New Method Based on Curvelet Transform for Image Denoising
基+于多尺度图像融合方法研+究
II
重庆大学硕士学位论文
英文摘要
transform which has the trait of shift invariance. the pyramid decompose of nonsubsampled is taken place by the newly obtained transform. Unifying this new transform with a certain fusion rule, a novel algorithm of image fusion based on the nonsubsampled lifting wavelet-Contourlet transform is obtained. Experimental results show that the new algorithm, compared with the nonsubsampled Contourlet transform method, can extract more useful information from the source image and pour the information into the fusion image, thus obtain a better performed fusion image.
③ A novel image fusion algorithm is proposed based on nonsubsampled Contourlet transform with the region contrast and spatial frequency ratio as fusion rules limit. For the fusion of visible and infrared image of the same scene, a weighted average scheme based on the physical features of images is proposed for choosing the low-frequency coefficients. With the high-frequency coefficients, a selection principle is proposed based on region contrast and spatial frequency ratio. The experimental results show that the method compared with the traditional simple fusion method and the regional energy fusion rule can be better in visual quality and objective evaluation criteria;
结合非下采样轮廓波变换和混合阶次图像扩散的图像去噪董超
的二阶偏微分方程去噪模型的时频域的局部表示能力比较 ［2 ］ 差， 而且处理边缘突变信息的效果不是很好 ． 小波变换具有多尺度 、 多分辨率、 时频聚焦等特点， 它能 很好地表述图像的时频局域化特性 ． 然而小波变换缺乏方向 性， 只能检测水平， 垂直， 对角方向上的信息， 所以不能充分表 ［3 ， 4 ］ ． 现图像的信息 为了克服小波变换的局限性， 近年来人们提出了多种新 ［5 ］ 的多 尺 度 图 像 变 换 方 法， 如 曲 波 （ Curvelet ） 变 换 、 脊波 ［6 ］ ［7 ］ （ Ridgelet） 变换 和轮廓波 （ Contourlet ） 变换 等． 其 中 Arthur 等人［8］ 提出的非下采样（ Nonsubsampled ） 轮廓波变换［9］ 采用非下采样金字塔分解及非下采样方向滤波器组为基础对 图像信号进行分析处理， 能更好的表现图像信号的细节特征 ． 本文通过研究非下采样轮廓波变换和偏微分去噪模型， 提出 了一种新的图像去噪方法， 首先对含噪图像进行非下采样轮 廓波变换， 之后通过不同阶次的偏微分方程对变换后得到的
小 型 微 型 计 算 机 系 统 Journal of Chinese Computer Systems
2013 年 2 月 第 2 期 Vol. 34 No． 2 2013
结合非下采样轮廓波变换和混合阶次图像扩散的图像去程学院，北京 100083 ） Email： dongchaodc100@ 126． com
Image Denoising Using Hybrid Image Diffusion and Nonsubsampled Contourlet Transform
DONG Chao， WANG Zhiming
（ School of Computer and Communication Engineering，University of Science and Technology Beijing，Beijing 100083 ，China）
WAVELET TRANSFORM MOMENTS FOR FEATURE EXTRACTION FROM TEMPORAL SIGNALS
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. 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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. 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一种基于曲波变换的图像增强方法
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。
频率切片地震数据重建_张恒琪
式，采用基于曲波变换的凸集投影算法进行数据插值重建；并在此基础上，提出直接对频率切片进行数据重建的思
想。实验表明：2 种方式都能很好的对缺失数据进行重建，但是相较于对时间切片循环处理重建，直接对频率切片
处理重建的方式重建效果更好，运算时间更快，信噪比更高。对工业化生产有一定的应用价值。
关键词：地震数据重建；凸集投影算法；时间切片；频率切片
1 方法原理
1.1 压缩感知 （CS） 采样原理 压缩感知 （CS） 理论能够在学术界的急速兴
起，利用了大部分信号具有稀疏性或可压缩性的特 点，可以利用极低的采样率来采集数据，为地震观 测系统的设计提供了强大的理论依据。在该理论框
开发与应用
能源研究与管理 2021 （1）
窑91窑
架下我们可以通过稀疏促进策略，适当的减少炮点 与检波点的布设，极大的降低采样成本 。 [15] 将缺失 的原始采集数据重建出近似完整的数据 0，该理论 计算公式如下：
中图分类号：P631.4+4
文献标志码：A
文章编号：2096－7705 （2021） 01－0089－05
ZHANG Hengqi 1, DIAO Su 2, *, PANG Yang 1, PENG Qing 1, LIANG Shuang 1
（1. School of Geophysics and Measurement-control Technology, East China University of Technology, Nanchang 330013, China; 2. Geological Exploration Technology Institute of Anhui Province· Energy Exploration Center, Hefei 230000, China）
基于Curvelet变换的图像压缩感知重构
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
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。
纹理物体缺陷的视觉检测算法研究--优秀毕业论文
摘 要
在竞争激烈的工业自动化生产过程中，机器视觉对产品质量的把关起着举足 轻重的作用，机器视觉在缺陷检测技术方面的应用也逐渐普遍起来。与常规的检 测技术相比，自动化的视觉检测系统更加经济、快捷、高效与 安全。纹理物体在 工业生产中广泛存在，像用于半导体装配和封装底板和发光二极管，现代 化电子 系统中的印制电路板，以及纺织行业中的布匹和织物等都可认为是含有纹理特征 的物体。本论文主要致力于纹理物体的缺陷检测技术研究，为纹理物体的自动化 检测提供高效而可靠的检测算法。 纹理是描述图像内容的重要特征，纹理分析也已经被成功的应用与纹理分割 和纹理分类当中。本研究提出了一种基于纹理分析技术和参考比较方式的缺陷检 测算法。这种算法能容忍物体变形引起的图像配准误差，对纹理的影响也具有鲁 棒性。本算法旨在为检测出的缺陷区域提供丰富而重要的物理意义，如缺陷区域 的大小、形状、亮度对比度及空间分布等。同时，在参考图像可行的情况下，本 算法可用于同质纹理物体和非同质纹理物体的检测，对非纹理物体 的检测也可取 得不错的效果。 在整个检测过程中，我们采用了可调控金字塔的纹理分析和重构技术。与传 统的小波纹理分析技术不同，我们在小波域中加入处理物体变形和纹理影响的容 忍度控制算法，来实现容忍物体变形和对纹理影响鲁棒的目的。最后可调控金字 塔的重构保证了缺陷区域物理意义恢复的准确性。实验阶段，我们检测了一系列 具有实际应用价值的图像。实验结果表明 本文提出的纹理物体缺陷检测算法具有 高效性和易于实现性。 关键字: 缺陷检测；纹理；物体变形；可调控金字塔；重构
Keywords: defect detection, texture, object distortion, steerable pyramid, reconstruction
II
基于区域特性的Curvelet变换图像融合算法
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
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）。
