Simultaneous speckle reduction and data compression using best wavelet packet bases with application to SAR based ATD/R D. Wei , H. Guo , J. E. Odegard , M. Lang , and C. S. Burrus ? Department of Electrical and Computer Engineering Rice University , Houston , TX 77251-1892 January 25 , 1995 Paper accepted for the Proceedings of SPIE , Mathematical Imaging : Wavelet Applications for Dual use -LRB- 2491 - 104 -RRB- , in SPIE Symposium on OE/Aerospace Sensing and Dual Use Photonics , 17-21 April 1995 , Orlando , FL . This is also Technical Report Rice University , CML TR95-02 . ABSTRACT We propose a novel method for simultaneous speckle reduction and data compression based on shrinking , quantizing and coding the wavelet packet coefficients of the logarithmically transformed image . A fast algorithm is used to find the best wavelet packet basis in the rate-distortion sense from the entire library of admissible wavelet packet bases . Soft-thresholding in wavelet domain can significantly suppress the speckles of the synthetic aperture radar -LRB- SAR -RRB- images while maintaining bright reflections for subsequent detection and recognition . Optimal bit allocation , quantization and entropy coding achieve the goal of compression while maintaining the fidelity of the SAR image . 1 INTRODUCTION Synthetic aperture radar -LRB- SAR -RRB- is an active coherent all-weather imaging system that operates in the microwave region of the spectrum . This imagery is well suited to the task of remote ground mapping in many applications , such as surveillance , oceanography , glaciology , and agriculture . There are two problems in the practical SAR applications . One is the speckle phenomena . Speckle results from the necessity of creating the image with coherent radiation . A fully developed speckle pattern appears chaotic and unordered . When the detail information in the image is important , speckle can be viewed as a noise that causes degradation of the image . Therefore , speckle reduction is an essential procedure before the procedures of automatic target detection and recognition . Another problem is the large amount of SAR data . The data collected and processed by a SAR system are inherently complex . Thus , data compression is desirable for quick transmission of the collected ? This work is supported in part by ARPA , BNR , TI and Alexander von Humboldt foundation . STFT tree Wavelet tree Arbitrary wavelet packet tree Figure 1 : Some wavelet packet decompositions of level 3 -LRB- one-dimensional case -RRB- information . The wavelet transform is a relatively new technique for multi-resolution decomposition of images and is widely used in both noise reduction and data compression of SAR images .6,11 Thus , it is very efficient to combine the procedures of noise reduction and data compression in a single process of decomposition and reconstruction . As a generalization of the wavelet basis , the wavelet packet -LRB- WP -RRB- bases , which is a rich family of orthonormal bases , can be expected to be more suited to match the non-stationary statistics of the images . Therefore , it is desirable to fast select an optimal wavelet packet basis under some criterion and to achieve better performances of de-noising and compression . We apply a fast dynamic programming pruning algorithm to efficiently choose the best basis that jointly optimize the bit allocation and the quantization in the rate-distortion sense . Our method can be viewed as not only a generalization but also a combination of the methods used by Guo et al6 and Werness et al. 11 Section 2 introduces fundamental backgrounds of image coding based on wavelet packet decomposition . In Section 3 , de-noising via wavelet shrinkage is described . We discuss our novel algorithm for de-speckling and compression of SAR images in Section 4 and Section 5 . Section 6 describes our experimental results . Finally , summary follows in Section 7 . 