS’99. Proceedings of the 21st Annual In-ternational Conference on Engineering in Medicine and Biology Society, ICEMBS Press.Englehart K., Hudgins B., Parker P., 2001. A Wavelet –Based Continuous Classification Scheme for Multi-fucntion Myoelectric Control. In IEEE Transactions on Biomedical Engineering, vol. 48, No. 3, pp. 302-311. Farry, K. A., Walker I. D., Baraniuk R. G., 1996. Myoelec-tric Teleoperation of a Complex Robotic Hand. IEEE Trans On Robotic and Automation, Vol. 12, No.5. Gunn S.R., 1997. Support Vector Machines for Classifica-tion and Regression. Technical Report, Image Speech and Intelligent Systems Research Group, University of Southampton.Hannaford B. and Lehman S., 1986. Short Time Fourier Analysis of the Electromyogram: Fast Movements and Constant Contraction. In IEEE Transactions On Bio-medical Engineering. BME-33,Han-Pan Huang H-P, Chen C_Y., 1999. Development of a Myoelectric Discrimination System for Multi-Degree Prosthetic Hand. Proc. of the 1999 International Con-ference on Robotics and Automation, Detroit, May pp.2392-2397.Hudgins, P. Parker and R.N. Scott, 1991. A Neural Net-work Classifier for Multifunctional Myoelectric Con-trol. Annual Int. Conf. Of the EMBS, Vol. 13, No. 3, pp.1454-1455.Hudgins B., Parker P. and Scott R. N., 1993. A New Strat-egy for Multifunctional Myoelectric Control. In IEEE Transactions on Biomedical Engineering, vol. 40, No.1, pp. 82-94.Saito N. and Coifman R. R., 1995. Local Discriminant Basis and their applications. J. Math. Imag. Vis., Vol.5, no 4, pp. 337-358.Nishikawa D, Yu W. Yokoi H, and Kakazu Y, 1991. EMG Prosthetic Hand Controller using Real-Time Learning Method. In Proc. of the IEEE Conf. on SMC, Vol. 1, pp. I 153-158. Carreño I. R., Vuskovic M., 2005. Wavelet-Based Feature Extraction from Prehensile EMG Signals. In 13th Nor-dicBaltic on Biomedical Engineering and Medical Physics (NBC'05 UMEA), Umea, Sweden, 13-17. Schlesinger, D., 1919. Der Mechanische Aufbau der Kunstlishen Glieder. In Ersatzglieder und Arbeitshil-fen, Springer, Berlin.Seber G.A.F., 1984. Multivariate Observations, John Wiley & Sons, pp 102-117.Uchida N. U., Hiraiwa A., Sonehara N., Shimohara K., 1992. EMG Pattern Recognition by Neural Networks for Multi Fingers Control. Proc. of the Annual Int.Conf. of the Engineering in Medicine and Biology So-ciety. Vol 14, Paris, pp.1016-1018.Vapnik V. N., 1998. Statistical Learning Theory. John Wiley & Sons.Vuskovic M., Pozos A. L., Pozos R, 1995. Classification of Grasp Modes Based on Electromyographic Patternsof Preshaping Motions. Proc. of the Internat. Confer-ence on Systems, Man and Cybernetics. Vancouver,B.C., Canada, pp. 89-95, 1995.Vuskovic M., Schmit J., Dundon B. Konopka C., 1996.Hierachical Discrimination of Grasp Modes Using Sur-face EMGs. Proc. of the Internat. IEEE Conference on Robotics and Automation, Minneapolis, Minnesota, April 22-28. 2477-2483.Yao, Y., 1965. An Approximate Degrees of Freedom Solu-tion to the Multivariate Behrens-Fisher Problem, Biometrica, Vol. 52, 139-147.。

一种基于曲波变换的图像增强方法作者：杨光韩耀平来源：《中国新技术新产品》2009年第14期摘要:本文提出了一种新的基于曲波变换的图像增强方法,文中首先介绍了曲波变换模型,采用曲波变换方法增强图像的原理。

然后提出新的算法:对含噪声图像进行曲波变换,得到曲波变换系数; 对图像的曲波变换后各尺度系数中的高频成分进行软阈值操做,而对低频成分作灰度拉伸; 对处理后的曲波变换系数进行曲波反变换,得到增强后的图像。

最后通过图像质量评价方法对实验结果作了分析,结果证明该方法能够有效抑制噪声。

关键词:图像增强; 曲波变换1 引言图像在采集过程中,由于受环境、设备等因素的影响,往往具有对比度低、信噪比低、细节模糊等特点,使人眼的视觉分辨或机器识别较为困难,不利于图像的后续处理。

图像增强就是提高图像的对比度,使处理后的图像比原始图像更适于人眼的视觉特性或适合机器自动识别。

基于小波变换的增强方法被证明是一种较好的图像增强算法,但它提高图像的对比度、抑制噪声的同时,会在边缘处引起失真。

为了克服小波的这一局限性,1999 年 Candes E J和 Donoho D L提出了曲波(Curvelet) 变换理论[1], 也就是第一代曲波变换。

2004年Candes E.J在原有曲波变换的基础上提出二代曲波理论[2],完成了 Curvelet 理论的简化和快速实现。

本文提出了一种基于第二代曲波变换的图像增强方法,利用曲波变换对图像几何特征更优的表达能力,有效地提取原始图像的特征,较好地区分图像的边缘和噪声。

实验表明,该方法能够提高图像的对比度,降低噪声,并且较好地保留边缘信息,具有良好的视觉效果,便于后续的处理。

2 曲波变换2.1 离散曲波变换以笛卡尔坐标系下的为输入,曲波变换的离散形式为:目前有两种快速离散曲波变换的实现方法,分别是USFFT算法和Wrap算法[3][4],本文采用了第一种算法,USFFT算法步骤为:对输入图像笛卡尔坐标系下的2 curvelet变换系数以一幅512*512的图像为例,如表1所列,它在经过曲波变换之后被划分为6个尺度层,最高层被称为Coarse尺度层,是由低频系数组成的一个32*32的矩阵,体现了图像的概貌;最外层称为Fine尺度层,是由高频系数组成的512*512的矩阵,体现了图像的细节、边缘特征;其他层被称为Detail尺度层,由中高频系数组成,每层系数被分割为四个大方向,每个方向上被划分为8个,8个,16个,16个小方向,体现了在各个方向上的图像细节、边缘。

