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The Non-local Means Based On Model For Polarization SAR Data Speckle Filtering

Guohui Yang1 , Shuang Wang2，Bo Zheng1， Wude Xu1

1.Institute of sensing technology, Gansu Academy of Sciences; Gansu Engineering Research Center of Sensor Technology; Gansu Senor and application Technology Center. Lanzhou, China 2.Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Institute of Intelligent Information Processing, Xidian University, Xi'an 710071, P.R. China

Abstract—The advancement of SAR technology with high problem by reducing the accuracy of image segmentation resolution and multiple polarization data demands better and and classification. The speckle reduction problem is more efficient speckle filtering algorithms. During the past few complicated for polarimetric SAR than a single polarization years, the non local(NL) means[1]have proved their efficiency SAR, because of the difficulties of preserving polarimetric for image denoising. In its original version, the NLmeans deal properties and of dealing with the cross-product terms. with additive white Gaussian noise, but several extensions have In this paper, we applies non local(NL) means based on been proposed for non-Gaussian noise. In this paper ,we Bayesian framework to Pol-SAR estimation. Our algorithm applies the Bayesian Nonlocal Means Filter proposed in [2]to speckle noise model of the polarimetric SAR data has been based on the speckle noise model of the polarimetric SAR multiplicative–additive speckle noise model[3]. The proposed for speckle filtering. Experimental results show the filter seems to deal well with the preservation of polarimetric effectiveness of our scheme presenting a good property properties and statistical characteristics of Pol-SAR data. exceeding current state-of-the-art methods. Results are given on synthetic and L-Band E-SAR data to validate the proposed method.The impact of using this polarimetric speckle filtering on terrain classification is quite Ⅱ. SPECKLE NOISE OF POLARIMETRY SAR dramatic in boosting classification performance. Airborne A. POLARIMETRIC SAR DATA DESCRIPTION polarimetric radar images are used for illustration. Index Terms—denoising, polarimetric SAR image, speckle The 2X2 complex scattering matrix [S] is employed to filtering characterize a resolution cell of polarimetric SAR data .The scattering matrix in the linear polarization base can be can be parameterized as follows Ⅰ. INTRODUCTION 轾S S [S ] = hh hv 犏S S PECKLE appearing in synthetic aperture radar (SAR) 臌vh vv S (1) images is due to the coherent interference of waves reflected where Shh is the scattering element of horizontal transmitting from many elementary scatterers. Speckle characteristics and horizontal receiving polarization, and the other three and speckle filtering for a single polarization SAR image elements are defined similiarly. For the backscattering have been investigated in many studies [1], [2]. Polarimetric direction,due to the reciprocity theorem under the BSA SAR responses for reciprocal media can be considered as S = S the interactions of three correlated coherent interference convention[4], hv vh . The scattering matrix [S] that processes: horizontal transmitting and horizontal receiving isexpressed vectorially by means of the lexicographic (HH), horizontal transmitting and vertical receiving (HV), orthogonal basis for 2 × 2 complex matrices [5],leading to and vertical transmitting and vertical receiving (VV) the target vector k polarization channels. Speckle noise not only appears in the k= [Shh , 2 S hv , Svv ] three intensity images, but also in the complex, cross- (2) product terms. Speckle complicates the image interpretation where the superscript T indicates the matrix transpose.The obtained at the expense of spatial resolution. The statistics

2 on the Shv term is instroduced to maintain balance in the of the covariance matrix Z have been found to be the span. For point scatters,[S] can characterize completely the Wishart pdf W([ C ], n ) [7], [8] scattering process of the deterministic target.Howrver, for n-3 distributed scatterers, it fails to characterize the scattering n3n Zexp(- ntr ([ C ]- 1 [ Z ])) p[z ] ([ Z ]) = process for distributed scatterers[6] i.e.,random targets. Cn G ( n ) Based on the SAR’s coherent nature, under the Gaussian 3 scatterer assumption,k can be modeled by a multivariate, (6) G (n ) complex, zero-mean, Gaussian probability density function where 3 is 3 (pdf) N(0,[ C ]) [6], [7] 3 G3 (n ) =p G( n - i + 1) i=1 1 H -1 pk (k)=3 exp( - k [ C ] k) (7) p [C ] tr(▪) is the matrix trace. (3) As given in(5),C is estimated from a finite number of where H represents the transpose complex conjugate of a samples L.Since the estimated covariance matrix Z is itself a multivariate random variable,it will present an error might vector,and [C ] denotes the determinant of [C ] . This pdf be considered as being produced by a noise component. The is completely determined by the 3X 3 complex, Hermitian, advantage of such a characterization is that an optimized covariance matrix[C ] , defined as filtering might be envisaged. In [3], the authors have shown [C ]= E { kk H } that speckle noise for PolSAR data, under the hypothesis