2 IMAGE CODING USING WAVELET PACKET TRANSFORM 2.1 Wavelet packets Wavelet packet bases2 were introduced recently as a collection of orthonormal bases for discrete functions of RN . This library contains the well-known wavelet basis , Short-Time-Fourier-Transform -LRB- STFT -RRB- - like basis , Walsh functions , and smooth versions of Walsh functions called wavelet packets . Wavelet packets represent a generalization of the method of multi-resolution decomposition , and comprise the entire family of subband coded -LRB- tree -RRB- decompositions . The library of wavelet packet bases organizes itself into a homogeneous tree -LRB- e.g. , for two-dimensional signals such as images , this library has a structure of a complete quad-tree . -RRB- , which can be efficiently searched for a \ best basis '' under some optimality criterion . The entire WP tree can be obtained by recursively decomposition of both the low-pass subband and the high-pass subband using the quadrature mirror filters -LRB- QMF 's -RRB- . Each admissible basis appears as a subtree formed by pruning the whole tree -LRB- See Figure 1 -RRB- . 2.2 Bit allocation In the wavelet packet decomposition of an image , in general , each subband may require a different relative accuracy of reproduction to achieve a desired overall quality of the reconstructed image . It is desirable to assign more bits to the relatively more \ important '' subband if the total bits budget is given . Consequently , a major concern of a subband coding system is bit allocation , the task of distributing a given quota of bits to subbands to optimize the overall coding performance . Let Wi -LRB- b -RRB- denote the distortion incurred in optimally quantizing the ith subband with b bits of resolution . Let B be the given fixed quota of available bits . We define the overall distortion , D , and the total bit rate , R as functions of the bit allocation vector , b = -LRB- b1 ; b2 ; : : : ; bk -RRB- according to D = D -LRB- b -RRB- = K X i = 1 Wi -LRB- bi -RRB- ; R = R -LRB- b -RRB- = K X i = 1 bi : The bit allocation problem is to find bi for i = 1 ; 2 ; ? ? ? ; K to minimize D -LRB- b -RRB- = PKi = 1 Wi -LRB- bi -RRB- subject to the constraint that R -LRB- b -RRB- = PKi = 1 bi < = B . The classical closed-form solution7 to this optimization problem using the assumption of high resolution quantization is bi = B K + 1 2 log2 oe2i ae2 ; where ae2 = -LRB- QKi = 1 oe2i -RRB- 1K is the geometric mean of the variances in all subbands . However , the main drawback of this solution is that the assumption may not be valid in practical subband coding . Shoham and Gersho10 proposed a fast algorithm that is not based on any assumption about the set of quantizers . We apply it to optimally assign a fixed quota of bits for wavelet packet subband coding . Assume that the wavelet packet decomposition is given . The \ hard '' constrained problem of minimizing the total distortion D for a target bits budget R -LRB- or vice versa -RRB- can be converted to a relatively \ easy '' equivalent unconstrained problem by using Lagrange multiplier > = . Thus the unconstrained problem becomes the minimization of the Lagrangian cost function defined as J -LRB->=-RRB- = D + > = R : It can be shown that the necessary condition for rate-distortion optimality is that all subbands have the same slope point > = . Based on this condition , we can sweep > = from zero . For a given > = , we compute all the bit rates bi 's of subbands resulting the minimum Lagrangian costs and sum them together to obtain the total bit rate . We iterate this process until we achieve the required bit rate . For most of applications , the rate-distortion functions are convex so that we can perform a fast bisection search of the optimal > = corresponding to the required bit rate . 2.3 Quantization In this paper , we only discuss uniform scalar quantizer -LRB- USQ -RRB- . There are several advantages of using USQ : 1 . It is very easy and fast to implement . 2 . It can be shown that , if the quantized variables are entropy coded by a variable-length coding scheme such as Huffman coding , then the USQ gives a better performance than the optimum mean square or Lloyd-Max quantizer .7 It has been found that the uniform quantizer is a quite good approximation of the \ optimum quantizer '' based on entropy versus mean square distortion criterion , if the quantization step size is optimized with respect to this criterion . 3 . It is easy to combine the procedure of wavelet domain soft-thresholding , which does good to both de-noising and data compression , with the procedure of quantization if we use USQ . 4 . In our application , each subband is assigned a finite set of quantizers with different quantization steps . Hence , the rate-versus-distortion functions of subbands can be easily computed . We use mean-squared error -LRB- MSE -RRB- and first-order entropy as the measures of distortion and rate , respectively . Obviously , they are both additive measures . Instead of using MSE criterion , we may also use weighted MSE criterion related to the model of human vision system -LRB- HVS -RRB- to further improve the visual quality of the reconstructed images . 