基于Curvelet变换的图像压缩感知重构叶慧;孔繁锵【摘要】Discrete Cosine Transform(DCT) and wavelet transform are used for sparse representation, but DCT can’t analyse well in domain of time and frequency. The directional selectivity of wavelet transform is poor and can’t reconstruct edge information well enough. Against the optimization of sparse representation, Curvelet transform has characters of multi-scale, singularity and more sparsity. This paper proposes a compressed sensing reconstruction algorithm based on Curvelet transform, which uses Curvelet transform for sparse representation and thresholding method in wavelet domain to solve the noise problem of signal reconstruction. Results demonstrate that the algorithm gets 1.86 dB higher Peak Signal to Noise Ratio(PSNR) and 1.15 dB higher PSNR compared with traditional wavelet transform and Contourlet transform. As Curvelet transform is applied to compressed sensing, optimal result of edge and smooth part of image are got, also the reconstructed quality of details is increased.%压缩感知主要采用离散余弦变换(DCT)和正交小波进行图像的稀疏表示，但是 DCT 时频分析性能不佳，小波方向选择性差，不能很好地表示图像边缘的信息。

一种基于Curvelet变换的图像放大方法研究李建平;张泠【摘要】Curvelet变换是近几年发展的一个新的多尺度几何分析方法.在分析主要的图像放大算法及Curvelet变换的基础上,提出了一种新的基于Curvelet变换的图像放大方法.为了验证算法,设计了仿真实验,与传统的二次线性和立方卷积图像放大方法进行比较.结果表明,在保持原始图像的结构信息方面,本文方法要优于其他图像放大方法.【期刊名称】《北京测绘》【年(卷),期】2014(000)001【总页数】4页(P4-7)【关键词】Curvelet变换;图像放大;结果评价;平均梯度【作者】李建平;张泠【作者单位】天津市勘察院,天津300191;天津市星际空间地理信息工程有限公司,天津300384;【正文语种】中文【中图分类】P208图像放大是图像处理的一个基本操作，是对二维数据的重采样，也可以说是对图像分辨率的改变。

图像放大在图像显示、分析、配准等图像处理中有着广泛的应用。

早期的图像放大技术主要有最近邻域法（nearest neighbor）、双线性内插法（bilinear interpolation）以及立方卷积法（cubic convolution）［1］。

后来学者们不断研究出新的方法，包括自适应重采样算法、高阶样条缩放法、多神经网络算法等。

这些方法多数因为运算量少，实现简单，因而得到了广泛的应用。

但由于在插值过程中没有针对图像的边缘和纹理特征进行特殊处理，所以导致放大后的图像难以保持物体边界清晰和轮廓分明。

小波变换在图像处理中的应用日益广泛，从频域角度上分析，图像放大中用到的一些函数表现为低通滤波器，经滤波后图像容易发生高频分量损失，因而出现边缘锯齿现象及高频细节模糊化［3－6］。

针对此特点，本文提出了一种基于Curvelet变换的图像放大方法，能够较多地保留图像边缘细节，尽可能少地减少高频分量的损失。

1 Ridgelet变换和Curvelet变换1.1 Ridgelet变换如果存在函数ψ满足条件其中）是函数ψ的傅立叶变换。

基于区域特性的Curvelet变换图像融合算法王坤臣;孙权森【摘要】In order to overcome the weakness of wavelet transform in 2D or higher dimensional spatial analysis and improve quality of image fusion,a modified image fusion algorithm based on Curvelet transform is proposed. Curvelet transform which can effectively analyze singularity of a curve and rationally process edge information in an image is introduced to decompose images. An adaptive threshold regional variance and Gaussian⁃weighted fusion algorithm is used in low⁃frequency region toen⁃hance the correlation between pixels in an image and preserve its details and edges effectively. The regional energy fusion method is applied in high⁃frequency region to reduce noise and enhance the details of the image. Many fusion experiments of different images were carried out with the algorithm. The fusion results were evaluated by information entropy,cross entropy,correlation coefficient and space frequency. The experiment results indicate that the proposed algorithm is more outstanding than the conven⁃tional fusion rules and methods,and can obtain better resolution and more rich image content.%为克服小波变换在二维或更高维度空间分析中的缺陷，提高图像融合质量，提出基于二代Curvelet变换的图像融合改进算法。

基于 Curvelet 变换的压缩传感超分辨率重建叶坤涛;郭振龙;贺文熙【期刊名称】《计算机应用与软件》【年(卷),期】2016(033)010【摘要】In order to improve the resolution of single-frame degraded images under the condition of no any training set,we implemented a compressed sensing super-resolution reconstruction algorithm,called Curvelet-FIST,which is based on Curvelet transform and fast iterative threshold-shrinkage (FIST)algorithm.First,the algorithm sets up a sampling mode of pseudo-star-shape sampling on low-resolution images. Then by making use of the theory of compressed sensing,and in Curvelet transform domain,it restores the high-resolution image from sampling values through FIST algorithm.Simulation experiment showed that this super-resolution reconstruction algorithm,compared with traditional interpolation algorithm and the compressed sensing reconstruction algorithm based on Wavelet transform and FIST (Wavelet-FIST), has higher peak signal-to-noise ratio (PSNR).%为了在无训练集的情况下，改善单帧退化图像的分辨率，实现了一种基于 Curvelet 变换和快速迭代收缩阈值法（FIST）的压缩传感超分辨率重建算法（Curvelet-FIST）。

基于Contourlet变换的纹理图像检索李丽君【摘要】提出了一种基于Contourlet变换的纹理图像检索算法.首先对纹理图像进行Contourlet变换,通过调整子带能量的次序,改进其旋转不变性,再提取不同尺度、不同方向上变换系数矩阵的均值和标准方差作为特征向量,最后采用Canberra 距离进行相似性度量.实验结果表明,此算法在对纹理图像检索时具有良好的检索性能,并具有旋转不变性.%A texture image retrieval algorithm based on Contourlet transform was proposed. Firstly, texture images were transformed based on Contourlet domain, then the rotation in variance was improved by adjusting sequence of sub - band energy parameters. Then means and standard deviation of coefficients matrix in different scale and various directions were extracted to form feature vectors. Finally, similarity measurement was implemented by using Canberra distance. Experimental results show that the new method has better retrieval performance and the properties of rotation invariance.【期刊名称】《科学技术与工程》【年(卷),期】2012(012)013【总页数】3页(P3245-3247)【关键词】图像检索;Contourlet变换;旋转不变性;Canberra距离【作者】李丽君【作者单位】辽宁石油化工大学理学院,抚顺113001【正文语种】中文【中图分类】TP391.41多媒体和通信技术，特别是网络技术的高速发展使得人们对图像的应用日益广泛，如何从海量图像库中快速提取有价值的信息已成为人们的迫切需求。