2 that data are distributed according to (3), must be considered 轾 E S2 E{ S SH} E{ S S H } 犏 { hh} hh hv hh vv as a combination of multiplicative and additive noise 犏 2 components, which represents a generalization of the fully = 2E S SH 2 E S 2 E S S H 犏 { hv hh} { hv} { hv vv } multiplicative noise model for the diagonal elements of C. 犏 2 犏E{ S SH} 2 E{ S S H } E S The model has also been generalized for multilook PolSAR 臌 vv hh vv vh{ vv } data [16]. (4) where E{x} indicates mathematical expectation x. Based on B. Polarimetric SAR Speckle Noise Model (3),the randomness of k is completely determined by C.C The PolSAR data speckle noise model represents a noise contains all the necessary information to characterize the model for Z. As one may observe in (4), all the elements of distributed scatter under observation. Most of natural scenes this matrix are obtained as the Hermitian product of two are considered as distributed scatters, therefore they are components of k. Hence, the final model for Z may be completely determined by[C] and not by [S].This difference derived from a generalization of a speckle noise model for H comes from the fact that [C] contains information the Hermitian product of two SAR images Su S v , where concerning the data’s correlation structure.[C] must u, v hh , hv , vv .The Hermitian product of two SAR estimated from the original data [S]. This process is alse { } { } considered as the preliminary PolSAR despeckle process. images, for single-lookSAR data, under the hypothesis that SAR data are frequently multilook-processed for speckle (2) is distributed as (3),may be modeled as [3] reduction by averaging several neighboring.Under the H assumption of statistical ergodicity and homogeneity, is Su S v=y z n N c n mexp( j f x ) obtained by substituting the statistical expectation by a +y ( r -N z )exp( j f )+ y ( n + jn ) spatial averaging c n x ar ai (8) The following enumeration details the different parameters 1 L Z= kkH = C of the above model. L l (5) L l =1 a) ψ represents the average power in the two channels 2 2 where L is the number of looks and Cl is the one-look y = E{ Su }E{ S v } C= k k H covariance matrix of the lth pixel defined as l l l .As (9) the sample covariance matrix Z is derived from [S ] ,it is b) rexp(j fx ) is the complex correlation coefficient that affected by speckle noise in such a way that the larger the S number of averaged pixels, the lower the speckle noise characterizes the correlation among the channels u and content[9], [10]. Sv In this case, speckle reduction (i.e., signal estimation) is H E{ Su S v } Ⅲ. NON LOCAL MEANS FILTER FOR r= rexp(j fx ) = 2 2 (10) SPECKLE NOSIING E{ Su } E { S v }

The amplitude of the complex correlation coefficients is A.NL MEANS FILTER normally referred to as coherence. c) Nc is a important filtering parameters The NL means filter, which denoising by finding high similarity regions to weighting, is not limited to local area of p 2 Nc = r2 F 1 0.5,0.5;2; r the image in comparison with the traditional spatial 4 ( ) (11) filter.The NL means filter is defined as where 2F 1( a, b ; c ; z) is the Gauss hypergeometric function. 1 NLz( x) = w( x, y) uy This parameter contains coherence information . C( x) y蜽

d) zn is the normalized Hermitian product amplitude (18) C( x) = w( x, y) H where denotes the weight normalized E{ Su S v } y蜽 zn = (12) y function. w( x, y) is the weight derived from computing the n e) m is the first speckle noise component characterized similarity,and the weight is computed as for presenting a multiplicative noise behavior with respect 骣 2 ux- u y to the information of interest. This noise component w( x, y) = exp琪 - 2 presents the following first- and second-order moments: 琪 h2 桫 E{ nm }= 1 (19) (13) where h means a smoothing parameter related to the noise var{nm }= 1 2 variance of image. u- u is the distance measurement for (14) x y 2 n+ jn additive Gaussian noise. This distance measurement often f) ar ai is the second complex speckle noise component presenting an additive nature with respect to used in nature images denoising. the information of interest. The components are characterized by B.NL MEANS FILTER FOR MULTIPLY NOISE