2.4 Best wavelet packet bases Due to the non-stationary behavior of image , any particular choice of subband decompositions , including the widely used wavelet decomposition , does n't provide the optimal wavelet packet bases that best adapt to the image . On the other hand , it can be shown that there are more than 2N wavelet packet bases for a given signal , where N is the size of the signal . Therefore , fast search algorithm is essential . A key point for this class of \ best-bases '' algorithms is that the cost functional M -LRB- ? -RRB- should be additive , i.e. , M -LRB- 0 -RRB- = and M -LRB- P i Xi -RRB- = P i M -LRB- Xi -RRB- , so that it can split nicely across Cartesian product . Thus the search is a fast divide-and-conquer . An entropy-based algorithm for best-basis selection was proposed .3 It chooses the basis with the minimum Shannon-Weaver entropy , which is a measure of the energy distribution of an unquantized vector and is not directly related to quantization and coding . Therefore , this criterion does not guarantee the optimality in the rate-distortion sense , which is the measure of the \ true '' performance of data compression . Ramchandran and Vetterli9 generalized the above algorithm using the rate-distortion framework . Their algorithm is indeed a combination of the concept of orthonormal tiling of the spatial-frequency plane using wavelet packets and the discipline of rate-distortion optimal bit allocation . However , neither of the two algorithms indicates how to efficiently construct the rate-distortion curves , whose computational complexity counts for a significant percentage of the total cost of the searching algorithm . In our method , we use the simplest quantizer , USQ , to alleviate the computational complexity for the ratedistortion functions . The trade-off between computational complexity and performance is very crucial in the applications of compression of very large data set . 3 WAVELET DOMAIN SOFT-THRESHOLDING Donoho and Johnstone4 proposed a nonlinear method for reconstructing an unknown function from noisy data . The method attempts to reject noise by damping or thresholding in the wavelet domain . Suppose we wish to recover an unknown function f on -LSB- 0 ; 1 -RSB- from noisy data di = f -LRB- ti -RRB- + oezi ; i = ; 1 ; : : : ; n ? 1 ; where ti = i = n ; zi iid ? N -LRB- 0 ; 1 -RRB- is a white Gaussian noise , and oe is a noise level . Let ^ f be the estimate of f . Our goal is to optimize the mean-squared error 1 nE k ^ f ? f k22 = 1 n n ? 1 X i = 0 E -LRB- ^ f -LRB- i = n -RRB- ? f -LRB- i = n -RRB- -RRB- 2 subject to the side condition that with high probability , ^ f is at least as smooth as f . The simple soft-thresholding method has three steps : 1 . Apply the wavelet transform to the measured data -LRB- di = pn -RRB- , obtaining empirical wavelet coefficients . 2 . Apply the soft-thresholding nonlinearity ? t -LRB- y -RRB- = sgn -LRB- y -RRB- -LRB- jyj ? t -RRB- + = 8 < : y ? t for y > t for ? t < = y < = t y + t for y < ? t coordinate-wise to the empirical wavelet coefficients with specially-chosen threshold tn = p2log -LRB- n -RRB- ? oe = pn , 3 . Apply the inverse wavelet transform , recovering ^ f ? -LRB- ti -RRB- ; i = ; 1 ; : : : ; n ? 1 . It has been shown that this method has three distinct features : 1 -RRB- The estimate ^ f ? achieves almost the minimax mean square error over every one of a wide range of smoothness classes , including many classes where traditional linear estimators do not achieve the minimax rate . 2 -RRB- This procedure maintains the sharp features of f -LRB- e.g. , the edges in images -RRB- , therefore , it provides better visual quality than procedures based on mean-squared error alone . 