- Email: [email protected]
A SAR image compression algorithm based on Mallat tower-type wavelet decomposition
Accepted Manuscript Title: A SAR image compression algorithm based on Mallat tower-type wavelet decomposition Author: Jia Li Liping Chang PII: DOI: Reference:
S0030-4026(15)00749-4 http://dx.doi.org/doi:10.1016/j.ijleo.2015.07.196 IJLEO 55912
To appear in: Received date: Accepted date:
11-7-2014 29-7-2015
Please cite this article as: J. Li, L. Chang, A SAR image compression algorithm based on Mallat tower-type wavelet decomposition, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.07.196 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A SAR image compression algorithm based on Mallat tower-
ip t
type wavelet decomposition Jia Li*, Liping Chang
cr
Institute of Fiber Optic Communication & Information Engineering, College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China Corresponding author: [email protected]
us
*
Abstract: Image compression algorithms based on wavelet transform can achieve ideal
an
effect on optical image, while cannot do well in SAR image compression. The paper introduces wavelet encoding in optical image compression into SAR image processing and designs an improved SAR compression algorithm based on Mallat tower-type wavelet
M
decomposition according to characteristics of SAR image. The algorithm firstly conducts noise reduction in wavelet domain on image and then performs differential pulse lossless
d
coding on low-frequency sub-band after removing speckle noise. The high-frequency is
te
scanned for important coefficients using spatial direction tree similar to SPIHT algorithm and processed with universal trellis coded quantization. The Euclidean distance among
Ac ce p
important coefficient can be expanded to improving quantification gain. It can be seen from simulation results analysis on algorithm compression performance that the proposed improved algorithm is more suitable for SAR image compression compared with SPIHT algorithm and JPEG2000 standard. Keywords: Mallat, Wavelet, SAR, Image compression
1. Introduction Wavelet transform has spatial frequency local characteristic and multi-resolution analyzing features. It is very close to human visual system. Compared with methods on DCT, image compression algorithms based on DWT can achieve higher compression ratio, which also has not box effect in low bit rate. In recent years, emergence of JPEG2000 standard enables further development of wavelet compression technologies. Research results show that such kind of
1
Page 1 of 14
algorithms can obtain good compression effect on optical images, while compression on SAR images is unsatisfactory. Nevertheless, we can still refer to part design idea of existing algorithms. The paper introduces advanced wavelet encoding methods in optical image compression into SAR image compression and designs an improved SAR compression algorithm
ip t
based on Mallat tower-type wavelet decomposition. Finally, simulation experiment is used to analyze compression performance of the proposed algorithm.
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This paper is organized as follows: section 2 gives improvement idea of algorithms; section 3 designs the improved algorithm in detail; section 4 performs simulation experiment and
an
us
analyzes results; section 5 concludes our work.
2. Algorithm for the design progress
M
2.1 Spatial orientation tree features of wavelet decomposition As to its space-frequency localized characteristics, wavelet coefficient on same spatial location under different scale is self-similarity. Seen from multi-resolution structure of wavelet,
d
there exists quad-tree relationship between low-frequency sub-band and adjacent high-frequency
te
sub-band of wavelet coefficient to describe same spatial location, as shown in Fig. 1. If such tree structure can be effectively utilized, image compression algorithm based on wavelet transform
Ac ce p
can then be designed.
Fig. 1 describes parent-children relationship among sub-band coefficients with different resolution after Mallat tower-type decomposition. As shown in the figure, each parent node contains four child nodes or none. The coefficients from low-frequency to corresponding highfrequency constitute a spatial orientation tree. The SPIHT algorithm proposed by Said and Pearlman took full advantage of features of above spatial orientation tree. It uses bit-plane scanning method to effectively mapping wavelet coefficients for some specified thresholds. Then, they are represented with effective values. The encoding process requires three auxiliary lists, namely List of Significant Pixels (LSP), List of Insignificant Pixels (LIP) and List of Insignificant Set (LIS). The encoded stream is sorted in accordance with importance and outputted in the manner of progressive transmission. The encoding process can be ended according to requirements of target bitrates or distortion at any time. We can know from
2
Page 2 of 14
experiment result in Ref. [1] that optical image compression with SPIHT algorithm can achieve
an
us
cr
ip t
good effect.
Fig. 1 Spatial orientation tree structure
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2.2 Improvement method
The SAR image compression algorithm refers to idea of SPIHT spatial orientation tree in
d
wavelet coefficient importance scanning. Based on characteristics of SAR image, we made
te
modifications on existing algorithm as follows: firstly, SAR image is affected by multiplicative speckle noise determined by its imaging mechanism. Compared with optical image, correlation
Ac ce p
among adjacent pixels in SAR image greatly reduces. Therefore, it is necessary to carry out noise reduction before wavelet coefficient scanning. The image correlation after removing speckle noise will be greatly enhanced, which is help for improving image compression quality. Secondly, SAR image data has large dynamic range. In general, each pixel is represented with 32bit or 16bit data. For this feature, we can conduct logarithmic transformation on SAR image data. Such processing manner can not only reduce data dynamic scope, but also translate multiplicative noise into additive noise so as to facilitate denoising in the wavelet domain. Thirdly, SAR image data has richer grey scales and contrasts than optical remote sensing images. Under influence of speckle noise, similarity among wavelet sub-bands is poor. The bitplane scanning and zero-tree encoding techniques used in SPIHT algorithm have ineffective compression on SAR image because number of zero-tree after SAR image wavelet decomposition has great different with optical images, when more wasted code stream is needed to encode large amount of important coefficients. In order to better adapt above features of SAR
3
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image data, different wavelet sub-band may be treated indifference in encoding. For example, low-frequency sub-band has less data amount and more energy. It has greater impact on SAR image quality. Thus, differential predictive coding method (DPCM) can be used for its lossless compression to prevent most important information been lost. In case of high-frequency
ip t
encoding, bit-plane scanning and quantitative techniques can be comprehensively used, namely, bit-plane scanning is used to find important coefficient, which is then quantified directly.