E{ nar }= E { n ai } = 0 Let v( x ) be the noisy data at pixel x , and u( x ) be the (15) noise free reflectance (intensity or amplitude for SAR 1 2 1.32 image). Also, we use v(x ) and u(x ) to denote the var{nar }= var{ n ai } = (1 - r ) 2 vectorized patches centered at pixel x with the size M M (16) . Then the BNL filter [12] is: Equation (8) states that, for the complex Hermitian product of two channels, speckle results from the combination of the p(v ( x ) | u ( y )) p ( u ( y )) u ( y ) y蜠 ( x ) multiplicative noise source nm and the complex additive uˆ( x ) = p(v ( x ) | u ( y )) p ( u ( y )) n n= n + jn noise source a ( m ar ai ). The combination of these y蜠 ( x ) noise components is determined by the complex correlation (20) coefficient (10). This parameter must be considered for a where uˆ( x ) is obtained pixel-wised as the weighted average correct use of the model (8) when filtering the multiplicative and the additive noise sources, in such a way that the better of all grey values u( y ) in the neighborhood D(x ) of x . the estimation of the complex coherence, the better the The term p(v ( x ) | u ( y )) p ( u ( y )) acts as the similarity filtering. v( x ) u( y ) Considering (8), the sample covariance matrix for the measure between and . single-look and the multi-look data cases may be written as Here we use intensity image as an example for the follows derivation of the new form of the BNL filter. For the

Z= C + Nm + N a amplitude case, similar expression can be obtained. (17) Consider the case of fully developed and independent p(v ( x ) | u ( y )) The matrix Nm contains the information concerning the speckle samples, the conditional distribution multiplicative speckle noise components, whereas Nm in (20) can be written as: contains the information about the additive ones M M model-based PolSAR(MBPolSAR) filtering technique, p(v ( x ) | u ( y ))= p ( vm ( x ) | u m ( y )) (21) improving the estimation of the off-diagonal elements of the m=1 covariance matrix. For this reason, it is important to detail u( y ) v( x ) where m and m represent the mth pixel in the first the origin of (8) with the objective to define in the corresponding patches, respectively. Assume that um ( y ) is following the process, based on this model, to filter speckle noise. among the dictionary of the possible reflectance of v( x ) , m Lee et al. established in [11] three principles for Pol- the conditional pdf p( vm ( x ) | u m ( y )) for an L -look intensity SAR data filtering which may be extended to any type of SAR image can be expressed as [13],[14]: multidimensional SAR scheme. These principles are: L-1 L 1) To avoid crosstalk between polarization v( x) 骣L 骣 Lv( x) p v x| u y =m exp - m channels,each element of the covariance matrix must be ( m( ) m ( )) 琪琪 G(L) 桫 um( y) 桫 u m ( y) filtered independently in the spatial domain. Filtering (22) algorithms exploiting the degree of statistical where G( ) is the gamma function. Based on (21), (22) can independence between elements of the covariance matrix will introduce crosstalk. be rewritten as [2] 2) To preserve polarimetric properties, each term of the p(v ( x ) | u ( y )) covariance matrix should be filtered in a manner similar M M to multi-look processing by averaging the covariance 骣 1骣vm ( x) L - 1 �exp琪+琪 - ln(um( y)) ln ( v m ( x)) matrices of neighboring pixels. All terms of 琪 h2 琪u y L 桫 m=1 桫m ( ) thecovariance matrix should be filtered by the same (23) amount. where h is the smoothing parameter. Let s 2 be the 3) To preserve features, edge sharpness, and point variance of speckle, the linear relation can be assumed, i.e. targets,the filtering has to be adaptive and should use a h= ks , as in [10]. It is confirmed that the constant k 2 is homogeneous area from selected neighboring pixels. The local statistics filter (or other adaptive filtering good choice for intensity image. For an L -look SAR algorithms) should be applied in edge-aligned windows. intensity image, we have s = 1/ L [15]. Note that in (4) These principles were established, in part, under the the weight for u(y ) depends on the ratio between v(x ) and assumption of a multiplicative noise model for speckle. As it u(y ) , which can better reflect the multiplicative has been shown previously, this model is not able to characterize speckle for the off-diagonal elements of the characteristics of speckle. sample covariance matrix. One of the objectives of the work Comparing Equations (19) and (23), it is easy to get a presented in this paper is to analyze in-depth the role of the new statistical distance measure for SAR image as follows: previous principles,showing at the end, that these principles M 骣vm ( x ) L - 1 may be relaxed without a loss of information. d( vm ( x ), v m ( y ))=琪 + In( v m ( y )) - In( v m ( x )) m=1桫vm ( y ) L The second of the previous principles states that the (24) which is a ratio based distance measure. By substituting the 2 NLM filtering u( y ) v( y ) 2 true value m back for m ,Equation (23) can be Sh 揶 E{ Sh rewritten as for multiply noise M 骣vm ( x ) L - 1 Fig,1. processing chain filtering for the diagonal d( vm ( x ), u m ( y ))=琪 + In( u m ( y )) - In( u m ( x )) m=1桫um ( y ) L elements of Zn (25) Therefore, w( x, y) is the weight derived from the new elements of the sample covariance matrix must be processed in the same manner. As it has been demonstrated, the nature distance measure,and the weight is rewritten as of speckle noise for the off-diagonal elements of the sample covariance matrix varies according to the complex 骣 d( vm ( x ), u m ( y )) coherence. Consequently, the elements should be processed w( x, y) = exp琪 - (26) 桫 h2 differently, according to this complex coherence parameter. Hence, the degree of correlation should be employed to Ⅳ. POLARIMETRIC SAR DATA optimize speckle noise reduction. As it will be demonstrated FILTERING at the end of this manuscript, the relaxation of the principles established in [11] will not translate into a loss of This section will detail and analyze the BNL in the Phase 