3 -RRB- The estimate does not exhibit any noise-induced structures , unlike most minimum mean square methods . The only parameter in this nonlinear operator , the threshold , has a significant effect on the de-noising performance . In practice , the unknown parameter oe , the variance of noise , must be estimated from the data . We found a simple and very effective way to estimate oe : taking the standard deviation of the high/high subband of the first level wavelet decomposition as the estimate .6 For i.i.d. Gaussian noise , we found that t 2 -LSB- 1:5 oe ; 3:0 oe -RSB- yields excellent results so that 86:6 % : : : 99:7 % of the noise values have been suppressed . Our thresholding scheme is different from the one in Donoho 's work which over-smoothes the image . Also , the subband of the lowest resolution is not thresholded so that the mean of the de-noised image is guaranteed to be the same as that of the original image . The speckle noise in SAR images can be approximately modeled as a multiplicative i.i.d. Gaussian noise with unit mean .5 Therefore , after the logarithmic operation , it becomes additive noise so that it can be suppressed via the above soft-threolding scheme . There is a particular advantage of combining the de-noising procedure and the compression procedure . In general , soft-thresholding does good to data compression . The wavelet shrinkage decreases the dynamic range of the coefficients and thresholds those insignificant coefficients . Therefore , the entropy of the quantized wavelet coefficients is lowered . Though we could put the de-noising procedure either in the coder or in the decoder , in our scheme , we put this procedure before the quantization according to the above argument . 4 BASIC IDEA OF THE ALGORITHM Our goal is to optimally compress the SAR image in rate-distortion sense while maintaining the performance of de-noising . Assume that the optimal wavelet packet subtree from node n \ onwards '' to the full tree-depth is known . Then by Bellman 's optimality principle ,1 we know that all possible paths passing through node n must invoke this same optimal \ finishing '' path . At each non-leaf node of the tree , there are two contenders for the \ surviving path '' , the parent and its children , with the winner having the lower Lagrangian cost . According to this sub-optimality , we can apply a dynamic programming to construct the optimal subtree starting from the leaf nodes upwards . When we reach the root node , the best basis with the mimimum Lagrangian cost is known . fxk ; mg - log j ? j - fyk ; mg Wavelet Packet Decomposition - fclk ; mg SoftThresholding f ~ clk ; mg - Quantization & Entropy Coding - bit stream Best Basis Selection - ? Figure 2 : Block diagram of the algorithm for simultaneous de-speckling and compression Figure 2 describes the basic structure of our algorithm . We perform the de-noising first . Then , we initialize a slope value > = and compute the rate-distortion relations and the minimum Lagrangian costs of all subbands with the admissible quantizer sets . We then start from the leave nodes of the entire WP tree and compare each node with its four child nodes . For each node , if its Lagrangian cost is less than the sum of those costs of its four child nodes , we mark this node as a \ merge '' node ; otherwise , we mark this node as a \ split '' node and update its information -LRB- Lagrangian cost and the corresponding rate and distortion -RRB- with those of its child nodes . We recursively do the comparisons until we reach the root node . Then , we can curve out the optimal sub-tree for this > = by pruning all the sub-trees rooted at those \ merge '' nodes and the corresponding rates and distortions . We can repeat the above process using a bisection search to find the proper > = so that the resulting rate is identical to the desired bit budget . Since we include the de-noising procedure in our coding algorithm , which obviously causes some error between the de-noised image and the original image , we use the MSE between the coefficients of the de-noised image and those of the decompressed image as the distortion . Since the orthonormal wavelet packets bases are used , the MSE 's of the spatial domain and the wavelet domain are identical . 5 IMPLEMENTATION OF THE ALGORITHM The following procedure is used to de-speckle and compress the images . Step 1 : Take the logarithmic operation on the magnitude of the complex SAR image fxk ; mg to obtain fyk ; mg . Step 2 : Perform a full wavelet packet decomposition -LRB- up to a desired level L -RRB- on the image y to obtain the coefficients fclk ; mg for the entire WP library . Step 3 : Apply soft-thresholding to fclk ; mg and obtain the De-noised WP coefficients f ~ clk ; mg . Step 4 : For each subband nli ; j , quantize the de-noised WP coefficients using all admissible quantizers , record the resulting rates fRq2Q -LRB- nli ; j -RRB- g , and the corresponding distortions fDq2Q -LRB- nli ; j -RRB- g , and form the rate-versus-distortion function . Step 5 : Initialize an operating slope value > = , populate all the nodes fnli ; jg of the WP tree with their minimum Lagrangian costs J li ; j -LRB->=-RRB- , i.e , J li ; j -LRB->=-RRB- = minq2Q -LSB- Dq -LRB- nli ; j -RRB- + > = Rq -LRB- nli ; j -RRB- -RSB- Step 6 : Initialize l L. Let q ? -LRB- nli ; j -RRB- = argmin -LRB- J li ; j -LRB->=-RRB- -RRB- . Initialize ~ J li ; j J li ; j ; ~ Rli ; j Rli ; j -LRB- where Rli ; j = Rq ? -LRB- nli ; j -RRB- -LRB- nli ; j -RRB- -RRB- , ~ Dli ; j Dli ; j -LRB- where Dli ; j = Dq ? -LRB- nli ; j -RRB- -LRB- nli ; j -RRB- -RRB- : Step 7 : l l ? 1 . If l < 0 , go to Step 10 . Step 8 : 8i ; j = 1 ; 2 ; ? ? ? ; 2l at the lth tree level : if -LRB- J li ; j -LRB->=-RRB- < ~ J l +1 2i ? 1 ; 2j ? 1 + ~ J l +1 2i ? 1 ; 2j + ~ J l +1 2i ; 2j ? 1 + ~ J l +1 2i ; 2j -RRB- then fsplit -LRB- nli ; j -RRB- FALSE ; ~ Rli ; j Rli ; j ; ~ Dli ; j Dli ; j ; ~ J li ; j J li ; j g else fsplit -LRB- nli ; j -RRB- TRUE ; ~ Rli ; j ~ Rl +1 2i ? 1 ; 2j ? 1 + ~ Rl +1 2i ? 1 ; 2j + ~ Rl +1 2i ; 2j ? 1 + ~ Rl +1 2i ; 2j ; ~ Dli ; j ~ Dl +1 2i ? 1 ; 2j ? 1 + ~ Dl +1 2i ? 1 ; 2j + ~ Dl +1 2i ; 2j ? 1 + ~ Dl +1 2i ; 2j ; ~ J li ; j ~ J l +1 2i ? 1 ; 2j ? 1 + ~ J l +1 2i ? 1 ; 2j + ~ J l +1 2i ; 2j ? 1 + ~ J l +1 2i ; 2jg : Step 9 : Go to Step 7 . Step 10 : Perform a tree-walk , starting from the root of the WP tree , n00 ; 0 , and carve out the optimal subtree corresponding to the best wavelet packet basis with its associated optimal choice of quantizer set . ~ R00 ; 0 and ~ D00 ; 0 are the rate and distortion of the optimal subtree for the given > = , respectively . Step 11 : Iterate from Step 5 to Step 10 towards the optimal operating slope > = ? until the given target bit rate is achieved . From the above implementation , we find that the computational complexity is mainly determined by the size of the nested loops . In most applications , we only decompose the image up to 4 ? 6 levels . The number of quantizers of each subband is also a small constant compared to the size of image . Also , the number of iterations for bisection search of a convex curve is relatively small . Therefore , the total computational complexity of our algorithm is O -LRB- N -RRB- , where N is the size of image . 6 EXPERIMENTAL RESULTS The data used here were collected near Stockbridge , NY , by the Lincoln Laboratory MMW SAR . We apply our algorithm to a polarimetric whitening filtered -LRB- PWF -RRB- image . We choose four types of clutter regions in the image : trees , scrub , grass and shadows . The following four statistics are used to evaluate the performance of our algorithm . Standard-deviation-to-mean ratio -LRB- s/m -RRB- : The quantity s = m -LRB- both in power -RRB- is a measure of image speckle in homogeneous region .5 We computed the s = m ratio for each type of clutter region to quantify the speckle reduction capacity of our algorithm . Log standard deviation8 : The standard deviation of the clutter data -LRB- in dB -RRB- . This is an important quantity that directly affects the target detection performance of a standard two-parameter constant-false-alarm-ratio -LRB- CFAR -RRB- algorithm . Target-to-clutter ratio -LRB- t/c -RRB- : The difference between the target and clutter means -LRB- in dB -RRB- . It measures how the target stands out of the surrounding clutter . Deflection ratio : This is the two-parameter CFAR detection statistic . M = y ? ^ ? y ^ oey where y is the scalar pixel value of the cell , ^ ? y is the estimated mean of y , and ^ oey is the estimated standard deviation of y . After speckle reduction , M should be higher at known reflector points and lower elsewhere . Table 1 , 2 and 3 show the numerical results of those four values for the original and the processed images at four typical regions . The large reductions of standard-deviation-to-mean ratio -LRB- s/m -RRB- and log-standard - deviation indicate that a significant amount of speckle has been removed . Figure 3 shows the perceptual results . We can see from the images that speckle is greatly reduced while sharp features are maintained . After speckle reduction , the deflection ratio should be higher at known reflector points and lower elsewhere . As shown in table 3 , the deflection ratio is much higher in the wavelet processed images than that in the PWF processed image . Elsewhere in the image , the deflection ratio values are roughly the same for both methods . This strongly indicates the advantage of our method , and suggests a big