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Finally, SAR image contains not only large number of flat slowly varying background regions, but also rich texture structures. It also has some geometric characteristics different from
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optical images. Generally, people paid more attention to edge, texture and detail in SAR image analysis, which are embodied in some coefficients with larger amplitude of high-frequency sub-
an
band. They are also important coefficients in the definition of SPIHT algorithm. Compared with optical image, important coefficient of high-frequency sub-band in SAR image has of importance value, which should be kept as possible meeting condition of compression ratio. To
M
address above application requirements, the technique as Universal Trellis Coded Quantization (UTCQ) can be used to substitute traditional scalar quantization. The quantized gain of important
d
coefficient can be further improved by expanding Euclidean distance among quantized
Spatial orientation tree scanning
UTCQ
Low-frequency Sub-band reconstruction
Differential Pulse Decoding
Low-frequency Sub-band reconstruction
Spatial orientation tree restoring
UTCQ antiquantization
Arithmetic decoding
Channel
High-frequency Sub-band
Arithmetic coding
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DPCM
Threshold denoising
Wavelet decomposition
Low-frequency Sub-band
Wavelet composition
Logarithmic transformation
Exponential
transformation
Reconstructed image
Ac ce p
Original image
coefficients, so as to better protect texture information.
Fig. 2 Principle diagram of improved algorithm
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2.3 Algorithm diagram In accordance with above design idea, the paper proposes an improved SAR image compression
algorithm
based
on
Mallat
tower-type
decomposition.
The
algorithm
comprehensively uses techniques as SAR image wavelet denoising, SPIHT spatial orientation
ip t
tree, bit-plane scanning as well as UTCQ. Before describing these key techniques in detail, the
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principle diagram of algorithm is given in Fig. 2.
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3. Approach for performing the algorithm 3.1 Wavelet coefficient scanning
The correlation among adjacent pixels in SAR image can be enhanced after speckle
an
denoising, which is facilitating for subsequent coding process. Based on SAR image data statistical features, the improved algorithm scans low-frequency sub-band and high-frequency
M
sub-band after wavelet decomposition with different methods. 3.1.1 Low-frequency sub-band
d
Low-frequency sub-band reflects overview of original SAR image, which contains most of energy of original image. Therefore, coefficient distortion of low-frequency sub-band will
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severely affect image quality. The algorithm uses Differential Pulse Coding Modulation technique to perform lossless compression and retain most important information. Wiener-Hopf
Ac ce p
equation has proved that for wide stationary random sequence, if the correlation function of signal r(k)=E[x(n)x(n-k)] is known, and mean value of signal is 0, there exists optimal predictor under mean square sense [2]. It must be noted that solving on optimal predictor will increase computation complexity. For simplicity, we use general linear predictor in the paper. Assuming
Xˆ N as pixel predicted value of the N-th point and eN as prediction error. If actual gray value of the N-point pixel is XN, there exists following equation on (N-1)-order linear prediction N 1
Xˆ N ai X i a1 X 1 a 2 X 2 a N 1 X N 1 ,
(1)
i 1
where prediction coefficient ai i 1,2, , N 1 is a constant to be determined, which shows no relation with Xi. 3.1.2 High-frequency sub-band
5
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Rich texture of SAR image is reflected as high-frequency coefficients with larger amplitude. In case of high-frequency sub-band scanning, spatial orientation tree scanning of SPIHT algorithm can be referred. The specific flow of high-frequency scanning is as following:
,
where is down integrated
ip t
Step1: initialization. Compute and output i log 2 max wxk
operator. Put coefficient coordinates (m, n) of future generations in set H as low-frequency subband into list of insignificant blocks (LIB) and treat it as D(m, n) set type. Then set list of
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classified pixels (LCP) as empty, namely LCP .
Step2: spatial orientation tree decomposition. For each D(m, n) in the LIB, compute Si(D). If
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Si(D)=0, output one bit 0 and continue scanning next D(m, n). If Si(D)=1, output one bit 1 and put child node O(m, n) of coefficient wx(m, n) into LCP list. Delete coordinate (m, n) in the LIB. list, when the L(m, n) is treated as D(k, l) set type.
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If the coefficient coordinate (k, l) corresponding to O(m, n) has posterity, put it in the end of LIB
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Step3: important coefficient quantization. Use universal trellis coded quantization on all wavelet coefficients to expand euclidean distance among all quantized coefficients so as to further improve quantization gain of important coefficients.
d
Step4: update. After all D(m, n) is the LIB been scanned, delete all coefficient coordinates in the
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LCP and reset LCP . Set i=i-1 and halve scanning threshold. Return to Step2 to conduct spatial orientation scanning again.