 2 Estimation Correlation Compute Sh correlation coefficient  Nc  2 Sv

NLM filtering for Calculate * Amplitude Complex  * S S multiply noise bias B E Sh S v  h v

Fig,2. processing chain filtering for the non-diagonal elements of Zn information, but into an improvement of the speckle noise amplitude of the corresponding Hermitian product of SAR filtering process. images. The speckle noise model established by (8) and (17) is Consequently, the last action to be performed is to now considered to define a novel strategy for PolSAR eliminate the multiplicative speckle noise component. This speckle noise filtering, which in the following will be is filtered considering the same spatial filtering employed referred as Model Based PolSAR (MB-PolSAR) speckle for the diagonal elements of Z . At this point, is is filter.The most relevant characteristic of this new approach n necessary to eliminate a slight bias as a consequence of is that it will adapt to the complex correlation characterizing eliminating the second additive term of (4). This bias, must each entry of the sample covariance matrix in order to be easily eliminated by multiplying the retrieved result by optimize speckle noise filtering. Consequently, as the [5] complex correlation differs in the different entries of the r sample covariance matrix, these entries shall be processed B = differently. Unlike other multidimensional speckle noise p 2 r2F 1(0.5,0.5;2; r ) filtering techniques, the MB-PolSAR approach will consider 4 specifically the additive speckle noise component. (27) The diagonal elements of the sample covariance matrix, Finally, due to the data variability, the retrieved presenting fully multiplicative speckle, are processed by any correlation is checked to avoid non valid values. At this of the alternatives present in the literature to reduce point, one may see that the filtered Hermitian product may multiplicative speckle. In the implementation of the MB- be employed to improve the estimation of complex PolSAR filter presented in this paper, for the sake of correlation. Consequently, the previous algorithm is iterated, simplicity, a multilook approach is considered, see Fig. 1. since the improved estimation coherence helps to improve The process to filter speckle noise in the off-diagonal the separation of the multiplicative and the additive speckle elements of Zn differs considerably from the process noise components. Tests performed on simulated and specified for the diagonal elements, as detailed in Fig. 2. experimental Pol-SAR data have shown that about three The main difference is that the filtering process must adapt iterations of the proposed scheme result into an improved to the complex correlation coefficient. Consequently, the capability of speckle noise reduction. first step to reduce speckle for these elements of Zn is to For Pol-SAR data, the filtering is based on the estimate the complex correlation coefficient. One alternative covariance matrix C . The diagonal and off-diagonal to perform such an estimation may be to consider an spatial elements of the sample covariance matrix are respectively averaging. Nevertheless, since the separation of the processed. multiplicative and the additive speckle noise components depends on a correct, i.e., unbiased, estimation of the sample Ⅴ.FILTER EVALUATION AND RESULTS correlation coefficient, the alternative presented in [7] is In order to quantify the proposed method, test images are considered. This coherence estimation approach is also acquired from the website:http://earth.eo.esa.int/polsarpro/. based on the model (4). After coherence is estimated, it is Three sets of POLSAR dates are used here for illustrating transformed to the parameter Nc through (7). the effectiveness of the our algorithm. The color images are At this point of the processing chain, the additive produced by using Pauli vector display method in Fig.3 and speckle noise component is eliminated since it is possible to 4 . synthesize the first additive term of (4) by considering the The importance of point targets preserving is introduced product of the complex term Nc exp( jf x ) by the in SAR filtering, so here we did three sets of experiments. In Table I Numerical Evaluation For Real Pol-SAR

