Multi-polarimetric SAR image compression based on sparse representation

Transcription

1 . RESEARCH PAPER. Special Issue SCIENCE CHINA Information Sciences August 2012 Vol. 55 No. 8: doi: /s Multi-polarimetric SAR based on sparse representation CHEN Yuan, ZHANG Rong &YINDong Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei , China Received November 28, 2011; accepted May 13, 2012; published online June 22, 2012 Abstract The use of sparse representation in signal and image processing has gradually increased over the past few years. Obtaining an over-complete dictionary from a set of signals allows us to represent these signals as a sparse linear combination of dictionary atoms. By considering the relativity among the multi-polarimetric synthetic aperture radar (SAR) images, a new compression scheme for multi-polarimetric SAR image based sparse representation is proposed. The multilevel dictionary is learned iteratively in the 9/7 wavelet domain using a single channel SAR image, and the other channels are compressed by sparse approximation, also in the 9/7 wavelet domain, followed by entropy coding of the sparse coefficients. The experimental results are compared with two state-of-the-art compression methods: (set partitioning in hierarchical trees) and Because of the efficiency of the coding scheme, our method outperforms both and 2000 in terms of peak signal-to-noise ratio (PSNR) and edge preservation index (). Keywords multi-polarimetric SAR, sparse representation, multilevel dictionary learning, edge preservation index () Citation Chen Y, Zhang R, Yin D. Multi-polarimetric SAR based on sparse representation. Sci China Inf Sci, 2012, 55: , doi: /s Introduction Synthetic aperture radar (SAR), as a high-resolution active microwave remote sensing tool, has been widely used in remote sensing applications, including military reconnaissance, terrain mapping, and target recognition. One of the most important research directions in this field is the development of multifrequency, multi-polarimetric and multi-mode SAR. Multi-polarimetric data consists of four channels, denoted by HH, HV, VH and VV. Because SAR can be working in all weathers during both day and night, the volume of data collected is increasing rapidly. The requirements for remote sensing information storage on the ground are therefore increasing, which makes it necessary to study more efficient compression algorithms for the multi-polarimetric SAR data. The traditional compression algorithms for SAR images were only concerned with intensity images and directly adopted wavelet [1,2] algorithms or other efficient methods for optical images. However, these methods are often overly complicated and inefficient for SAR images, because of their unique features when compared with optical images. First, the SAR images are contaminated with multiplicative noise, Corresponding author ( johnson@mail.ustc.edu.cn; zrong@ustc.edu.cn) c Science China Press and Springer-Verlag Berlin Heidelberg 2012 info.scichina.com

2 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No which is generated in the process of imaging with coherent radiation. Second, the images contain large homogeneous regions and regions with detailed texture. It would be appropriate to allocate relatively few bits for the homogeneous regions, while preserving as much edge information as possible at the textured regions. Third, SAR images have a wide dynamic range, which gives image data that is unlike that provided by other Earth-imaging sensors. This paper is concerned with multi-polarimetric SAR intensity images only. Wavelet packet embedded block (WPEB) coding for multi-polarimetric SAR data compression has been proposed [3], but WPEB compresses all of the polarimetric channels without considering the redundancy among them. A 3D-matrix transform based method was also investigated [4], which removes not only the redundancies within the image but also the redundancies among the polarimetric channels. In this paper, a new sparse representation based compression method is proposed which also considers the redundancy among the polarimetric channels. The multi-polarimetric SAR images represent the same texture information over the same area, which means that there is some redundancy present among the polarimetric channels. The redundancies in the polarimetric images and among the polarimetric channels should therefore be considered in polarimetric SAR. The redundancy of an over-complete dictionary can effectively capture the structural characteristics of the images [5], and has been successfully used for facial [6,7]. These approaches use a piecewise-affine warping of the face, and this pre-processing ensures that the various facial features coincide with the pre-specified facial template. Each block of the facial template defines a class of signals and the warping parameters are sent as side information. However, this affine warping procedure cannot be used for SAR with sparse representation. We therefore approximate the SAR images as a sparse representation over the multilevel over-complete dictionary in the transform domain. First, a single polarimetric channel SAR image is chosen to be compressed losslessly and is then prepared for training. This SAR image is then downsampled to different resolutions, and is blocked into overlapping patches, which removes blocking artifacts. Then, the dictionaries are learned in the 9/7 wavelet domain with the RLS-DLA (recursive least squares dictionary learning algorithm) [8], which improves the sparse representation capabilities. To increase the structural characteristics of the dictionaries, a hierarchical energy based learning approach is used to learn each multilevel dictionary. Finally, the coefficients are processed using uniform quantization followed by entropy coding. The results are compared with those of, (wavelet packet transform) and the state-of-the-art (set partitioning in hierarchical trees) and 2000 compression methods, and show that the proposed method provides better performance in terms of both peak signal-to-noise ratio (PSNR) and edge preservation index (). The paper is organized as follows. In Section 2, we present the proposed sparse representation algorithm. Section 3 describes the quantization and entropy coding of the compression scheme. Section 4 incorporates the simulation results and Section 5 gives our conclusions. 2 The basic idea of sparse representation is the use of a redundant basis instead of the traditional orthogonal basis, and the over-complete dictionary should include as many of the signal s information structures as possible. We define a dictionary D = {d k } K k=1,whered k R N, with N<Kimplying the redundancy. Every d k is a column vector and D R N K. We can approximate the signal vector x as a sparse representation over the over-complete dictionary as x = K w k d k + x r (1) k=1 with a constraint on the coefficients w 0 s. The joint optimization of the sparse representation and the dictionary learning is proposed as min x w,d Dw 2 F, subject to w 0 s, (2)