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The above high-frequency coefficient scanning method integrates techniques of spatial orientation tree, bit-plane encoding as well as UTCQ. At the same time of taking full advantage of successful experience of SPIHT algorithm, the feature of rich texture in SAR image should be also considered so as to retain useful high-frequency information as possible. 3.2 Improvement on universal Trellis Coded Quantization In the case of high-frequency sub-band scanning, important coefficients are placed into LCP list. The improved algorithm uses UTCQ to quantify LCP. UTCQ is an improved method based on TCQ. 3.2.1 Improvement The Trellis Coded Modulation (TCM) is an technique combining convolution code and modulation. It refers to ideas of TCM set extension and set partitioning. Usually, quantization of m bits corresponds to 2m quantization levels. TCQ firstly doubles 2m quantization levels with
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Page 6 of 14
~/m ~ 11 m ~ m . It then selects subset where quantized coefficient convolution encoder of m ~ 1 bit date. The remaining m m ~ bit date is used to select quantization value. The located with m ~ 1 , namely 1/2 manner increase the Euclidean distance among quantized coefficients. If m
convolution encoder is used, as to TCQ quantization with m encoding rate, the outputted
ip t
quantized coefficient will be selected from Llody_Max quantizer with m+1 rate. However, quantization directly using 2m+1 levels will cause rise of compression bit rate. In order to solve
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the problem, we should divide expanded signal sets. The above 2m+1 levels are usually divided into 4 subsets that represented with (D0, D1, D2, D3). Each subset is equal to an independent
us
quantizer, including 2m-1 quantization levels. Four subsets are divided into two supersets, where S 0 D0 D2 and S1 D1 D3 . The supersets S0 and S1 are actually doubled uniform
an
quantizer with two quantization intervals. Each one contains 2m quantization levels. The output quantization indexing from TCQ quantizer is selected from above two supersets. As the
M
quantization levels of subset D0 and D2 of superset S0 do not overlap, the quantization levels of subset D1 and D3 of superset S1 also not overlap. Therefore, we can determine the corresponding set that coefficient belongs to by judging superset.
d
We can see from above superset division method that superset S1 D1 D3 of TCQ has not
te
zero output value. If a long zero sequence emerges, the quantization value will retain at state 0. In case of non-zero value quantized with subset D2, the grid state will transit, resulting in poor
Ac ce p
mean square error characteristic. To address the problem, UTCQ improves subset division method of TCQ, as shown in Fig. 3.
…
D3
D0
D1
D1 D2
D3
D0
… -5 -4 -3 -2
-
0
D2
D3
D0
D1
D2 …
2
3
4
5 …
Fig. 3 UTCQ subset division
Comparatively, the superset definition of UTCQ is same as that of TCQ, division of which is shown in Fig. 4.
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… D0
D2
D0
D2
D0
D2 …
… -4
-2
0
3
5 …
… -2
-1
0
1
2
S0
Superset indexing value
3
…
D1
D3
D0
D3
D1
…
-3
-
0
2
4
…
-2
-1
0
1
2
Quantified reconstructed value
… …
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Superset indexing value
…
cr
S1
…
ip t
Quantified reconstructed value
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Fig. 4 UTCQ superset division
From UTCQ subset and superset division we can know that subset indexing of UTCQ positive part shift one bit left compared with TCQ, so that supersets S0 and S1 have zero
M
quantization values. Assume the quantized coefficients have symmetrical probability, there exists the following equations [4]:
(2)
d
CWiS0 CWiS1 ,
te
p S0 CWi p S1 CWi , (3) where i is the superset index; CW is quantized constructed value, p is probability of superset
Ac ce p
indexing. According to above equations, the superset S0 and S1 can be quantized with the same method. The outputs are superset index of S0 and negate S1. The quantization method of UTCQ on single coefficient or coefficient sequence is similar with that of TCQ, while UTCQ quantization uses uniform quantization step but not record and search for other thresholds. If the quantization step is known, the reconstructed value can be regarded as intermediate value of subset where quantization coefficient located. The UTCQ inverse quantizer uses the following manner to obtain reconstructed value of quantization coefficient. Concerning CWi : i 2, the quantization reconstructed value takes intermediate value of subsets. Concerning CWi : i 0, the quantization reconstructed value is zero. Concerning CWi : i 1,2 , the quantization reconstructed value is mean of quantization coefficients in superset indexes i S 0 and i S1 . 3.2.2 Mathematical analysis
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From work principle of trellis coding we can know that the implementation idea of UTCQ comes from TCM. Contrast to traditional scalar quantization, the advantage of UTCQ mainly lies in improving quantization gain while not increase compression bit rate. We try to explain the
white noise additive channel, the channel capacity is
R P E C B log 2 1 s B log 2 1 b b , N0B B N0
(4)
cr
.
ip t
phenomenon referring to Shannon channel coding theory [5]. Firstly, to band-limited Gaussian
where C is the channel capacity; Ps is average signal power; B is channel bandwidth; N0B is
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average power of Gaussian white noise in bandwidth B; Rb is bit rate of signal transmission; Eb is average energy contained in each bit data of signal. The above equation can be rewritten as:
R E C log 2 1 b b , B B N0
an
.
(5)
M
generally speaking, in actual TCM system, Rb
(6)
te
d
Rb B , Rb Eb log 2 1 B N 0
Ac ce p
define k Rb / B k Rb / B . If k keeps unchanged, the above equation can be further rewritten as: .
k E log 2 1 k b N0
,
(7)
Define as utilization of channel capacity. From analyzing on above equation we can see that decrease of Eb/N0 will induce the increment of . TCM is based on this principle that under the condition of not changing signal transmission rate Rb and channel bandwidth B, the Euclidean distance among modulated signals is increased to achieve the decay of Eb/N0, improve channel capacity utilization , and finally achieve target of improving encoding gain. UTCQ uses same techniques with TCM, namely trellis coding, set division, set partitioning, Viterbi decoding. They are all used to increase Euclidean distance among modulated or quantized signals. Therefore, the principle to generate quantization gain of UTCQ is consistent with that of TCM. 3.3 Mathematic coding
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The code stream after low-frequency sub-band prediction coding, high-frequency spatial orientation tree decomposition as well as UTCQ can be directly outputted. If the compression ratio needs to be further improved, the adaptive mathematic coding can be used to re-compress obtained code stream, as shown in broken lines in Fig. 1. The implementation details of
ip t
mathematic coding can refer to [6] and [7]. The usage of adaptive mathematic coding will increase complexity, which is a burden to system with high requirement on coding and decoding
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time. Therefore, whether to using mathematic coding is determined by different applications. If high compression ratio is wanted without real-time demanding, the coding method can be used.