基于改进Curvelet变换的地震数据重建方法侯文龙;贾瑞生;孙圆圆;俞国庆【摘要】由于采集环境及仪器性能的限制,采集到的地震勘探数据经常是不规则和不完整的,进而影响到地震数据后续处理及反演,因此在对地震数据进行下一步分析处理前有必要先重建出完整的地震数据,提出了一种基于改进的Curvelet域的地震数据压缩重建算法.首先在压缩感知理论的框架下,利用Curvelet的稀疏特性,建立缺失地震数据重建模型;然后在CRSI(Curvelet Recovery by Sparsity-Promoting Inversion,CRSI)算法框架基础上,采用改进的指数阈值方法,对缺失地震数据进行恢复重建.使用了4层水平均匀介质模型和Marmousi模型模拟的地震数据进行了随机稀疏采样和重建的数值实验.实验结果表明,与传统重建算法比较,该方法不仅加快了原有算法的收敛速度,同时保证了重建数据的高信噪比,验证了所提方法的可行性和有效性.【期刊名称】《煤炭学报》【年(卷),期】2018(043)009【总页数】9页(P2570-2578)【关键词】压缩感知;地震数据重建;Curvelet变换;指数阈值【作者】侯文龙;贾瑞生;孙圆圆;俞国庆【作者单位】山东科技大学计算机科学与工程学院,山东青岛266590;山东科技大学山东省智慧矿山信息技术省级重点实验室,山东青岛266590;山东科技大学计算机科学与工程学院,山东青岛266590;山东科技大学山东省智慧矿山信息技术省级重点实验室,山东青岛266590;山东科技大学计算机科学与工程学院,山东青岛266590;山东科技大学山东省智慧矿山信息技术省级重点实验室,山东青岛266590;山东科技大学计算机科学与工程学院,山东青岛266590;山东科技大学山东省智慧矿山信息技术省级重点实验室,山东青岛266590【正文语种】中文【中图分类】P631.4煤层气主要储集在煤层裂隙中，煤层中裂隙的存在使煤储层表现为方位各向异性和双相介质特征[1-3]，这些特征为利用地震技术预测煤层气奠定了基础。

基于曲波变换的图像压缩算法研究的开题报告一、研究背景和意义随着科技的不断发展，数字图像处理已经成为了当今互联网时代不可或缺的一部分。

然而，数字图像处理可靠性和效率的问题一直是图像处理领域的研究热点。

在数字图像的处理中，图像的压缩一直是一个重要的环节，这是因为数字图像存储需要大量的存储空间，而图像压缩可以使存储变小同时还能实现更快的传输速度。

因此，图像压缩算法的研究是数字图像处理的重要方向之一。

本研究将基于曲波变换，通过对曲波变换和其在图像处理中的应用研究，提出一个新的图像压缩算法，希望能够通过实现这一算法，为数字图象处理领域的研究和实际应用提供一种有效的高效的压缩方法。