improvement in detection performance . Also , cleaner images suggest potential improvements for classification and recognition . Comparing the de-noised and uncompressed image to the denoised and decompressed image , we achieve very high peak-signal-to-noise-ratio -LRB- PSNR -RRB- and good visual quality . Thus , we have compressed the original image at very low bit rate , 0.2 bits per pixel -LRB- bpp -RRB- , while maintaining the high performance of de-noising . Table 1 : s = m for clutter data Trees Scrub Grass Shadow Original Image 1.3025 0.8229 0.6545 0.7020 De-noised Image 0.9245 0.4478 0.3098 0.3611 De-noised & Decompressed Image 0.9251 0.4470 0.3101 0.3618 Table 2 : log ? std for clutter data Trees Scrub Grass Shadow Original Image 4.9398 3.4301 2.9543 2.9008 De-noised Image 3.8335 1.8822 1.3271 1.3081 De-noised & Decompressed Image 3.8337 1.8825 1.3268 1.3077 Table 3 : Target-to-clutter ratio -LRB- t/c -RRB- and Deflection ratio for clutter data Target-to-clutter ratio -LRB- t/c -RRB- Deflection ratio Original Image 34.0269 11.1842 De-noised Image 30.1037 18.2339 De-noised & Decompressed Image 30.0989 18.2401 100 200 300 400 500 50 100 150 200 250 300 350 400 450 500 -LRB- a -RRB- 100 200 300 400 500 50 100 150 200 250 300 350 400 450 500 -LRB- b -RRB- 100 200 300 400 500 50 100 150 200 250 300 350 400 450 500 -LRB- c -RRB- Figure 3 : SAR images of a farm area -LRB- PWF processed -RRB- -LRB- a -RRB- Original image -LRB- b -RRB- De-noised image -LRB- c -RRB- De-noised and decompressed image -LRB- Rate :0.2 bpp ; PSNR :39.6 dB -RRB- 7 SUMMARY An efficient algorithm is proposed for representing the SAR image with the optimal wavelet packet basis in the rate-distortion framework , and simultaneously suppress the speckle noise and compress the image data , while maintaining the resolution of the original image . The reconstructed image will significantly improve the performances of detection , classification and recognition , and require much fewer number of bits for transmission and storage . The future work includes : 1 . to achieve higher compression ratio , combine the de-noising and the vector quantization using optimal wavelet packet basis . 2 . use space-varying wavelets to de-noise and compress the SAR images . 3 . use weighted MSE measure related to the model of HVS instead of the unweighted MSE to improve the visual quality of the reconstructed images . 8 REFERENCES -LSB- 1 -RSB- R. Bellman . Dynamic Programming . Princeton University Press , 1957 . -LSB- 2 -RSB- R. Coifman , Y. Meyer , S. Quake , and M. V. Wickerhauser . Signal processing and compression with wave packets . preprint . -LSB- 3 -RSB- R. R. Coifman and M. V. Wickerhauser . Entropy-based algorithms for best basis selection . IEEE Trans . Inform . Theory , 38 -LRB- 2 -RRB- :1713 -LCB- 1716 , 1992 . -LSB- 4 -RSB- D. L. Donoho . De-noising by soft-thresholding . IEEE Trans . Inform . Theory , 1994 . Also Tech . Report , Department of Statistics , Stanford University ,1992 . -LSB- 5 -RSB- J. W. Goodman . Some fundamental properties of speckle . J. Opt . Soc . Am. , 66:1145 -LCB- 1150 , November 1976 . -LSB- 6 -RSB- H. Guo , J. E. Odegard , M. Lang , R. A. Gopinath , I. Selesnick , and C. S. Burrus . Speckle reduction via wavelet shrinkage with application to SAR based ATD/R . In SPIE Math . Imagaging : Wavelet Applications in Signal and Image Processing , volume 2303 , San Diego , CA , July 1994 . Also Tech report CML TR94-03 , Rice University , Houston , TX . -LSB- 7 -RSB- A. K. Jain . Fundamentals of Digital Image Processing . Prentice Hall , Englewood Cliffs , NJ , 1989 . -LSB- 8 -RSB- L. M. Novak , M. C. Burl , and W. W. Irving . Optimal polarimetric processing for enhanced target detection . IEEE Trans . AES , 29:234 -LCB- 244 , January 1993 . -LSB- 9 -RSB- K. Ramchandran and M. Vetterli . Best wavelet packets in a rate-distortion sense . IEEE Trans . on Image Processing , 1993 . 160-175 . -LSB- 10 -RSB- Y. Shoham and A. Gersho . Efficient bit allocation for an arbitrary set of quantizers . IEEE Trans . ASSP , 36:1445 -LCB- 1453 , September 1988 . -LSB- 11 -RSB- S. A. Werness , S. C. Wei , and R. Carpinella . Experiments with wavelets for compression of SAR data . IEEE Trans . Geoscience and Remote Sensing , 32 -LRB- 1 -RRB- :197 -LCB- 201 , January 1994 .

O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

noise reduction
synthetic aperture radar
wavelet packet
wavelet shrinkage

image data compression
rate-distortion function