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Otherwise, neglect the step and directly output code stream after scanning and quantization.
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4. Simulation Experiment and Result Analysis 4.1 Experiment Conditions
To verify performance of proposed improved algorithm, we conducted simulation
M
experiment and compared result with that of SPIHT algorithm and JPEG2000. The 8bits/pixel (bpp) original SAR amplitude image as shown in Fig. 5 was used. The resolution is
256 256 .
We
d
can see from the figure that the scene of original image is foothill, which has rich texture
Ac ce p
te
information. The image has been polluted by speckle noise.
Fig. 5 Original image
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The experiment was carried out in the following manner. Firstly, the denoising method of improved algorithm is used to remove speckle noise in original SAR image. Then the coefficient scanning, quantization method, SPIHT algorithm and JPEG2000 standard were used to encode denoised image. The hardware experiment conditions include Intel Pentium 3.0G dual-core
ip t
processor; 1G DDR2 memory and 160G hard drive. Software conditions are Windows XP SP3; Matlab R2008b and VC++ 6.0. The improved algorithm was implemented by Matlab
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programming. Simulation program of SPIHT algorithm comes from Ref. [8] and JPEG2000 algorithm from Ref. [9]. The experiment parameters were set as following. An improved manner
us
was used to conduct four levels db9/7 wavelet decomposition. The low-frequency sub-band does not denoise. The adjustment factor of other levels of high-frequency sub-band to remove speckle noise is cy=0.8. The prediction coefficient of improved algorithm to perform DPCM lossless
an
compression on low-frequency was set {3/4,-1/4,3/8,1/8}. The quantization step 2 i . Where 0.3 and i is bit plane where bit plane scans. In order to reduce computation
M
complexity, the code stream obtained with three algorithms is directly outputted without mathematic coding.
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4.2 Experiment result analysis
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The SPIHT algorithm, JPEG2000 and proposed algorithm are used to perform variable compression ratio coding. From the results we can know that in case of les compression ratio, the
Ac ce p
subjective visual experience on obtained three reconstructed image has not difference. In case of larger compression ratio, the reconstructed image obtained from improved algorithm retains more details and texture information compared with that obtained from SPIHT algorithm as well as JPEG2000 standard, which also has better visual effect. Except for above subjective evaluations on improved algorithm, we also use objective evaluation standard to compare and verify algorithms. The objective evaluation indexes used in experiment are signal-to-noise ratio (SNR) and peak signal-to-noise ratio (PSNR). They are computed as following: M 1 N 1 xm, n 2 , SNR 10 lg M 1 N m10 n 0 2 xm, n xˆ m, n m 0 n0
2 nmax 1 PSNR 10 lg , MSE
11
(8)
(9)
Page 11 of 14
1 MSE MN
M 1 N 1
xm, n xˆm, n , 2
(10)
m 0 n 0
where xm, n and xˆ m, n are respectively gray value of original image and reconstructed image on coordinate (m, n). The image size is M N ; nmax is binary bits used by pixel. As to 8bpp
ip t
image, nmax=8. Fig. 6 and Fig. 7 give statistical result of SNR and PSNR under different compression ratio by different algorithms of SPIHT, JPEG2000 and improved algorithm,
cr
respectively. From figures we can see that SNR and PSNR of improved algorithm are higher than that of SPIHT and JPEG2000 under various compression ratios. The statistical result
Ac ce p
te
d
M
an
us
difference under low bit rate is more apparent than that in high bit rate.
Fig. 6 SNR comparison under different compression ratio
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Fig. 7 PSNR comparison under different compression ratio
5. Conclusion Aiming at shortcomings of existing SAR image wavelet compression algorithm with simple
ip t
bit-plane scanning and zero trees encoding, this paper proposes an improved algorithm based on Mallat tower-type wavelet and UTCQ. It firstly performs logarithmic transformation to translate
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multiplicative speckle noise into additive noise. The wavelet denoising was then conducted before encoding on different sub-band with different strategies. Low-frequency reflects overview
us
of SAR image, so DPCM was used. High-frequency mainly embodies edge texture of SAR. The spatial orientation tree similar to SPIHT algorithm was used to scan important coefficients,
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which were then conduct UTCQ to further improving quantization gain by expanding Euclidean distance among quantized coefficients. Experiment results show that the improved algorithm is more suitable for SAR image compression. The algorithm also lost large amount high-frequency
M
information in case of Mallat tower-type wavelet decomposition. The proposed algorithm will be
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Acknowledgement
d
further improved in the future so as to obtain compression ratio with better performance.
This work is supported by the National Natural Science Foundation of China (NSFC) under the
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Grant No. 61205121, 61304124, the Natural Science Foundation of Zhejiang Province under the Grant No. LY13F010009, the Scientific Research Fund of Zhejiang Provincial Education Department under the Grant No. Y201225146 and the Natural Science Foundation of Zhejiang University of Technology under the Grant No. 2013XZ003.
References
1. A. Said, W.A. Pearlman, “A new fast and efficient image codec based on set partitioning in hierarchical tree,” IEEE Transactions on Information Theory, 1995, 41(3): 613-627. 2. J. G. Garbas, B. P. Popescu, M. Trocan, A. Kaup, “Wavelet-based multi-view video coding with joint best basis wavelet packets,” Proceedings of IEEE International Conference on Image Processing (ICIP’08), 2008, 1232-1235.
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3. M. J. Tsai, C. H. Shen, “Wavelet tree group modulation (WTGM) for digital image watermarking,” Proceedings of IEEE International conference on Acoustics, Speech and Signal Processing, 2007, 173-176. 4. W. H. A. Beaudot, K. T. Mullen, “Orientation selectivity in luminance and color vision
ip t
assessed using 2-D band-pass filter spatial noise,” Vision Research, 2005, 45(6): 687-696.