3 1890 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 Table 1 Correlation coefficients of four polarimetric SAR images HH HV VH VV Average HH HV VH VV where s is the sparsity of the representation of the training vector, F is the Frobenius norm and 0 is the l 0 norm. 2.1 Generation of the training vectors The polarimetric SAR image consists of four channels, which are denoted by HH, HV, VH and VV. To choose which of the channels is to be trained, the correlation coefficients of these four images are considered, as shown in Table 1. We calculate the average correlation coefficients for each row and find the maximum to be that of the HV channel (the HV channel and the VH channel are very similar, so selection of either channel is acceptable). The HV channel SAR image is therefore chosen for training for dictionary learning. Sparse approximation using a dictionary can also be done in the transform domain, and provides better compression performance than in the pixel domain [9]. All of the dictionaries have N = 64, corresponding to 8-by-8 patches with three-level dyadic 9/7 wavelets in the wavelet domain. To remove the blocking artifacts, overlapping patches are prepared. Each patch is made into a training vector simply by lexicographical ordering. The training vectors are then picked in a random order from the training wavelet coefficients, and are presented for the dictionary learning algorithm. Wavelet transforms can give multi-scale and multi-resolution structural information, but the 8-by-8 patch is only suitable for three-level wavelet transforms. For greater structural detail for the learned dictionary, the HV channel SAR image must deal with down-sampling. One HV channel SAR image with a size of was down-sampled to pixels, and was then down-sampled again to achieve a pixel image. All three images with different resolutions were prepared for wavelet transform processing and generated 8-by-8 patches in the 9/7 wavelet domain. 2.2 Multilevel dictionary learning When the training vectors of the HV channel SAR image in the 9/7 wavelet domain are generated, dictionary learning is used. The success of the wavelet coding methods is mainly attributed to the data organization and representation strategies that use the nature of the wavelet coefficients well. When sparse representation is used in an scheme, both the quantized coefficients and the positions of these coefficients should be entropy coded. The learned dictionary should therefore give a sufficiently sparse approximation of the images with moderate size. The training vectors are usually collected as columns in a matrix X of size N L. The aim of dictionary learning is to find a dictionary D of size N K with unit l 2 -norm columns, and a coefficient matrix W of size K L such that the representation error, R = X DW, is minimized and W fulfills a sparseness criterion. The dictionary learning problem can therefore be formulated as follows: {D opt,w opt } =argmin D,W L w l 0 + γ X DW 2 F. (3) l=1 An optimization strategy, which does not necessarily lead to a global optimum, can be found by splitting the problem into two parts, which are solved alternately within an iterative loop. The two parts of the problem are: 1) keeping D fixed, and finding W ; and 2) keeping W fixed, and finding D. The first part, where the dictionary D is fixed, is known to be an NP-hard problem [10]. There are many modified versions of the matching pursuit (MP) methods for this problem, including basic MP (BMP)