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R 2 as:
R ψ A , B ,θ
2
CRT f (a, b, θ ) = ∫
f ( x)dx
(3)
For a straight line with the singular multi-variable function, Ridgelet transform has a good approximation performance. [4] I. INTRODUCTION Wavelet transform has been widely used in the traditional methods to remove noise from image .Though the wavelet transform have the best bases when it represents target functions which has dot singularity[1], it can hardly get the best bases when it present the singularity of line and hyper-plane. This makes the traditional two-dimensional wavelet transform in dealing with the image have some limitations. To overcome the above-mentioned shortcomings that Wavelet transform has, Donoho, and some other people promote Curvelet transform theory. The anisotropy of Curvelet transform theory is very conducive to present the edge of the image. B. Radon transform Ridgelet transform is achieved by Radon transform in the domain of one-dimensional wavelet transform. For the function f ( x, y ) in ( x, y ) ∈ R 2 plane, the Radon transform is the function at all angles on a straight line projection,[5] [6]as:
R f ( ρ , θ ) = ∫∫ f ( x, y )δ ( ρ − x cos θ − y sin θ )d ρ dθ ( ρ , θ ) ∈ [0, 2π ) × R
(4)
δ is the unit pulse function, and the Ridgelet transform coefficient R f ( a, b, θ ) of f ( x, y ) can be carried out on its
Ridgelet function
ψ a ,b,θ : R 2 → R 2
of the two-dimensional
space is defined as:
ψ a , b ,θ ( x ) = a 1 / 2ψ ( x1 co s θ + x 2 sin θ − b ) / a
(2) And, a represents the Ridgelet scale; b stands for Ridgelet position; θ represents Ridgelet direction. Given the dual integral function f ( x ) , we can define continuous Ridgelet transform (CRT) in the
components [7]. For the N × N image f, the first break will be:
transformed noise-image, choose a
f = P0 f + ∑ Δi f 0
i =1
I
×n as the processing subject.Fig.2 indicate the windowShink when n is 3. Each of d i , j following the processing below (If
∑Δ f
i i =1
I
windowshrink
3× 3
Smooth Partitioning [8]. Each sub-band highfrequency sub-divided into a number of pieces, with different sub-component division of the sub-block size can be different.
of removing noise. This article applies the theory in Curvelet parameter processing for removing noise. Set
d i , j is the parameter which is from curveletd i , j centered window of n
2010 International Conference on Measuring Technology and Mechatronics Automation
New Method Based on Curvelet Transform for Image Denoising
Donglei Li, Zhemin Duan, Meng Jia
(6)
∫
ψ (ξ ) ξ
2
2
dξ < ∞
(1)
C. Curvelet transform The edges of natural image are almost in curve, so the Ridgelet analysis of the images of the entire single-scale is not very effective. To singular curve with the multi-variable function, its performance is only close to the equivalent of wavelet transform. In order to solve the singular curve with the multi-variable function of the sparse approximation problem,
Abstract—A new method to remove noise form image is described in the article. Curvelet transform that combines both WindowShrink and BayesShrink can be used to complete the processing. Though the Wavelet transform can do the job well, it has low Resolving rate in high frequency area and it also lacks of the direction in dealing with images. Curvelet transform have an efficient way of representing the line and surface property of image. If the WindowShrink theory and BayesShrink theory are combined, the results are better. Firstly, the image should be done by Curvelet transform, then, the noise should be declined basing on Wavelet theory and the combination of WindowShrink and BayesShrink. The results of the method described in the article are better from both PSNR and the disposed image. Keywords-curvelet transform; image denoise; hard threshold; adaptive coefficient
Radon transform coefficients of wavelet transform to be
II.
THEORIES OF CURVELET TRANSFORM
Cruvelet Transform was developed on the basis of Ridegelet Transform, and Ridgelet transform is introduced first. A. Ridgelet transform Ridgelet transform (RIT) overcomes the weakness of wavelet transform representing in two or higher dimensions[2]. Ridgelet transform can be defined as follows [3]. Set satisfying the conditions:
Department of Electronics and Information Northwestern Polytechnical University Xi’An, China 20003214@; zhemind@; tiansh ∫ R f ( ρ , θ )a −1 2ψ [( ρ − b) / a ]d ρ