5. S. D. Servetto, M. T. Orchard, K. Ramchandran, “Image coding based on a morphological
cr
representation of wavelet data,” IEEE Transactions on Image Processing, 1999, 8(9): 11611174.
us
6. F. Lazzaroni, R. Leonardi, A. Signoroni, “High-performance embedded morphological wavelet coding,” IEEE Signal Processing Letters, 2003, 10(10): 293-295.
an
7. E. Candès, L. Demanet, D. L. Donoho, L. X. Ying, “Fast discrete curvelet transforms,” Multiscale Modeling & Simulation, 2006, 5(3): 861-899.
8. C. Xiu, H. Zhu, “A modified SPIHT algorithm based on wavelet coefficient blocks for robust
M
image transmission over noisy channel,” Proceedings of 2010 International Symposium on Information Science & Engineering (ISISE), 2010, 58-61.
d
9. C. Christopoulos, A. Skodras, T. Ebrahimi, “The JPEG2000 still image coding system: an
Ac ce p
te
overview,” IEEE Transactions on Consumer Electronics, 2004, 11(4): 1103-1127.
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S0030-4026(15)00749-4 http://dx.doi.org/doi:10.1016/j.ijleo.2015.07.196 IJLEO 55912
To appear in: Received date: Accepted date:
11-7-2014 29-7-2015
Please cite this article as: J. Li, L. Chang, A SAR image compression algorithm based on Mallat tower-type wavelet decomposition, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.07.196 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A SAR image compression algorithm based on Mallat tower-
ip t
type wavelet decomposition Jia Li*, Liping Chang
cr
Institute of Fiber Optic Communication & Information Engineering, College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China Corresponding author: [email protected]
us
*
Abstract: Image compression algorithms based on wavelet transform can achieve ideal
an
effect on optical image, while cannot do well in SAR image compression. The paper introduces wavelet encoding in optical image compression into SAR image processing and designs an improved SAR compression algorithm based on Mallat tower-type wavelet
M
decomposition according to characteristics of SAR image. The algorithm firstly conducts noise reduction in wavelet domain on image and then performs differential pulse lossless
d
coding on low-frequency sub-band after removing speckle noise. The high-frequency is
te
scanned for important coefficients using spatial direction tree similar to SPIHT algorithm and processed with universal trellis coded quantization. The Euclidean distance among
Ac ce p
important coefficient can be expanded to improving quantification gain. It can be seen from simulation results analysis on algorithm compression performance that the proposed improved algorithm is more suitable for SAR image compression compared with SPIHT algorithm and JPEG2000 standard. Keywords: Mallat, Wavelet, SAR, Image compression
1. Introduction Wavelet transform has spatial frequency local characteristic and multi-resolution analyzing features. It is very close to human visual system. Compared with methods on DCT, image compression algorithms based on DWT can achieve higher compression ratio, which also has not box effect in low bit rate. In recent years, emergence of JPEG2000 standard enables further development of wavelet compression technologies. Research results show that such kind of
1
Page 1 of 14
algorithms can obtain good compression effect on optical images, while compression on SAR images is unsatisfactory. Nevertheless, we can still refer to part design idea of existing algorithms. The paper introduces advanced wavelet encoding methods in optical image compression into SAR image compression and designs an improved SAR compression algorithm
ip t
based on Mallat tower-type wavelet decomposition. Finally, simulation experiment is used to analyze compression performance of the proposed algorithm.
cr
This paper is organized as follows: section 2 gives improvement idea of algorithms; section 3 designs the improved algorithm in detail; section 4 performs simulation experiment and
an
us
analyzes results; section 5 concludes our work.
2. Algorithm for the design progress
M
2.1 Spatial orientation tree features of wavelet decomposition As to its space-frequency localized characteristics, wavelet coefficient on same spatial location under different scale is self-similarity. Seen from multi-resolution structure of wavelet,
d
there exists quad-tree relationship between low-frequency sub-band and adjacent high-frequency
te
sub-band of wavelet coefficient to describe same spatial location, as shown in Fig. 1. If such tree structure can be effectively utilized, image compression algorithm based on wavelet transform
Ac ce p
can then be designed.
Fig. 1 describes parent-children relationship among sub-band coefficients with different resolution after Mallat tower-type decomposition. As shown in the figure, each parent node contains four child nodes or none. The coefficients from low-frequency to corresponding highfrequency constitute a spatial orientation tree. The SPIHT algorithm proposed by Said and Pearlman took full advantage of features of above spatial orientation tree. It uses bit-plane scanning method to effectively mapping wavelet coefficients for some specified thresholds. Then, they are represented with effective values. The encoding process requires three auxiliary lists, namely List of Significant Pixels (LSP), List of Insignificant Pixels (LIP) and List of Insignificant Set (LIS). The encoded stream is sorted in accordance with importance and outputted in the manner of progressive transmission. The encoding process can be ended according to requirements of target bitrates or distortion at any time. We can know from
2
Page 2 of 14
experiment result in Ref. [1] that optical image compression with SPIHT algorithm can achieve
an
us
cr
ip t
good effect.