4 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No Table 2 Algorithm for building a multilevel dictionary Input X = {x i } L i=1,ann L matrix of training vectors. S, the number of levels of the dictionary. K s, the number of atoms in a level s dictionary, where s = {1, 2,...,S}. ε s, the error goal of level s representation. Output D = {ψ s} S s=1, an adaptive multilevel dictionary for all S levels. Algorithm Initial: s =1andR 1 = X, andd is made by the first K random training vectors. While s S {ψ s,w s} =RLS DLA(R s,k s,ε s) R s+1 = R s ψ sw s s = s +1 end while [11], orthogonal MP (OMP) [12], and order recursive MP (ORMP) [13,14], which is used in this work. In the second part of the problem, W is fixed and the update problem is reduced to the minimization of X DW 2 F. Three methods - the iterative least squares dictionary learning algorithm (ILS-DLA) [15], the K-SVD algorithm [16], and the RLS-DLA [8] - are often used for this problem. The real advantage of the RLS-DLA when compared to the ILS-DLA and K-SVD comes with the flexibility introduced by the forgetting factor λ. The search-then-converge scheme is particularly favorable, and the idea is to forget more quickly at the beginning of the process, and then forget less as the learning process proceeds. The final outcome of the process provides greater confidence in the quality of the dictionary. The performance of the RLS-DLA was also shown to be excellent in [8], and that is why it was selected for use in our work. In the derivation of the RLS-DLA, a time step i is introduced and C i = (W i Wi T) 1 is defined, along with the dictionary, which is the least squares minimization solution of X i DW i 2 F, i.e., D i = (X i Wi T)(W iwi T) 1. At each update step, a new training vector x i is supplied, and the corresponding weights w i are found using the previous dictionary D i 1 and a vector selection algorithm. The matrix inversion lemma on C i is used, and simple updating rules are defined as follows: C i = C i 1 αuu T, (4) D i = D i 1 + αr i u T, (5) u = C i 1 w i and α =1/(1 + wi T u) in these equations. r i = x i D i 1 w i is the representation error. An adaptive forgetting factor λ i 1instepi makes the dictionary significantly less dependent on the initial dictionary and improves the convergence properties. This is the search-then-converge scheme, which changes the update in (4) to C i =(λ 1 i C i 1 ) αuu T with u =(λ 1 i C i 1 )w i, while (5) and α are also changed. It is obvious that the patches used as the training data are highly self-similar because they can be represented by a small set of shared element features. To increase the structural characteristics of the dictionaries, a hierarchical energy based learning approach is used to achieve a multilevel dictionary. Atoms that contribute the greatest energy to the representation are learned at the first level, followed by the next set of highest energy contributing atoms, which are trained using the residual part of the representation from the previous level, and this process continues until all levels are complete. The multilevel dictionary is denoted by D = {ψ s } S s=1, and the coefficient matrix is denoted by W = {w s} S s=1. The approximation in level s is expressed as R s 1 = R s + ψ s w s, for s =1, 2,...,S, (6) where R s 1 and R s are the residuals for the levels s 1andsrespectively. Table 2 lists the complete multilevel dictionary learning algorithm, and a particularly detailed depiction of RLS-DLA is given in [8].

5 1892 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No Sparse approximation When the multilevel dictionary D for the HV channel SAR image in the 9/7 wavelet domain is prepared, sparse approximation is carried out for the other three SAR images to obtain the sparse coefficient matrix W. Most sparse representation algorithms in the sparse approximation part subtract the mean pixel value for each patch, so that the dictionary represents the image textures rather than the absolute intensities. In this work, the 9/7 wavelet transform is used for each image patch, so the large amplitudes of the DC components are not suitable for sparse approximations. To avoid large sparse coefficients, we only use the sparse approximation for the AC components of the patches to achieve W, which then minimizes X DW 2 F. The problem of finding W is an NP-hard problem, which can be written as w opt =argmin w w 0 + γ x Dw 2 2. (7) As γ increases, the solution becomes increasingly dense. Here, the l 0 norm is used, which means that the number of non-zero coefficients to be coded is as small as possible. Solutions can be found using MP algorithms. OMP is computationally more expensive than the non-orthogonal version, but typically gives significantly better results in the context of coding. ORMP is another MP variant. Like OMP, it keeps the residual orthogonal to all of the selected frame vectors. The only difference between ORMP and OMP is that ORMP adjusts the inner products before the largest value is found. The details of the ORMP algorithm are given as follows. The subspace of R N is spanned by the selected frame vectors, and is given as A, and its orthogonal complement is given as A. A frame vector, f j, can be divided into two orthogonal components, with one in A of length a(j), and the other in A of length u(j) with uniform frame equality. Lengthening each remaining or not yet selected frame vector f j by a factor of 1/u(j) makes the lengths of the components in A equal to 1. Thus, selecting the next frame is like selecting a vector from a uniform frame in A, which means that the frame vectors are orthogonalized to A and are normalized. ORMP generally selects different frame vectors than those selected by OMP, and the total residual will often be smaller. 3 The compression scheme When sparse representation is used, entropy coding of the quantized coefficients and their positions should follow the sparse representation part. The entire scheme can thus be described using the following steps. 1. The DC components of the wavelet transform patches are quantized and compressed separately. The differential pulse code modulation (DPCM) predictive method, followed by Huffman coding of the prediction errors, is used here, as in the scheme. 2. The sparse coefficients matrix W is found using ORMP based on the learned dictionary for the AC part of the wavelet coefficients. The representation error corresponds to a given target PSNR, and is treated as a stopping criterion. 3. The sparse coefficients are then quantized by simply rounding the real values into the nearest integers. 4. The quantized non-zero entries of W and their position information are put into two sequences. The position indexes are also coded using DPCM, while the quantized coefficients are processed using both end-of-block coding and run-length coding to exploit the structure. Both sequences are then coded using the recursive splitting Huffman algorithm [17]. The proposed compression scheme is shown in Figure 1. The HV image is compressed losslessly and is used for dictionary training, while the other three SAR images are compressed. Because of the relativities between the multi-polarimetric SAR images, the dictionary trained using the HV image contains characteristics which also exist in the other three channel SAR images. The advantage of our compression scheme is that the representation is sparser when compared with a wavelet transform in the same PSNR. This means that the energy in the sparse coefficients is much more concentrated. Otherwise, the coefficients matrix is larger, with almost random