二、研究内容和方法1. 研究曲波变换的原理和算法，包括离散曲波变换和连续曲波变换。

2. 研究曲波变换在图像处理中的应用，探究曲波变换在图像压缩中的有效性。

3. 设计并实现基于曲波变换的图像压缩算法。

在该算法中，首先对图像进行曲波变换，然后根据曲波变换系数的大小和位置，进行量化和编码，最终得到压缩后的图像。

4. 通过实验分析和对比，验证所提出的基于曲波变换的图像压缩算法的有效性和优越性。

三、预期研究成果和创新点1. 提出一种基于曲波变换的图像压缩算法，通过该算法，能够高效地实现对数字图像的压缩，并减少存储空间和传输带宽。

2. 实验分析和对比验证该算法的有效性和优越性。

3. 对于数字图像处理领域提供一种新的压缩方法，为数字图像处理领域的研究和实际应用提供一种有效的高效的压缩方法。

四、研究进度安排1. 阅读相关文献资料，了解现有的数字图像压缩算法和曲波变换的基本原理和应用，完成文献综述，所需时间1个月。

2. 研究离散曲波变换和连续曲波变换的原理和算法，研究曲波变化在图像处理中的应用，所需时间2个月。

3. 设计并实现基于曲波变换的图像压缩算法，所需时间3个月。

4. 进行实验验证并对比分析，完成论文撰写，所需时间4个月。

5. 修稿完善，所需时间1个月。

㊀第３８卷第１期物㊀探㊀与㊀化㊀探Ｖｏｌ．３８，Ｎｏ．１㊀㊀２０１４年２月ＧＥＯＰＨＹＳＩＣＡＬ＆ＧＥＯＣＨＥＭＩＣＡＬＥＸＰＬＯＲＡＴＩＯＮＦｅｂ．，２０１４㊀ＤＯＩ：１０．１１７２０／ｊ．ｉｓｓｎ．１０００－８９１８．２０１４．１．１４改进的曲波变换及全变差联合去噪技术薛永安，王勇，李红彩，陆树勤（中国石油化工股份有限公司江苏油田分公司物探技术研究院，江苏南京㊀２１００４６）摘要：运用常规的基于曲波变换和全变差的联合去噪技术，可以有效地衰减随机噪声，较好地克服使用曲波变换带来的强能量团以及在同相轴边缘产生的不光滑现象，但是这种常规的联合去噪方法对有效信号有一定的损害㊂笔者采用一种多尺度多方向改进的Ｄｏｎｏｈｏ阈值去噪思想，较好地克服了常规的联合去噪方法的缺陷，保护了有效信号㊂该方法在应用曲波变换去噪时，对每一个尺度的每一个方向都选取一个合适的阈值因子，而不是常规的方法对整个曲波系数矩阵只选取一个固定比例的阈值因子㊂理论模型与实际资料的处理结果表明，该技术最大限度地保留了地震数据的有效信号，在地震资料处理中具有较好的应用前景㊂关键词：曲波变换；全变差；随机噪声；多尺度中图分类号：Ｐ６３１．４㊀㊀㊀文献标识码：Ａ㊀㊀㊀文章编号：１０００－８９１８（２０１４）０１－００８１－０６㊀㊀在地震勘探中，常规的随机噪声衰减方法对有效信号的损害比较大㊂为了较好地去除随机噪声，Ｎｅｅｌａｍａｎｉ等人在２００８年引入了一种多尺度的变换方法 曲波变换来衰减随机噪声，取得了较好的效果［１］㊂但是曲波变换不可避免地会产生伪影现象，同时在同相轴边缘产生不光滑现象，为了克服曲波变换的这个缺点，２０１０年，清华大学的唐刚在其博士论文中详细介绍了基于曲波变换和全变差的联合去噪技术，在压制随机噪声的同时，较好地保护了有效信号，同时该方法较好地压制了单独使用曲波变换去噪时产生的伪影现象［２］㊂此前，２００８年卢成武㊁２００９年倪雪，都相继介绍过基于曲波变换和全变差的联合去噪技术［３－４］㊂曲波变换最先由Ｃａｎｄèｓ和Ｄｏｎｏｈｏ等人于１９９９年在Ｒｉｄｇｅｌｅｔ变换的基础上提出［５］，随后几年，Ｃａｎｄèｓ等人对第一代Ｃｕｒｖｅｌｅｔ变换作了比较大的改进［６］㊂２００４年，ＨｅｒｒｍａｎｎＦ等最先将Ｃｕｒｖｅｌｅｔ变换应用到地震数据处理领域，成功地将Ｃｕｒｖｅｌｅｔ变换应用到多次波的衰减中［７－８］㊂虽然运用的联合去噪技术较好地克服了单独使用曲波变换去噪带来的强能量团以及在同相轴边缘产生的不光滑现象，同时较好地保护了有效信号，但是这种联合去噪方法对有效信号有一定损害［９］，笔者对该方法进行了一定的改进，通过模型数据和实际数据的测试表明，该技术较好地衰减了随机噪声，同时最大限度地保留了地震数据的有效信号，是一种值得推广的随机噪声压制方法㊂１㊀第二代曲波变换及全变差技术简介１．１㊀第二代曲波变换简介连续曲波变换属于稀疏理论的范畴，可以采用基函数与信号的内积形式实现信号的稀疏表示，因此曲波变换可以表示为ｃ（ｊ，ｌ，ｋ）＝ ｆ，φｊ，ｌ，ｋ⓪＝ʏＲ２ｆ（ｘ）φｊ，ｌ，ｋ（ｘ）ｄｘ，（１）式中：φｊ，ｌ，ｋ表示曲波函数，ｊ，ｌ，ｋ分别表示尺度㊁方向㊁位置参数㊂因为数字曲波变换是在频域进行的，根据Ｐｌａｎｃｈｅｒｅｌ定理，可以将这种内积形式表示成频域的积分形式ｃ（ｊ，ｌ，ｋ）＝１（２π）２ʏ＾ｆ（ω）φｊ，ｌ，ｋ（ω）ｄω＝１（２π）２ʏ＾ｆ（ω）Ｕｊ（Ｒθｌω）ｅｊ ｘ（ｊ，ｌ）ｋ，ω⓪ｄω，（２）经过变换后得到Ｃ｛ｊ｝｛ｌ｝（ｋ１，ｋ２）结构的系数，ｊ表示尺度，ｌ表示方向，（ｋ１，ｋ２）表示尺度层上的矩阵坐标［６］㊂１．