Fig. 1 Spatial orientation tree structure
M
2.2 Improvement method
The SAR image compression algorithm refers to idea of SPIHT spatial orientation tree in
d
wavelet coefficient importance scanning. Based on characteristics of SAR image, we made
te
modifications on existing algorithm as follows: firstly, SAR image is affected by multiplicative speckle noise determined by its imaging mechanism. Compared with optical image, correlation
Ac ce p
among adjacent pixels in SAR image greatly reduces. Therefore, it is necessary to carry out noise reduction before wavelet coefficient scanning. The image correlation after removing speckle noise will be greatly enhanced, which is help for improving image compression quality. Secondly, SAR image data has large dynamic range. In general, each pixel is represented with 32bit or 16bit data. For this feature, we can conduct logarithmic transformation on SAR image data. Such processing manner can not only reduce data dynamic scope, but also translate multiplicative noise into additive noise so as to facilitate denoising in the wavelet domain. Thirdly, SAR image data has richer grey scales and contrasts than optical remote sensing images. Under influence of speckle noise, similarity among wavelet sub-bands is poor. The bitplane scanning and zero-tree encoding techniques used in SPIHT algorithm have ineffective compression on SAR image because number of zero-tree after SAR image wavelet decomposition has great different with optical images, when more wasted code stream is needed to encode large amount of important coefficients. In order to better adapt above features of SAR
3
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image data, different wavelet sub-band may be treated indifference in encoding. For example, low-frequency sub-band has less data amount and more energy. It has greater impact on SAR image quality. Thus, differential predictive coding method (DPCM) can be used for its lossless compression to prevent most important information been lost. In case of high-frequency
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encoding, bit-plane scanning and quantitative techniques can be comprehensively used, namely, bit-plane scanning is used to find important coefficient, which is then quantified directly.
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Finally, SAR image contains not only large number of flat slowly varying background regions, but also rich texture structures. It also has some geometric characteristics different from
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optical images. Generally, people paid more attention to edge, texture and detail in SAR image analysis, which are embodied in some coefficients with larger amplitude of high-frequency sub-
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band. They are also important coefficients in the definition of SPIHT algorithm. Compared with optical image, important coefficient of high-frequency sub-band in SAR image has of importance value, which should be kept as possible meeting condition of compression ratio. To
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address above application requirements, the technique as Universal Trellis Coded Quantization (UTCQ) can be used to substitute traditional scalar quantization. The quantized gain of important
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coefficient can be further improved by expanding Euclidean distance among quantized
Spatial orientation tree scanning
UTCQ
Low-frequency Sub-band reconstruction
Differential Pulse Decoding
Low-frequency Sub-band reconstruction
Spatial orientation tree restoring
UTCQ antiquantization
Arithmetic decoding
Channel
High-frequency Sub-band
Arithmetic coding
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DPCM
Threshold denoising
Wavelet decomposition
Low-frequency Sub-band
Wavelet composition
Logarithmic transformation
Exponential
transformation
Reconstructed image
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Original image
coefficients, so as to better protect texture information.
Fig. 2 Principle diagram of improved algorithm
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2.3 Algorithm diagram In accordance with above design idea, the paper proposes an improved SAR image compression
algorithm
based
on
Mallat
tower-type
decomposition.
The
algorithm
comprehensively uses techniques as SAR image wavelet denoising, SPIHT spatial orientation
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tree, bit-plane scanning as well as UTCQ. Before describing these key techniques in detail, the
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principle diagram of algorithm is given in Fig. 2.
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3. Approach for performing the algorithm 3.1 Wavelet coefficient scanning
The correlation among adjacent pixels in SAR image can be enhanced after speckle
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denoising, which is facilitating for subsequent coding process. Based on SAR image data statistical features, the improved algorithm scans low-frequency sub-band and high-frequency
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sub-band after wavelet decomposition with different methods. 3.1.1 Low-frequency sub-band
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Low-frequency sub-band reflects overview of original SAR image, which contains most of energy of original image. Therefore, coefficient distortion of low-frequency sub-band will
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severely affect image quality. The algorithm uses Differential Pulse Coding Modulation technique to perform lossless compression and retain most important information. Wiener-Hopf
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equation has proved that for wide stationary random sequence, if the correlation function of signal r(k)=E[x(n)x(n-k)] is known, and mean value of signal is 0, there exists optimal predictor under mean square sense [2]. It must be noted that solving on optimal predictor will increase computation complexity. For simplicity, we use general linear predictor in the paper. Assuming
Xˆ N as pixel predicted value of the N-th point and eN as prediction error. If actual gray value of the N-point pixel is XN, there exists following equation on (N-1)-order linear prediction N 1
Xˆ N ai X i a1 X 1 a 2 X 2 a N 1 X N 1 ,
(1)
i 1
where prediction coefficient ai i 1,2, , N 1 is a constant to be determined, which shows no relation with Xi. 3.1.2 High-frequency sub-band
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Rich texture of SAR image is reflected as high-frequency coefficients with larger amplitude. In case of high-frequency sub-band scanning, spatial orientation tree scanning of SPIHT algorithm can be referred. The specific flow of high-frequency scanning is as following:
,
where is down integrated
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Step1: initialization. Compute and output i log 2 max wxk
operator. Put coefficient coordinates (m, n) of future generations in set H as low-frequency subband into list of insignificant blocks (LIB) and treat it as D(m, n) set type. Then set list of
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classified pixels (LCP) as empty, namely LCP .
Step2: spatial orientation tree decomposition. For each D(m, n) in the LIB, compute Si(D). If
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Si(D)=0, output one bit 0 and continue scanning next D(m, n). If Si(D)=1, output one bit 1 and put child node O(m, n) of coefficient wx(m, n) into LCP list. Delete coordinate (m, n) in the LIB. list, when the L(m, n) is treated as D(k, l) set type.
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If the coefficient coordinate (k, l) corresponding to O(m, n) has posterity, put it in the end of LIB
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Step3: important coefficient quantization. Use universal trellis coded quantization on all wavelet coefficients to expand euclidean distance among all quantized coefficients so as to further improve quantization gain of important coefficients.
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Step4: update. After all D(m, n) is the LIB been scanned, delete all coefficient coordinates in the
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LCP and reset LCP . Set i=i-1 and halve scanning threshold. Return to Step2 to conduct spatial orientation scanning again.
