Applcation of Multiscale Edge Detection to Speckle Reduction of SAR images
2.1 Region Segmentation
The method of multiscale edge detection described in (Mallat, 1992) is used to find the edges. This wavelet is nonsubsampled wavelet decomposition essentially implement the discretized gradient of the image at different scales. Assume f(x,y) is a given SAR image of size MxN. At each scale j with j>0 and sof = f(x,y), the wavelet transform decomposes
S
j-1 f into three wavelet bands: a lowpass band
S
if, a horizontal highpass band W
Hi f and a vertical highpass band W
Hi f.
The three wavelet bands (S
iF, W
Hi f, W
vi f) at scale j are of size MxN, which is the same as the original image, and all filters used at scale I(j>0) are upsampled by a factor of 2j compared with those at scale zero. Form traditional edge detection, one knows that the points of sharp variations occur at local maxima (cal modulus maxima) of the gradient nrom in the direction of the gradient. However, we do not perform maximum detection and whenever processing has the drawback of high computation complexity that requires a long computation
time. This evaluation can be characterized by the local Lipschitz regularity. That is, if the function f is Lipschitz
a at point
(x
o, y
o), then for (x,y) in its neighborhood,
Mjf(x,y)£K ja (1)
where M
jf denotes the modulus of the wavelet transform at scale j and k is a constant. Since the Lipschitz exponents show the local regularity of image, it is a good indicator to identify these regions. We refer to edge regions that have high local variation of regularity among the neighboring coefficients. From (1), if the modulus of the wavelet transform is greater than K j
a it belongs to edge region, otherwise belongs to non-edge region.
2.2 Wavelet Thresholding
The theoretical formalizationof thresholding in the context of removing noise via thrsholding wavelet was presented by (Donoho, 1995). The idea is that, in wavelet domain, coefficients insignificant relative to the threshold are likely due to noise, whearas significant coefficients are important signal structures. There are two thresholding schemes: (a) hard-thresholding and (b) soft-thresholding. The operations are defined by:
where the threshold
d is an estimation from the wavelet coefficients Y. There is such a simple relative to find a threshold
d. We consider
d from estimation for the standard deviation of SAR image in wavelet domain as:
fs
where N is the number of the iamge data, m the mean of wavelet coefficients
C
h,l, with b represents each of the high-frequency bands
(W
Hj f and W
vj f) and l the number of decomposition levels, and
f
a a thrshold factor systermatically assigned to 1/l.
Donoho shown that if the error or noise is bound then soft-thresholding is optimal. In general, the noise is uniformly distributed over all levels and show clearly in high-frequency bands. The soft-thresholding is therefore should be performed on all bands, with the exception to the lowest-frequency band. In this paper, we performed soft-thresholding in non-edge region of highpass bands. Moreover, in our method, the thresholding is applied only to the coefficients corresponding to non-edge regions.
3.Comparison Measures
Two quantitative measures were used to evaluate speckle reduction techniques, including the signal to mean square error ratio (S/MSE) and the mean value.
The Signal to Mean Square Error Ratio (S/MSE) ratio is used to quantify degradation in resolution. For best results, the S/MSE ratio should be maximized. If the original is l1 and the despeckled image is l2, it is defined as:
Another measure is the mean of the image. Speckle reduction techniques should in the image with a mean value close to that of the original one.