２㊀全变差技术简介全变差最小化技术最先由Ｒｕｄｉｎ等人于１９９２年提出（通常称ＲＯＦ模型），经过全变差的定义及化简，得数据ｆ的全变差收稿日期：２０１２－１２－２５物㊀探㊀与㊀化㊀探３７卷㊀ＴＶ（ｆ）＝ʏΩ｜∇ｆ｜ｄｘ，（３）其中：Ω为图像ｆ的支撑区间，ｘɪΩ为图像的坐标向量㊂基于全变差的去噪方法可归结为最小化问题Ｅ（ｆ）＝ʏΩ｜ｆ－ｆ０｜２ｄｘ＋λＴＶ（ｆ），（４）其中：第一项为逼近项，使去噪后的图像依然能够较好地逼近原始图像；第二项是全变差正则化项；λ是拉格朗日常数，在逼近项和正则化项之间起着重要的平衡作用㊂上述目标函数Ｅ（ｆ）是ｆ的凸函数，其存在极值的充分必要条件是∇Ｅ（ｆ）＝０，由此可以得到其对应的Ｅｕｌｅｒ⁃Ｌａｒｇｒａｎｇｅ方程为－∇∇ｆ｜∇ｆ｜æèçöø÷＋λ（ｆ－ｆ０）＝０㊂（５）该方程为非线性方程，假设方程满足Ｎｅｕｍａｎｎ边界条件，通过梯度下降法对数据进行反复迭代得到一个稳定解，从而得到去噪后的数据，其迭代公式ｆｎ＋１＝ｆｎ－ｔｎ［ｇＴＶ（ｆｎ）］㊂（６）其中：令初始值ｆ１＝ｆ０，ｔｎȡ０表示迭代步长，ｇＴＶ（ｆ）＝－∇∇ｆ｜∇ｆ｜æèçöø÷表示全变差函数在ｆ处的次梯度［１０］㊂２㊀改进的联合去噪策略常规的联合去噪技术通常在曲波变换后对每一个尺度都选取一个相同比例的阈值，该选取方法不能够充分利用曲波变换的优点，会造成某些区域有效信号损害相对较大，而某些区域，去噪效果不理想的情况，传统的联合去噪流程见图１㊂通常地震数据经过曲波变换后，被划分成多个尺度层，最内层，也就是第一层称为Ｃｏａｒｓｅ尺度层，是由低频系数组成的；最外层，称为Ｆｉｎｅ尺度层，是由高频系数组成；中间的尺度层称为Ｄｅｔａｉｌ尺度层，是由中高频系数组成的㊂在不同的尺度层，有效信号和随机噪声的系数分布是不一样的，在Ｃｏａｒｓｅ尺度层它们之间的系数没有较明显分界，同时随机噪声在这个尺度层占的比重不大，因此，在这个尺度层可以多保留一些系数，同时在Ｄｅｔａｉｌ尺度层和Ｆｉｎｅ尺度层选择合适的阈值㊂通过该方法的的处理，能够更好地保留有效信号，达到高保真处理的目的，改进的联合去噪流程见图２㊂对于多尺度阈值的选取，主要是在Ｄｏ⁃ｎｏｈｏ阈值处理的基础上进行一定的改进，尽管Ｄｏ⁃ｎｏｈｏ阈值法一开始主要是用于小波阈值的求取，但是对Ｃｕｒｖｅｌｅｔ变换的阈值求取也有一定的借鉴意义，Ｄｏｎｏｈｏ提出的阈值求取方法如Ｔｔｈｒｅｓｈｏｌｄ＝σ２ｌｏｇＮ（７）所示㊂其中：σ＝Ｍｅｄｉａｎ（Ｃｉ，ｊ）／０．６７４５，Ｎ为地震数据的长度㊂在Ｄｏｎｏｈｏ阈值处理的基础上，在Ｃｕｅｖｅｌｅｔ变换的每一个尺度和方向上选取一个合适的阈值（即对Ｄｏｎｏｈｏ阈值进行一定的调整，针对不同的尺度特点，为更好地去噪，０．６７４５可以改为更大或更小），能够在消除随机噪声的同时，较好地保留有效信号㊂图１㊀传统的联合去噪流程图２㊀改进的联合去噪流程㊃２８㊃㊀６期薛永安等：改进的曲波变换及全变差联合去噪技术３㊀模型及实际数据测试选取两个模型数据和一块江苏油田工区的实际数据进行测试㊂首先选取一个双曲模型进行测试（胡天乐提供），该模型为５１２道，２０４８个采样点，１ｍｓ采样㊂图３ａ为原始不含噪声数据，图３ｂ是加入随机噪声以后的数据得到的信噪比为０．２８的含噪剖面，这里的信噪比用信号振幅的平方和与噪声振幅的平方和的比值来表示㊂图３ｃ是传统的联合去噪方法去噪后的结果，信噪比为２㊂图４ａ和图４ｂ为Ｄｏｎｏｈｏ阈值法及改进的Ｄｏｎｏｈｏ阈值去噪后的结果，信噪比分别达到３．８５和４．２，图４ｃ和４ｄ分别为这两种方法去噪后的结果与不含噪数据的误差剖面，从图中可以看到，改进Ｄｏｎｏｈｏ阈值法要优于前两种方法，同时保护了有效信号㊂ａ 原始不含噪数据；ｂ 加入随机噪声后的数据；ｃ 传统的联合去噪方法去噪后图３㊀原始模型数据（一）ａ Ｄｏｎｏｈｏ阈值去噪后结果；ｂ 改进Ｄｏｎｏｈｏ阈值去噪后结果；ｃ Ｄｏｎｏｈｏ阈值去噪后的误差剖面；ｄ 改进Ｄｏｎｏｈｏ阈值去噪后的误差剖面图４㊀Ｄｏｎｏｈｏ阈值去噪及改进后的Ｄｏｎｏｈｏ阈值去噪㊀㊀第二个模型数据为６４道，５０１个采样点，时间采样间隔为１ｍｓ，不含噪声的数据见图５ａ，加入随机噪声，得到信噪比为０．３的含噪数据（图５ｂ）㊂经过传统的联合去噪方法压制后，地震资料的信噪比得到了较大的提高，信噪比为１．８２，随机噪声得到了有效的压制，同时曲波变换的伪影现象也得到了有效克服（图６ａ）㊂但是，从图６ｂ中的误差数据可以看到，运用常规的联合去噪方法对有效信号有一定损害，特别是对近偏移距数据和弯曲同相轴损害较大㊂运用笔者提出的改进方法，去噪结果见图７ａ，误差结果见图７ｂ，信噪比达到１．９５㊂从图中可以看到，运用改进的方法，有效信号得到了很好的保真，特别是近偏移距附近的同相轴得到了很好的保留，因此改进的方法是一种有效的高保真的处理方法㊂㊃３８㊃物㊀探㊀与㊀化㊀探３７卷㊀ａ 原始不含噪声数据；ｂ 加入噪声后的数据图５㊀原始模型数据（二）及原始含噪数据ａ 常规联合去噪方法去噪后的数据；ｂ 误差数据图６㊀模型数据处理结果ａ 本文方法去噪后的数据；ｂ 误差数据图７㊀模型数据处理结果㊃４８㊃㊀６期薛永安等：改进的曲波变换及全变差联合去噪技术㊀㊀研究中，选取江苏油田某工区的偏移数据作为测试对象（图８ａ），本数据有２００道，４５１个采样点，采样间隔为１ｍｓ，运用笔者提出的改进的联合去噪方法和常规的联合去噪方法进行对比，图８ｂ和图８ｃ分别为常规的联合去噪方法去噪后的结果及误差剖面㊂从图中可以看到本地区资料断层发育，由于随机噪声的存在，影响了资料的品质㊂经过联合去噪以后，原始剖面中的随机噪声得到了很好的压制，剖面的信噪比得到了明显的提高，同时剖面的断点得到了很好的保留㊂但是在构造复杂区，有效信号受到一定的损害，在误差剖面中可以明显看到有效信号的存在，同时在深部弱信号区域，对有效信号的保真比较差㊂图９显示的是用笔者提出的方法处理的结果，从去噪剖面和误差剖面中可以看到，随机噪声不但得到了很好的压制，在构造复杂区的有效信号也得到了很好的保留，深部弱信号也得到了很好的保留㊂ａ 实际原始数据；ｂ 常规方法去噪后的结果；ｃ 误差剖面图８㊀实际数据常规处理结果ａ 本文方法去噪后的结果；ｂ 误差数据图９㊀实际数据改进的方法处理结果４㊀结论改进的联合去噪技术，经过模型数据和实际数据的测试表明，该方法可以有效的压制地震数据中的随机噪声，对于复杂构造区的资料，在去噪的同时，更好地保留了有效信号，使反射波同相轴更加清晰㊁连续性更好，同时较好地保留了断点㊁断面的信息，是一种值得推广的随机噪声压制技术㊂㊃５８㊃物㊀探㊀与㊀化㊀探３７卷㊀参考文献：［１］㊀ＮｅｅｌａｍａｎｉＲ，ＢａｕｍｓｔｅｉｎＡＩ，ＧｉｌｌａｒｄＤ，ｅｔａｌ．