6 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No Figure 1 Compression scheme with sparse representation. Figure 2 Four channel multi-polarimetric city SAR images. (a) HH; (b) HV; (c) VH; (d) VV. non-zero positions in W. The success of the wavelet coding methods is mainly attributed to strategies that use the nature of the coefficients well for their advanced entropy coding, such as and EBCOT (embedded block coding with optimized truncation). 4 Experimental results and related analysis In this section, we tested the performance of our proposed method with the,, [18] and 2000 algorithms on four channel multi-polarimetric SAR images. Although these comparative coding algorithms have not been optimized for SAR image data, they are still typically representative of current image coding algorithms. More reasonable comparative algorithms may be found in [3] and [19] for SAR in particular. Unfortunately, the executable codes were not available to us. The four channel multi-polarimetric SAR images with sizes of pixels came from RADARSAT- 2, which has 6.25 m resolution. The four images, which contain the surface features of a city, are shown in Figure 2.

7 1894 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 HH VH VV Table 3 PSNR of city SAR images for five methods at different bit rates Sparse method (db) (db) (db) (db) (db) Sparse method (db) (db) (db) (db) (db) Sparse method (db) (db) (db) (db) (db) Figure 3 HH channel of mountain SAR image and of port SAR image. (a) Mountain; (b) port. After downsampling the images twice, the three images obtained with sizes of , and pixels are used to generate the training vectors. The atoms of the multilevel dictionary are 8 8 patches in the 9/7 wavelet domain. The multilevel dictionary has 4 levels and each level has 110 atoms. The multilevel dictionary D is thus sized The target PSNR for the training is 38 db. Compression is performed on the three SAR images (HH, VH, and VV), and the PSNR results are compared with those of the wavelet based schemes and the method in Table 3. Compared to the, and algorithms, our sparse method shows excellent performance, and is also slightly better than the state-of-the-art 2000 method, except at the rate of 1.5 bpp for the VH SAR image. To show the universality of our proposed method, we tested it on two other groups of multi-polarimetric SAR images, which contain a mountain and a port. Figure 3 shows the HH channels for these two group SAR images. The PSNR results for all of the compression schemes are shown in Tables 4 and 5. From these experimental results, we see that our sparse method achieves the highest R-D performance, except for several results that are still slightly lower than those for 2000 at 1.5 bpp. The edge preservation index (), defined in (11) below, is often used to evaluate the results of SAR. Larger values signify higher compression quality, i.e., this means that the compression method for SAR images is more suitable for human eyes. Here, P e (i, j) andp b (i, j) arethe

8 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No HH VH VV Table 4 PSNR of mountain SAR images for five compression schemes Sparse method (db) (db) (db) (db) (db) Sparse method (db) (db) (db) (db) (db) Sparse method (db) (db) (db) (db) (db) HH VH VV Table 5 PSNR of port SAR images for five compression schemes Sparse method (db) (db) (db) (db) (db) Sparse method (db) (db) (db) (db) (db) Sparse method (db) (db) (db) (db) (db) original and compressed pixels of the edges in the SAR images. Figure 4 shows the performance curves for the of three group multi-polarimetric SAR images that were compressed using the five compression methods. It was found that our proposed method outperforms the other methods, especially at low bit rates. ( Pe (i, j) P e (i +1,j) + P e (i, j) P e (i, j +1) ) = ( Pb (i, j) P b (i +1,j) + P b (i, j) P b (i, j +1) ). (8) The dictionary size also affects the compression efficiency, where a larger dictionary corresponds to sparser coefficients. However, a large dictionary means that it needs more bits for the index of the nonzero coefficients. For the first group of multi-polarimetric SAR images with the city region, Figure 5 shows the compression performance with different numbers of atoms. We chose bpp as the bit rate