Re-Lee Im-Sigma BM-NLM Idea Value

Mean 1.0296 0.9838 1.0224 1 Ottawa Ratio image Var 0.1563 0.1142 0.1675 0.2438

(span) HD 0.8530 0.8669 0.8685 1 EPD-ROA VD 0.8443 0.8897 0.8388 1

Mean 1.0094 0.9983 0.9747 1 Fland Ratio image Var 0.2426 0.1978 0.2436 0.3466

(span) HD 0.7496 0.8158 0.8390 1 EPD-ROA VD 0.7710 0.7758 0. 7870 1 the First one, we explain the disadvantages of the improved m ˆˆ sigma POLSAR filter and the NL means filter for XD1( i) X D 2 ( i) multiplicative noise in POLSAR. Then a comparison of the i=1 EPD- ROA = m (29) filtering results in computing filtering weight with or Y i Y i without the SPAN data is made in the second experiment. O1( ) O 2 ( ) i=1 And in the last experiment, our filtering algorithm in ˆ POLSAR data is compared with other classical filter. where m is the pixel number of the selected area. XD1 ( i) A. Evaluation Index-of Despeclking Performance and ˆ represent the adjacent pixel values of the In this paper, EPD-ROA and ENL of quantitative XD2 ( i) evaluation indicators are applied to evaluate the despeckled image along a certain direction. Similarly, performance of the different filters. Y i and Y i represent the adjacent pixel values of 1) ENL: ENL measures the degree of speckle reduction in O1 ( ) O2 ( ) a homogeneous region, which is given by [16] the original noise image. EPD-ROA for the whole image 骣I along the horizontal (HD) and vertical (VD) directions. mean when the EPD-ROA is closer to one, it means better ability ENL= Hc 琪桫Is of edge preservation.

1, intensity image H = (28) c 4p - 1, amplitude image where Imean and Is are the mean and the standard deviation of a chosen homogeneous region, respectively. In practice, we should artificially choose the homogeneous (a) (b) region to be as large as possible. A large ENL value corresponds to better speckle suppression. 2) EPD-ROA:EPD-ROA measures the edge-preservation degree of speckle reduction in a whole image, which is given by [17]