Ｃｏｈｅｒｅｎｔａｎｄｒａｎｄｏｍｎｏｉｓｅａｔｔｅｎｕａｔｉｏｎｕｓｉｎｇｔｈｅｃｕｒｖｅｌｅｔｔｒａｎｓｆｏｒｍ［Ｊ］．ＴｈｅＬｅａｄｉｎｇＥｄｇｅ，２００８：２４０－２４８．［２］㊀唐刚．基于压缩感知和稀疏表示的地震数据重建和去噪［Ｄ］．北京：清华大学，２０１０．［３］㊀卢成武，宋国乡．带曲波域约束的全变差正则化抑噪方法［Ｊ］．电子学报，２００８，３６（４）：６４６－６４９．［４］㊀倪雪，李庆武，孟凡，等．基于Ｃｕｒｖｅｌｅｔ变换和全变差的图像去噪方法［Ｊ］．光学学报，２００９，２９（９）：２３９０－２３９４．［５］㊀ＣａｎｄèｓＥＪ，ＤｏｎｏｈｏＤＬ．Ｃｕｒｖｅｌｅｔｓ：Ａｓｕｒｐｒｉｓｉｎｇｌｙｅｆｆｅｃｔｉｖｅｎｏｎ⁃ａｄａｐｔｉｖｅｒｅｐｒｅｓｅｎｔａｔｉｏｎｆｏｒｏｂｊｅｃｔｓｗｉｔｈｅｄｇｅｓ［Ｍ］．ＮａｓｈｖｉｌｌｅＴＮ：ＶａｎｄｅｒｂｉｌｔＵｎｉｖｅｒｓｉｔｙＰｒｅｓｓ，２０００：１０５－１２０．［６］㊀ＣａｎｄèｓＥＪ，ＤｅｍａｎｅｔＬ，ＤｏｎｏｈｏＤ，ｅｔａｌ．Ｆａｓｔｄｉｓｃｒｅｔｅｃｕｒｖｅｌｅｔｔｒａｎｓｆｏｒｍｓ［Ｃ］／／ＡｐｐｌｉｅｄａｎｄＣｏｍｐｕｔａｔｉｏｎａｌＭａｔｈｅｍａｔｉｃｓ，Ｃａｌｉｆｏｒ⁃ｎｉａＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，２００５：１－４３．［７］㊀ＨｅｒｒｍａｎｎＦＪ，ＶｅｒｓｃｈｕｕｒＥ．Ｃｕｒｖｅｌｅｔｄｏｍａｉｎｍｕｌｔｉｐｌｅｅｌｉｍｉｎａｔｉｏｎｗｉｔｈｓｐａｒｓｅｎｅｓｓｃｏｎｓｔｒａｉｎｔｓ［Ｊ］．ＳｏｃｉｅｔｙｏｆＥｘｐｌｏｒａｔｉｏｎＧｅｏｐｈｙｓｉ⁃ｃｉｓｔｓ，ＥｘｐａｎｄｅｄＡｂｓｔｒａｃｔｓ，２００４：１３３３－１３３６．［８］㊀ＨｅｒｒｍａｎｎＦＪ，ＶｅｒｓｃｈｕｕｒＥ．Ｓｅｐａｒａｔｉｏｎｏｆｐｒｉｍａｒｉｅｓａｎｄｍｕｌｔｉｐｌｅｓｂｙｎｏｎ⁃ｌｉｎｅａｒｅｓｔｉｍａｔｉｏｎｉｎｔｈｅｃｕｒｖｅｌｅｔｄｏｍａｉｎ［Ｃ］／／ＥＡＧＥ６６ｔｈＣｏｎｆｅｒｒｅｎｃｅ＆ＥｘｈｉｂｉｔｉｏｎＰｒｏｃｅｅｄｉｎｇｓ，２００４．［９］㊀俞华，薛永安，王勇，等．曲波变换及全变差最小化技术联合去噪［Ｊ］．石油物探，２０１２，５１（４）：３５０－３５５．［１０］ＲｕｄｉｎＬＩ，ＯｓｈｅｒＳ，ＦａｔｅｍｉＥ．Ｎｏｎｌｉｎｅａｒｔｏｔａｌｖａｒｉａｔｉｏｎｂａｓｅｄｎｏｉｓｅｒｅｍｏｖａｌａｌｇｏｒｉｔｈｍｓ［Ｊ］．ＰｈｙｓｉｃａＤ．，１９９２，６２（１）：６３－６７．ＡＮＩＭＰＲＯＶＥＤＲＡＮＤＯＭＡＴＴＥＮＵＡＴＩＯＮＭＥＴＨＯＤＢＡＳＥＤＯＮＣＵＲＶＥＬＥＴＴＲＡＮＳＦＯＲＭＡＮＤＴＯＴＡＬＶＡＲＩＡＴＩＯＮＸＵＥＹｏｎｇ⁃ａｎ，ＷＡＮＧＹｏｎｇ，ＬＩＨｏｎｇ⁃ｃａｉ，ＬＵＳｈｕ⁃ｑｉｎ（ＧｅｏｐｈｙｓｉｃａｌＰｒｏｓｐｅｃｔｉｎｇＴｅｃｈｎｏｌｏｇｙＲｅｓｅａｒｃｈＩｎｓｔｉｔｕｔｅ，ＪｉａｎｇｓｕＯｉｌＦｉｌｅｄＢｒａｎｃｈｏｆＳｉｎｏｐｅｃ，Ｎａｎｊｉｎｇ㊀２１００４６，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｒａｎｄｏｍｎｏｉｓｅｃａｎｂｅｅｆｆｅｃｔｉｖｅｌｙａｔｔｅｎｕａｔｅｄｂａｓｅｄｏｎｃｏｎｖｅｎｔｉｏｎａｌｃｏｍｂｉｎａｔｉｏｎｏｆｃｕｒｖｅｌｅｔｔｒａｎｓｆｏｒｍａｎｄｔｏｔａｌｖａｒｉａｔｉｏｎｔｅｃｈ⁃ｎｏｌｏｇｙ．Ｔｈｉｓｃｏｍｂｉｎａｔｉｏｎｔｅｃｈｎｏｌｏｇｙｃａｎｒｅｄｕｃｅｔｈｅｐｓｅｕｄｏ⁃ｇｉｂｂｓｅｆｆｅｃｔｓａｎｄｔｈｅａｌｉａｓｅｄｃｕｒｖｅｓｒｅｓｕｌｔｉｎｇｆｒｏｍｕｓｉｎｇｃｕｒｖｅｌｅｔｔｒａｎｓｆｏｒｍ，ｂｕｔｔｈｉｓｍｅｔｈｏｄｉｓｎｏｔｃｏｎｄｕｃｉｖｅｔｏｔｈｅｆｉｄｅｌｉｔｙｏｆｓｅｉｓｍｉｃｄａｔａｐｒｏｃｅｓｓｉｎｇ．Ｉｎｔｈｉｓｐａｐｅｒ，ａｒａｎｄｏｍｎｏｉｓｅａｔｔｅｎｕａｔｉｏｎｍｅｔｈｏｄｉｓｐｕｔｆｏｒ⁃ｗａｒｄｂａｓｅｄｏｎｍｕｌｔｉ⁃ｓｃａｌｅａｎｄｍｕｌｔｉ⁃ｄｉｒｅｃｔｉｏｎｉｍｐｒｏｖｅｄＤｏｎｏｈｏｔｈｒｅｓｈｏｌｄｓ，Ｔｈｉｓｉｍｐｒｏｖｅｄｃｏｍｂｉｎａｔｉｏｎｔｅｃｈｎｏｌｏｇｙｃａｎｖｅｒｙｅｆｆｅｃｔｉｖｅｌｙｏｖｅｒｃｏｍｅｔｈｅｄｉｓａｄｖａｎｔａｇｅｓｏｆｃｏｎｖｅｎｔｉｏｎａｌｃｏｍｂｉｎａｔｉｏｎｔｅｃｈｎｏｌｏｇｙａｎｄｂｅｔｔｅｒｐｒｅｓｅｒｖｅｔｈｅｓｉｇｎａｌｏｆｓｅｉｓｍｉｃｄａｔａ．Ｗｈｅｎｔｈｉｓｍｅｔｈｏｄｉｓｕｓｅｄｔｏａｔｔｅｎｕａｔｅｒａｎｄｏｍｎｏｉｓｅ，ｗｅｍｕｓｔｃｈｏｏｓｅａｐｐｒｏｐｒｉａｔｅｔｈｒｅｓｈｏｌｄｆａｃｔｏｒｓａｔｅｖｅｒｙｓｃａｌｅａｎｄｉｎｅｖｅｒｙｄｉｒｅｃｔｉｏｎ，ａｎｄｉｔｉｓｕｎｌｉｋｅｃｏｎ⁃ｖｅｎｔｉｏｎａｌｔｅｃｈｎｏｌｏｇｙｗｈｉｃｈｏｎｌｙｃｈｏｏｓｅｓｏｎｅｆｉｘｅｄｐｒｏｐｏｒｔｉｏｎｔｈｒｅｓｈｏｌｄｆａｃｔｏｒｓｏｆａｌｌｃｕｒｖｅｌｅｔｃｏｅｆｆｉｃｉｅｎｔｓ．Ｔｈｅｏｒｅｔｉｃａｌｍｏｄｅｌａｎｄｒｅａｌｄａｔａｐｒｏｃｅｓｓｉｎｇｒｅｓｕｌｔｓｓｈｏｗｔｈａｔｔｈｉｓｔｅｃｈｎｏｌｏｇｙｃａｎｍａｘｉｍａｌｌｙｐｒｅｓｅｒｖｅｔｈｅｓｉｇｎａｌｏｆｓｅｉｓｍｉｃｄａｔａ，ｓｏｉｔｈａｓａｇｏｏｄｐｒｏｓｐｅｃｔｉｎｔｈｅｓｅｉｓｍｉｃｄａｔａｐｒｏｃｅｓｓｉｎｇ．Ｋｅｙｗｏｒｄｓ：ｃｕｒｖｅｌｅｔｔｒａｎｓｆｏｒｍ；ｔｏｔａｌｖａｒｉａｔｉｏｎ；ｒａｎｄｏｍｎｏｉｓｅ；ｍｕｌｔｉ⁃ｓｃａｌｅ作者简介：薛永安（１９８４－），男，工程师，２０１０年硕士毕业于中国石油大学（北京），现在江苏油田物探技术研究院从事采集㊁处理以及方法方面的研究工作㊂㊃６８㊃。