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The above high-frequency coefficient scanning method integrates techniques of spatial orientation tree, bit-plane encoding as well as UTCQ. At the same time of taking full advantage of successful experience of SPIHT algorithm, the feature of rich texture in SAR image should be also considered so as to retain useful high-frequency information as possible. 3.2 Improvement on universal Trellis Coded Quantization In the case of high-frequency sub-band scanning, important coefficients are placed into LCP list. The improved algorithm uses UTCQ to quantify LCP. UTCQ is an improved method based on TCQ. 3.2.1 Improvement The Trellis Coded Modulation (TCM) is an technique combining convolution code and modulation. It refers to ideas of TCM set extension and set partitioning. Usually, quantization of m bits corresponds to 2m quantization levels. TCQ firstly doubles 2m quantization levels with
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~/m ~ 11 m ~ m . It then selects subset where quantized coefficient convolution encoder of m ~ 1 bit date. The remaining m m ~ bit date is used to select quantization value. The located with m ~ 1 , namely 1/2 manner increase the Euclidean distance among quantized coefficients. If m
convolution encoder is used, as to TCQ quantization with m encoding rate, the outputted
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quantized coefficient will be selected from Llody_Max quantizer with m+1 rate. However, quantization directly using 2m+1 levels will cause rise of compression bit rate. In order to solve
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the problem, we should divide expanded signal sets. The above 2m+1 levels are usually divided into 4 subsets that represented with (D0, D1, D2, D3). Each subset is equal to an independent
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quantizer, including 2m-1 quantization levels. Four subsets are divided into two supersets, where S 0 D0 D2 and S1 D1 D3 . The supersets S0 and S1 are actually doubled uniform
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quantizer with two quantization intervals. Each one contains 2m quantization levels. The output quantization indexing from TCQ quantizer is selected from above two supersets. As the
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quantization levels of subset D0 and D2 of superset S0 do not overlap, the quantization levels of subset D1 and D3 of superset S1 also not overlap. Therefore, we can determine the corresponding set that coefficient belongs to by judging superset.
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We can see from above superset division method that superset S1 D1 D3 of TCQ has not
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zero output value. If a long zero sequence emerges, the quantization value will retain at state 0. In case of non-zero value quantized with subset D2, the grid state will transit, resulting in poor
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mean square error characteristic. To address the problem, UTCQ improves subset division method of TCQ, as shown in Fig. 3.
…
D3
D0
D1
D1 D2
D3
D0
… -5 -4 -3 -2
-
0
D2
D3
D0
D1
D2 …
2
3
4
5 …
Fig. 3 UTCQ subset division
Comparatively, the superset definition of UTCQ is same as that of TCQ, division of which is shown in Fig. 4.
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… D0
D2
D0
D2
D0
D2 …
… -4
-2
0
3
5 …
… -2
-1
0
1
2
S0
Superset indexing value
3
…
D1
D3
D0
D3
D1
…
-3
-
0
2
4
…
-2
-1
0
1
2
Quantified reconstructed value
… …
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Superset indexing value
…
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S1
…
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Quantified reconstructed value
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Fig. 4 UTCQ superset division
From UTCQ subset and superset division we can know that subset indexing of UTCQ positive part shift one bit left compared with TCQ, so that supersets S0 and S1 have zero
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quantization values. Assume the quantized coefficients have symmetrical probability, there exists the following equations [4]:
(2)
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CWiS0 CWiS1 ,
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p S0 CWi p S1 CWi , (3) where i is the superset index; CW is quantized constructed value, p is probability of superset
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indexing. According to above equations, the superset S0 and S1 can be quantized with the same method. The outputs are superset index of S0 and negate S1. The quantization method of UTCQ on single coefficient or coefficient sequence is similar with that of TCQ, while UTCQ quantization uses uniform quantization step but not record and search for other thresholds. If the quantization step is known, the reconstructed value can be regarded as intermediate value of subset where quantization coefficient located. The UTCQ inverse quantizer uses the following manner to obtain reconstructed value of quantization coefficient. Concerning CWi : i 2, the quantization reconstructed value takes intermediate value of subsets. Concerning CWi : i 0, the quantization reconstructed value is zero. Concerning CWi : i 1,2 , the quantization reconstructed value is mean of quantization coefficients in superset indexes i S 0 and i S1 . 3.2.2 Mathematical analysis
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From work principle of trellis coding we can know that the implementation idea of UTCQ comes from TCM. Contrast to traditional scalar quantization, the advantage of UTCQ mainly lies in improving quantization gain while not increase compression bit rate. We try to explain the
white noise additive channel, the channel capacity is
R P E C B log 2 1 s B log 2 1 b b , N0B B N0
(4)
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.
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phenomenon referring to Shannon channel coding theory [5]. Firstly, to band-limited Gaussian
where C is the channel capacity; Ps is average signal power; B is channel bandwidth; N0B is
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average power of Gaussian white noise in bandwidth B; Rb is bit rate of signal transmission; Eb is average energy contained in each bit data of signal. The above equation can be rewritten as:
R E C log 2 1 b b , B B N0
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.