(a) (b) Fig.3.Experimental resultes for Ottawa (a) Original Pol- SAR image (b) Re-Lee (c) Im-Sigma (d) BM-NLM contrasts for these results indicate that the speckle B. Experimental Results for Real Pol- SAR Data suppression ability of BM-NLM is excellent. The improve In this section, two real SAR images are tested; visual sigma filter produces distinct speckle residue within uniform and numerical results are obtained for these images. Two regions and edge. In contrast, BM-NLM effectively removes real SAR images are tested: 1) a L-band experimental Pol- the speckle in uniform regions, for example, the left area of SAR dataset acquired by the CONVAIR sensor of the Ottawa[Fig . 3 (d)] and the middle area of Flevoland [Fig. Canadian Space Agency(CSA) on the Ottawa test site(Fig. 4(d)].Compared with the other filters, BM-NLM possesses 3(a) and named Ottawa); 2) a L-band experimental PolSAR the best smoothing ability in uniform regions, which is dataset acquired by the AIRSAR sensor of the Jet validated by the ENL comparisons in Table II. In terms of Propulsion Laboratory (JPL) on the Flevoland test site(Fig. edge and texture preservation, Improved sigma filter can 4(a) and named Flevoland). The results of our approach retain the edge and texture features to a certain extent, but (non-local means based on model for Pol-SAR speckle, the results for uniform regions are not satisfactory. Refined named BM-NLM)for these real Pol-SAR data have been Lee filter keeps the texture features while resulting in edge compared with those of refined lee filter(named Re-Lee) blurring. In contrast, our algorithm preserves the texture [11]and improved sigma filter(named Im-Sigma)[14]. The well and produces clear edges and without artifacts .Table I sliding window size was 7x 7 for the Re-Lee and Im-Sigma reveals that BM-NLM obtained the optimal EPD-ROA in filters. The despeckled results for the different algorithms most cases compared with the other algorithms. are shown in Figs. 3–4. Table I gives all the numerical evaluation results for these images, which contains the evaluations of ratio image and EPD-ROA along HD and VD for the whole image. Equivalent Number of Looks(ENL) is computed to compare the proposed method with other denoising algorithms .Table II gives the ENL evaluation results for these images,which can estamte smoothing ability in uniform regions[note: three uniform regions are chosen in Ottawa and Flevoland, respectively, which are marked in Figs. 3(a) and 4(a)]. Table II ENL Comparison (a) (b) Im- BM- Original Re-Lee Sigma NLM Region 1 44.7458 136.5988 262.9014 308.5393 Region 2 11.4444 51.1404 52.2230 114.3741 Region 3 40.2410 323.7267 246.0496 552.4424 The visual effect comparison of the various filtering results in Figs. 3 and 4 indicates that BM-NLM is free of fuzzy in edge regions , and the performance smoothness markedly precedes the Re-Lee and Im-Sigma filter .A very clean uniform region is obtained with our approach, while speckle residue appears in the results obtained with the other (a) (b) filters .In terms of edge preservation, BM-NLM produces Fig.4.Experimental resultes for Fleveland (a) Original Pol- clear and sharp edges, while “blur” phenomenon appears in SAR image (b) Re-Lee (c) Im-Sigma (d) BM-NLM the results obtained with the Re-Lee. The Le-Lee possesses nicer ability of smoothing , but it produces edge and point Ⅵ. CONCLUSIONS target blurring, particularly, in Fig. 4(b).The results shown In this paper, we develop a non local means based on in Fig. 3 show that BM-NLM possesses comparative maodel for the reduction of speckle noise in Pol-SAR. Our preserving performance of texture and small objects method can amplify the important features, and distinguish compared with the Re-Lee and Im-Sigma. Im-Sigma can edge structures better from noise. As can be seen, the hold a similar preserving performance, whereas edge despckled images using our scheme have better results in blurring is unavoidable. Therefore, the finer performance of the visual effect and the property of numerical evaluation . edge preservation is one of the virtues of our method compared with the other filters. In the Figs. 3 and 4 show the visual results for the tow real Pol-SAR images. The [10] R. Touzi and A. Lopes, “The principle of speckle filtering in polarimetric SAR imagery,” IEEE Trans. Ⅶ. ACKNOWLEDGMENTS Geosci. Remote Sensing, vol. 32,pp. 1110–1114, Sept.  1994. This work was supported by the Gansu Academy of [11] Jong-Sen Lee, M.R. Grunes, and G. de Grandi, Sciences Youth Foundation under Grant 2015-QN-06, the “Polarimetric SAR speckle filtering and its implication National Natural Science Foundation of China (Grant No. for classification,” IEEE Trans. Geosci.Remote 60702062, 60971128), the National High Technology Sensing, vol. 37, no. 5, pp. 2363–2373, Sept. 1999. Research and Development Program (863 Program) of [12] C. Kervrann, J. Boulanger, and P. Coupe, “Bayesian China (Grant No. 2008AA01Z1 25), the China Postdoctoral non-local means filter, image redundancy and adaptive Science Foundation Special funded project (No. 200902587) dictionaries for noise removal,” in Int. Conf. Scale ，the China Postdoctoral Science Foundation funded project Space Methods and Variational Methods in Computer (No. 20090461285) and the Program for Cheung Kong Vision, pp. 520–532, 2007. 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