基于Curvelet的边缘纹理特征提取及表情识别林克正;王浩【期刊名称】《计算机工程与应用》【年(卷),期】2013(000)016【摘要】针对小波变换在提取图像边缘特征上的局限性，提出一种使用Curvelet变换进行边缘纹理特征提取的表情识别方法。

Curvelet变换在表达图像的边缘曲线上的奇异性时比小波变换更能得到稀疏的图像表示。

在表情识别中，对表情图像使用Curvelet变换得到Curvelet系数作为边缘纹理特征能更好地反映表情的变化，使用K最邻近结点算法进行了识别。

结果表明在表情识别中该方法比小波变换更有效。

%For the wavelet transform has limitations to extract features of the edge of the images, a method of the facial expres-sion recognition is proposed that using curvelet transform to extract features of the edge of the images. The curvelet transform can get more representation of sparse images than the wavelet transform on the representation of the singular of the edges of the image curve. The curvelet coefficient that can be got by using the curvelet transform on the facial images as the edge of the tex-ture features can better reflect the changes in the facial expression, andthe k-nearest neighbor algorithm is used to recognition different expression in this paper. The result shows that the method proposed inthis paper is more effective than the wavelet transform in the expression recognition.【总页数】4页(P151-154)【作者】林克正;王浩【作者单位】哈尔滨理工大学计算机科学与技术学院，哈尔滨 150080;哈尔滨理工大学计算机科学与技术学院，哈尔滨 150080【正文语种】中文【中图分类】TP391.4【相关文献】1.卫星图像边缘与纹理特征提取 [J], 邹旭兴;谢明元2.基于边缘系数增强和对比度提升的Curvelet变换图像增强方法 [J], 徐效文;王阿记3.基于Curvelet变换和SVM的人脸表情识别方法研究 [J], 薄璐;周菊香4.基于Curvelet变换的边缘锐化方法 [J], 张恒娟;胡燕翔5.基于边缘云框架的高效安全人脸表情识别 [J], 张娴静;褚含冰;刘鑫因版权原因，仅展示原文概要，查看原文内容请购买。

CurveLab Toolbox,Version2.0Emmanuel Cand`e s,Laurent Demanet,Lexing YingApplied and Computational MathematicsCalifornia Institute of Technology,Pasadena CA911251IntroductionCurveLab is a collection of Matlab and C++programs for the Fast Discrete Curvelet Transform in two and three dimensions.For the2d curvelet transform,the software package includes two distinct implementations: the wrapping-based transform and the transform using unequally-spaced fast Fourier trans-form(USFFT).Both variants are based on the Curvelet transform as described in‘New Tight Frames of Curvelets and Optimal Representations of Objects with Piecewise C2Sin-gularities’,Comm.Pure Appl.Math.57(2004)219-266.The implementation is also discussed in detail in the Technical Report“Fast Discrete Curvelet Transforms”available at .We advise users to become familiar with these references. The two implementations differ by the choice of spatial grid used to translate curvelets at each scale and angle.•The USFFT version uses a decimated rectangular grid tilted along the main direction of each curvelet.There is one such grid per scale and angle,and this implementation is therefore very close to the definition given in the above reference.For the digital transform,tilting the grids induces a resampling of the Fourier transform on semi-regular grids,hence the use of a(perhaps novel)USFFT routine.For the inversion,a conjugate-gradient solver rapidly converges to the solution.•The wrapping version uses,instead,a decimated rectangular grid aligned with the image axes.For a given scale,there are essentially two such grids(decimated mostly horizontally or mostly vertically).The resulting sampling is not as faithful to the original transform,but the basis functions are curvelets as much as in the USFFT-based implementation.Since no interpolation is necessary in the frequency plane,the transform is a numerical isometry and can be inverted by its adjoint.For the3d curvelet transform,the software in this package is an extension of the wrapping version in2d.Due to the large size of the3d data and the increasing redundancy of the curvelet transform,three different implementation are included:•The in-core implementation which stores both the input data and the curvelet coef-ficients in the memory(suitable for small size data set).•The out-core implementation which stores the input data in the memory and most of the curvelet coefficients on the disc(suitable for medium size data set).•The MPI-based parallel implementation which distributes both input data and the curvelet coefficients on multiple nodes.The parallel implementation can successfully handle input data of size1k×1k×1k.The C++part of the package includes all the2d and3d implementations listed above. The Matlab part of this package includes only the2d implementations of the USFFT and wrapping transforms.2InstallationCopy the CurveLab package into the directory of your choice,and expand it withgunzip CurveLab-2.0.tar.gztar xvf CurveLab-2.0.tarMatlab part.The Matlab implementation for the wrapping-based transform can be found in the directory fdct_wrapping_matlab,while the Matlab implementation for the USFFT-based transform is in the directory fdct_usfft_matlab.In both directories,you canfind several demos which explain how to use the software.Before running the demos in the directory fdct_usfft_matlab,run fdct_usfft_path.m to set the correct searching path.C++part.The C++implementation is tested on Unix-type platforms,including Linux, SunOS and MacOS.The installation requires g++(version3.2or3.3),and Matlab’s mex compiler(version13or14)if you want to compile the mexfiles for the C++implementation (make sure the mex compiler use the same version of g++).To install the C++implementations(except the MPI-based parallel3d transform)•Download and install FFTW version2.1.5().•In makefile.opt,set FFTW_DIR to be the path in which FFTW2.1.5is installed,andset MEX to be the correct mex compiler(optional).•While in the main toolbox directory,typemake lib to build the libraries for both the wrapping-based and the USFFT-based implementationsmake test to test the C++installation(optional)make matlab to generate the mexfiles of the C++implementation,which can be called in Matlab(optional)For the2d implementations,the sourcefiles of the“C++wrapping version”are in the directory fdct_wrapping_cpp/src,while the mexfiles generated by make matlab and the demofiles are in the directory fdct_wrapping_cpp/mex.The sourcefiles of the“C++ USFFT version”are in the directory fdct_usfft_cpp/src,while the mexfiles and the demofiles are in the directory fdct_usfft_cpp/mex.For the3d implementations,the sourcefiles for the in-core version are in the directory fdct3d/src,while the mexfiles(generated by make matlab and the Matlab demofiles arein the directory fdct3d/mex.The sourcefiles of the out-core version are in the directory fdct3d_outcore.To install the MPI-based parallel3d implementation,please read fdct3d_mpi/src/README for details.3List of Matlab FilesThe2d Matlab implementations of the wrapping-based and USFFT-based transforms are in the directories fdct_wrapping_matlab and fdct_usfft_matlab respectively.The Matlab interface of the C++implementations is located in the directories fdct_wrapping_cpp/mex and fdct_usfft_cpp/mex.Below is a(non-exhaustive)list of the.mfiles which can be found in those directories.xxx stands for either wrapping or usfft.Basic transform functions.•fdct_xxx.m–forward curvelet transform.•ifdct_xxx.m–inverse curvelet transform.•afdct_xxx.m–adjoint curvelet transform(only for USFFT-based transform since for the wrapping transform,the adjoint is the same as the inverse transform).•fdct_xxx_param.m–this function returns the location of each curvelet in phase-space. Useful Tools.•fdct_xxx_dispcoef.m–a function which returns a matrix containing all the curvelet coefficients.•fdct_xxx_pos2idx.m–for afixed scale and afixed direction,this function returns the index of the curvelet closest to a certain point in the image.Demo Files.•fdct_xxx_demo_basic.m–displays a curvelet in the spatial and frequency domain.•fdct_xxx_demo_recon.m–partial reconstruction using the curvelet transform.•fdct_xxx_demo_disp.m–displays all the curvelet coefficients of an image.•fdct_xxx_demo_denoise.m–image denoising via curvelet shrinkage.There are two extra Matlab demos in fdct_wrapping_matlab directory:•fdct_wrapping_demo_denoise_enhanced.m–further techniques for image denoising.•fdct_wrapping_demo_wave.m–wave propagation using curvelets.An extrafile fdct_usfft_path.m is included in fdct_usfft_matlab directory.This script needs to be called to append several subdirectories into the searching path.The Matlab code for the in-core3d transform are located in fdct3d/mex.Below is a list of the.mfiles.Basic transform functions.•fdct3d_forward.m–forward3d curvelet transform.•fdct3d_inverse.m–inverse3d curvelet transform.•fdct3d_param.m–a function which returns the location of each curvelet in phase-space.Demo Files•fdct3d_demo_basic.m–displays a curvelet in the spatial and frequency domain. Extra information for each of these functions can be obtained by typing(at the Matlab prompt)help followed by the name of the function.4Changes•Fixed bug in the2d C++implementation for input data with odd size.•In2d C++implementation,changed the input data structure from class CpxOffMat to class CpxNumMat.The testfiles are changed accordingly.5Copyright and ContactCurveLab is copyright c California Institute of Technology,2005.For further information, please contact curvelab@.。