(5)
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generally speaking, in actual TCM system, Rb
(6)
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Rb B , Rb Eb log 2 1 B N 0
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define k Rb / B k Rb / B . If k keeps unchanged, the above equation can be further rewritten as: .
k E log 2 1 k b N0
,
(7)
Define as utilization of channel capacity. From analyzing on above equation we can see that decrease of Eb/N0 will induce the increment of . TCM is based on this principle that under the condition of not changing signal transmission rate Rb and channel bandwidth B, the Euclidean distance among modulated signals is increased to achieve the decay of Eb/N0, improve channel capacity utilization , and finally achieve target of improving encoding gain. UTCQ uses same techniques with TCM, namely trellis coding, set division, set partitioning, Viterbi decoding. They are all used to increase Euclidean distance among modulated or quantized signals. Therefore, the principle to generate quantization gain of UTCQ is consistent with that of TCM. 3.3 Mathematic coding
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The code stream after low-frequency sub-band prediction coding, high-frequency spatial orientation tree decomposition as well as UTCQ can be directly outputted. If the compression ratio needs to be further improved, the adaptive mathematic coding can be used to re-compress obtained code stream, as shown in broken lines in Fig. 1. The implementation details of
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mathematic coding can refer to [6] and [7]. The usage of adaptive mathematic coding will increase complexity, which is a burden to system with high requirement on coding and decoding
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time. Therefore, whether to using mathematic coding is determined by different applications. If high compression ratio is wanted without real-time demanding, the coding method can be used.
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Otherwise, neglect the step and directly output code stream after scanning and quantization.
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4. Simulation Experiment and Result Analysis 4.1 Experiment Conditions
To verify performance of proposed improved algorithm, we conducted simulation
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experiment and compared result with that of SPIHT algorithm and JPEG2000. The 8bits/pixel (bpp) original SAR amplitude image as shown in Fig. 5 was used. The resolution is
256 256 .
We
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can see from the figure that the scene of original image is foothill, which has rich texture
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te
information. The image has been polluted by speckle noise.
Fig. 5 Original image
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The experiment was carried out in the following manner. Firstly, the denoising method of improved algorithm is used to remove speckle noise in original SAR image. Then the coefficient scanning, quantization method, SPIHT algorithm and JPEG2000 standard were used to encode denoised image. The hardware experiment conditions include Intel Pentium 3.0G dual-core
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processor; 1G DDR2 memory and 160G hard drive. Software conditions are Windows XP SP3; Matlab R2008b and VC++ 6.0. The improved algorithm was implemented by Matlab
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programming. Simulation program of SPIHT algorithm comes from Ref. [8] and JPEG2000 algorithm from Ref. [9]. The experiment parameters were set as following. An improved manner
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was used to conduct four levels db9/7 wavelet decomposition. The low-frequency sub-band does not denoise. The adjustment factor of other levels of high-frequency sub-band to remove speckle noise is cy=0.8. The prediction coefficient of improved algorithm to perform DPCM lossless
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compression on low-frequency was set {3/4,-1/4,3/8,1/8}. The quantization step 2 i . Where 0.3 and i is bit plane where bit plane scans. In order to reduce computation
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complexity, the code stream obtained with three algorithms is directly outputted without mathematic coding.
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4.2 Experiment result analysis
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The SPIHT algorithm, JPEG2000 and proposed algorithm are used to perform variable compression ratio coding. From the results we can know that in case of les compression ratio, the
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subjective visual experience on obtained three reconstructed image has not difference. In case of larger compression ratio, the reconstructed image obtained from improved algorithm retains more details and texture information compared with that obtained from SPIHT algorithm as well as JPEG2000 standard, which also has better visual effect. Except for above subjective evaluations on improved algorithm, we also use objective evaluation standard to compare and verify algorithms. The objective evaluation indexes used in experiment are signal-to-noise ratio (SNR) and peak signal-to-noise ratio (PSNR). They are computed as following: M 1 N 1 xm, n 2 , SNR 10 lg M 1 N m10 n 0 2 xm, n xˆ m, n m 0 n0
2 nmax 1 PSNR 10 lg , MSE
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(8)
(9)
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1 MSE MN
M 1 N 1
xm, n xˆm, n , 2
(10)
m 0 n 0
where xm, n and xˆ m, n are respectively gray value of original image and reconstructed image on coordinate (m, n). The image size is M N ; nmax is binary bits used by pixel. As to 8bpp
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image, nmax=8. Fig. 6 and Fig. 7 give statistical result of SNR and PSNR under different compression ratio by different algorithms of SPIHT, JPEG2000 and improved algorithm,
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respectively. From figures we can see that SNR and PSNR of improved algorithm are higher than that of SPIHT and JPEG2000 under various compression ratios. The statistical result
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d
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an
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difference under low bit rate is more apparent than that in high bit rate.
Fig. 6 SNR comparison under different compression ratio
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Fig. 7 PSNR comparison under different compression ratio
5. Conclusion Aiming at shortcomings of existing SAR image wavelet compression algorithm with simple
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bit-plane scanning and zero trees encoding, this paper proposes an improved algorithm based on Mallat tower-type wavelet and UTCQ. It firstly performs logarithmic transformation to translate
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multiplicative speckle noise into additive noise. The wavelet denoising was then conducted before encoding on different sub-band with different strategies. Low-frequency reflects overview
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of SAR image, so DPCM was used. High-frequency mainly embodies edge texture of SAR. The spatial orientation tree similar to SPIHT algorithm was used to scan important coefficients,
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which were then conduct UTCQ to further improving quantization gain by expanding Euclidean distance among quantized coefficients. Experiment results show that the improved algorithm is more suitable for SAR image compression. The algorithm also lost large amount high-frequency
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information in case of Mallat tower-type wavelet decomposition. The proposed algorithm will be
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Acknowledgement
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further improved in the future so as to obtain compression ratio with better performance.
This work is supported by the National Natural Science Foundation of China (NSFC) under the
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Grant No. 61205121, 61304124, the Natural Science Foundation of Zhejiang Province under the Grant No. LY13F010009, the Scientific Research Fund of Zhejiang Provincial Education Department under the Grant No. Y201225146 and the Natural Science Foundation of Zhejiang University of Technology under the Grant No. 2013XZ003.
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9. C. Christopoulos, A. Skodras, T. Ebrahimi, “The JPEG2000 still image coding system: an
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