9 1896 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 of city HH channel SAR of mountain HH channel SAR of port HH channel SAR of city VH channel SAR of mountain VH channel SAR of port VH channel SAR of city VV channel SAR 2000 (a) 2000 (b) 2000 (c) (d) 2000 (e) 2000 (f) of mountain VV channel SAR of port VV channel SAR (g) 2000 (h) (i) Figure 4 The of five methods used for three group SAR images. (a) City HH; (b) city VH; (c) city VV; (d) mountain HH; (e) mountain VH; (f) mountain VV; (g) port HH; (h) port VH; (i) port VV. Figure 5 PSNR (a) Number of atoms EFI 70 (b) Number of atoms The PSNR and results with different numbers of atoms for the HH channel (bpp). (a) PSNR; (b). for the HH SAR. The PSNR increases with growth of the atom numbers, but when the number of dictionary atoms is larger than 220, PSNR tends to be stable. The graph of the growth curve shows that our multilevel dictionary training method can enable us to detect the image edge structure, which approaches the sensitivity of human eyes.

10 Chen Y, et al. Sci China Inf Sci August 2012 Vol. 55 No Conclusion In this paper, a new multi-polarimetric SAR algorithm based on sparse representation is proposed. The redundancy among the polarimetric channels is considered, and to enhance the structural characteristics of the training dictionaries, a hierarchical energy based learning approach is used to achieve a multilevel dictionary in the 9/7 wavelet domain. The experimental results show that our proposed compression method performs quite well when compared with and several other wavelet based algorithms, in terms of both PSNR and. In future work, improvement of the entropy coding part must be considered, and we will also look for a usable structure for the coefficient position information. Acknowledgements This work was supported by National Grand Fundamental Research Program of China (Grant No. 2010CB731904) and National Natural Science Foundation of China (Grant No ). References 1 Shapiro J M. Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans Signal Process, 1993, 41: Said A, Pearlman W A. A new, fast, and efficient image codec based on set partitioning in hierarchical trees. IEEE Trans Circ Syst Video Technol, 1996, 6: Cumming V I, Wang J. Polarmetric SAR data compression using wavelet packets in a block coding scheme. In: IEEE International Geoscience and Remote Sensing Symposium, Toronto, Zhang W C, Wang Y F, Hu G H. Compression of multi-polarimetric SAR intensity images based on 3D-matrix transform. IET Image Process, 2008, 2: Skretting K, Engan K, Husoy J, et al. of images using overlapping frames. In: 12th Scandinavian Conference on Image Analysis, Bergen, Bryt O, Elad M. Compression of facial images using the K-SVD algorithm. J Visual Commun Image Represent, 2008, 19: Zepeda J, Guillemot C, Kijak E. Image compression using sparse representations and the iteration-tuned and aligned dictionary. IEEE J Sel Top Signal Process, 2010, (99): Skretting K, Engan K. Recursive least squares dictionary learning algorithm. IEEE Trans Signal Process, 2010, 58: Skretting K, Engan K. Image compression using learned dictionaries by RLS-DLA and compared with K-SVD. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Davis G. Adaptive nonlinear approximations. Ph.D. dissertation. New York: New York University, Zhang S M Z, Mallat S. Matching pursuit with time-frequency dictionaries. IEEE Trans Signal Process, 1993, 41: Pati Y C, Rezaiifar R, Krishnaprasad P. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In: Proc Asilomar Conference on Signals Systems and Computers, Gharavi-Alkhansari M, Huang T S. A fast orthogonal matching pursuit algorithm. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Seattle, Chen S, Wigger J. Fast orthogonal least squares algorithm for efficient subset model selection. IEEE Trans Image Process, 1995, 43: Engan K, Skretting K, Husoy J H. A Family of iterative LS-based dictionary learning algorithms, ILS-DLA, for sparse signal representation. Digital Signal Process, 2007, 17: Aharon M, Elad M, Bruckstein A. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans Image Process, 2006, 54: Skretting K, Hus y J H, Aase S O. Improved Huffman coding using recursive splitting. In: NORSIG99, Asker, Sprljan N, Grgic S, Grgic M. Modified algorithm for wavelet packet image coding. Real-Time Imag, 2005, 11: Hou X, Liu G, Zou Y. SAR image data compression using wavelet packet transform and universal-trellis coded quantization. IEEE Trans Geosc Rem Sens, 2004, 42: