Abstractâ SAR images are corrupted by multiplicative noise (speckle) which limits the performance of the classical coder/decoder (codec) in the spatial domain.
A combined Speckle noise reduction and compression of SAR images using a multiwavelet based method to improve codec performance
J. Mvogo , G. Mercier , V.P. Onana , J.P. Rudant , E. Tonye , H. Trebossen
École Polytechnique de Yaoundé, Cameroun École Nationale Supérieure des Télécommunications de Bretagne, Brest, France Laboratoire de Géomatériaux - IFG - Université de Marne-La-Vallée, France
Abstract— SAR images are corrupted by multiplicative noise (speckle) which limits the performance of the classical coder/decoder (codec) in the spatial domain. Our objective is to give an evaluation of the efficiency of a multiwavelet transform coding algorithm . We use the additional degree of freedom offered by multiwavelets to fine tune the number of vanishing moments and the approximation order of ours basis functions. Once the multiwavelet transform is performed, we apply an optimal bit allocation scheme on the subbands data using a set of Vector Quantizers. The quantization of the high frequencies multiwavelets coefficients may be though of as a hard thresholding algorithm. A measure of the equivalent number of looks is performed in the reconstructed SAR image in order to evaluate the impact of the codec in the noise reduction process. We compare our method with classical algorithm (baseline scalar wavelet transform followed by an optimal scalar quantization). The codec achieves comparable SNR, but performs surprising speckle noise reduction. Some results are presented with ERS-PRI images of Cameroon which can be compressed at while still remaining of sufficient quality for visual interpretation, segmentation and land use monitoring.
I. I NTRODUCTION In this paper a lossy compression scheme of SAR images based on the multiwavelet transform is evaluated. We investigate the reconstructed image quality using the SNR and the speckle reduction induced by the hard thresholding of the high frequencies during the quantization stage and the effects of the postfilter. We use a set of vector quantizers according to the vectorial output stream of the multiwavelet coefficients and the existence of fast quantizing and decoding algorithms [1]. In order to adjust the quantizers precision for each subband, a preliminary bit allocation scheme is performed. Section II presents a brief review on multiwavelet theory and mainly the multiwavelet transform algorithm. Section III describes the lossy compression algorithm based on optimal vector quantization of multiwavelet coefficients. Some experimental results are presented in Section IV.
A biorthogonal multiwavelet system consists of two multiscaling function vectors: !"$# and & % % ' % ( ) % !*"+# where ,- . is an integer. The case ,/ reduced the system to scalar wavelets. The multiscaling functions generate a multiresolution analysis pair consisting of 0214365 387 9 and 0 14% 3:5;3$7 9 of < =8>@?A! . and % satisfy the refinement equations: E'FHG
E F
F
F
G % %
=JI'BLKNMO? =JI'BLKNMO?
(1)
(2)
F
5 7 9 and 0 G % 5 7 9 are finite length real-valued where 02G matrix sequences. We can associated to and % , the biorthogonal multiwavelets vectors: PQR ST SU( SU!"$# and PQ % S % S % S % ! "$# . They are called multiwavelet functions. P and P % satisfy the following equations:
E FWV
F
=CBD?
P% F
F
=CBD? P
E'F
F
F
V%
%
=JI'BXKNMO? =JI'BXKNMO?
(3)
(4)
F
where 0 V 5 7 9 and 0 V % 5 7 9 are finite length real-valued matrix sequences. F F F F The sequences 02G 5 7 9 and 0 G % 5 7 9 are called F F the matrix F coefficients of lowpass multifilters whereas
0 V 5 7 9 and F 0V % 5 7 9 constitute the corresponding matrix coefficients of highpass multifilters. The biorthogonal property of the multiscaling and multiwavelet functions implies the following relations: F
FE
G % #
G 7 9
II. T HE M ULTIWAVELET F ORMALISM FE
A. Scalar to vector wavelet We mainly present here the multiwavelet theory which is a review of the scalar wavelets. It was shown in [2] that symmetry, orthogonality, compact support and approximation order can be simultaneously achieved for multiwavelets.
F
=CBD? %
F
F
=CBD?
F
7 9 FE [
Z
F V
7 9
where
V% #
G
V% #
FY
Z
I'[
FY Z
Z
FY
ZC\]D
^O I'[
(5) (6)
Z \]D
if _`a^ and ^ otherwise in b .
(7)
and the global rate are given by:
B. Multiwavelet decomposition and reconstruction This subsection describes the multiwavelet decomposition and reconstruction algorithms. One have to note that the Mallat multiresolution algorithm [3] for scalar wavelet can not be used directly for multiwavelet filters, the matrix-based coefficients requires a vectorial input signal. The problem of obtaining the vector input streams from a given signal is known as prefilterI ing. In this paper, we use the multiwavelets of multiplicity ,c and achieve presentation F in oneF dimension.F F F F Denote 38d e d 3$d f*d 3$d "+# and % 38d g % d 3$dF % *d F 3$d "+# the multiscaling functions. Likewise, we define P 38d P % 38d the multiwavelet functions. Considering the multiresolution Y Y subspaces 14% 3 and 14% 3 , we have for each h-ijb : 14% 3lk 14% 3 and m (=8>@?! . 14% 3no< 3$7 9
Let consider now a vector signal pqi have: =uBD? p
#rd
FE 7 9wv
F
7 9wv F 3$d
where D/ . v
F
F
F
38d
p
P
v
F
7 9 V 7 9
3$ d
=CBD?A=CBD? F
3$d
F 3$ d
F
3$d
=CBD?
Using the refinement relation, for the following:
F v
E
7 9
F G % #
38 d v
¥§¦
¥§¦
¤
¨E
©wª
¦¬«®
¢.E
,/f
(15)
In this paper, we consider a set of ¯ vector quantizers. Let be the total number of the multiwavelet subbands. For the C± C± quantizer and the subband we have the distorsion d Z _ obtained from the rate , d Z . We choose for each multiwavelet subband an optimal quantizer ²( ³ in such a way that we minimize the global distortion: °
°µE ´ d Z }
(16)
=CBD?s=uBD?
hw
G
F F v
|
(11)
, we can deduce 3$d
3$d
(12)
7 9
F V% #
38 d
(13)
BIT ALLOCATION SCHEME
¥§¦
=
|·¶
?
|·¶ ? ¥§¦ = d Z E´ ,d Z ¸J¹ z º E ´ }` }
(18)
We describe succinctly the allocation algorithm: 1. Perform the multiwavelet transform of the SAR image with levels of decomposition. Multiwavelet coefficients are denoted » Z8d . 2. For each subband , evaluate the rate/distortion curve for the whole set of vectors quantizers ¼ . ¸ 7(½ = ? and ¸ 7(½ = ? denote the corresponding rates and distortions. ¶ 3. Allocate a value of . 4. For each subband , determine the minimum of the Lagrangian function: ² ³
=J¶s?
¸ = ?n|·¶
¥§¦ o'¾¿ ¸ 7(½
A. Allocation procedure The codec is assigned to a target rate , let us define , , the rate and the distortion associated to the coding of multiwavelet coefficients for the subband . The global distortion
(17)
Some minimization algorithms for this type of problems have been introduced in [4]. To solve this problem, we use the lagrangian multiplier rule:
v
E
°µE ´ ,*d ZU¤f }
¢
The multiwavelet filter bank uses a vectorial input stream p from a given one dimensional signal . In out case, , I F F*input Y and p is issued from ' f f"$# multiplied by the prefilter matrix . III. O PTIMAL
(14)
where the are evaluated with the MSE and the , with first order entropy. The objective of the allocation scheme is to distribute the ,* between subbands in such a way that the distorsion reaches the minimum while maintening the constraint £a¤ . We can formally write:
(10)
These relations represent the decomposition algorithm. Conversely, the reconstruction algorithm is defined as ( h` ): 3$d
,(
F F =uBD? (9) $3# d P % 3$d
¢.E
while maintaining the constraint: =CBD?
p
r E FE ~3 })r y 7 9
' 4¡
are defined by:
3$d
(8)
=uBD?n|
F 38d
v
and
, then we
=uBD?
% r d x
#rOyz { % r y d
FE
=C>@?A!
1s% rtkq
