A Fuzzy Filter for SAR Image De-noising Yilun Chen, Fuyue Huang, Jian Yang Department of Electronic Engineering, Tsinghua University [email protected]

Abstract In this paper, a new approach to speckle filtering of synthetic aperture radar (SAR) data is presented. The idea of fuzzy window is firstly presented, where the similarity of scattering mechanism between the central filtered pixel and neighborhood pixel is depicted by the fuzzy membership function. Based on the fuzzy window, a fuzzy filter is proposed; where different pixels in the neighborhood window make different contribution to the local statistics calculation, according to their membership function value. By paying more attention to the similar scattering mechanism pixels, better performance is achieved, especially for edge/structure preserving. The effectiveness of the fuzzy filter is demonstrated using the National Aeronautics and Space Administration Jet Propulsion Laboratory airborne SAR data.

1. Introduction Synthetic aperture radar (SAR) is attracting more and more attention recently for its all-day and allweather earth observation capability. However, the coherent speckle noise which is caused by SAR imaging brings many difficulties to its applications, such as terrain classification, target detection and etc. For years many approaches have been developed for speckle noise reduction. In particular, Lee proposed a de-noise algorithm based on MMSE (minimum mean square error) criteria [1], Novak and Burl [2] derived the polarimetric whitening filter (PWF) by optimally combining all elements of the polarimetric covariance matrix, which generates a minimum-speckle intensity image from polarimetric SAR data. Speckle filters using different criteria such as maximum a posteriori (MAP) [5] probability and the maximum likelihood (ML) [4] have also been well studied. The existing techniques need to estimate the local statistical characteristics from the neighboring region of the filtered pixel, and most of them choose the simplest “box-car” way, i.e., using all the pixels in a square area centered at the one to be filtered. However,

the “box-car” based filters can not achieve satisfied performance on edge/structure preservation, in other words, edges or other structures (such as roads) are easily blurred or lost during the filtering process. This is because in a square area, not all pixels have the same scattering mechanism, whereas using pixels with a different scattering mechanism to calculate local statistics is somewhat inappropriate. To solve this problem, Lee refined his original method by changing square windows to several typical non-square windows [6]. Then the one in which elements have the most uniform scattering characteristics is chosen and applied to the local statistics estimation. This method achieves some improvement in edge preservation; but for many cases, the distribution of terrain targets are too complex to be described by only a few typical nonsquare windows. Moreover, as the window size increases, the filtering performance would further deteriorate. To solve this tradeoff problem between the filtering performance and the window choice, Jing Gu et al. [7] presented a method based on subspace decomposition to generate a speckle-reduced image. In this method, detailed statistics and selection of a special window shape are not necessary. However, it's only applicable to polarimetric SAR images. Based on above observations, we provide a new algorithm in this paper. The basic idea is inspired from fuzzy mathematics. We aim to keep uniform scattering mechanisms between pixels concerned in the filtering process. Firstly, we introduce the concept of fuzzy window, where fuzzy membership function is employed to measure the similarity of the scattering mechanism between two pixels. The formula of membership function is presented, regarding the multiplicative property of noises in SAR images. Then a fuzzy filter is proposed for SAR image filtering. The experimental results demonstrate the proposed fuzzy filter's capability of selection uniform-scatteringmechanism neighbor windows adaptively, and its advance in both noise reduction and edge/structure preservation as well.

2. The Fuzzy Filtering Framework In this section, we first introduce the concept of fuzzy window, which is a more generalized form comparing to the traditional edge-aligned-window. Then we provide a formula to calculate the fuzzy membership function. Finally, we derive the proposed filtering method based on the fuzzy-weighted MMSE rule.

2.1 Edge-aligned-window v.s Fuzzy window To preserve edges during speckle reduction, the common approach is to use edge-aligned-window for filtering (see Fig. 1), where only a portion of pixels in the neighborhood window take account in the filtering process. For instance, Lee introduced a series of edgealigned-windows and selected the most appropriate one for filtering. However, this kind of edge-alignedwindow may lead to several problems. For example, the distribution of the terrain targets is always too complex for a simple shape window to cover the same scattering-mechanism pixels.

Denote the central pixel (the pixel to be filtered) in the fuzzy window as ( i , j ) , the neighborhood pixel as

( i + m, j + n )

, the fuzzy membership value is

( m,n )

denoted as w( i , j ) . As mentioned in the above section, ( m,n )

w( i , j ) measures the possibility that the two pixels

( i, j )

and

( i + m, j + n )

belongs to the same

scattering mechanism. Given a SAR image with its power intensity denoted as z , we would like to define the membership function based on the intensity value. ( m ,n )

The calculation formula of w( i , j ) is given by

⎛

w( i , j ) = exp ⎜ − β log ( m,n )

⎝

2

⎛ 1 + zi , j ⎞ ⎞ ⎜ ⎟⎟ , ⎝ 1 + zi + m , j + n ⎠ ⎠

(1)

where zi , j and zi + m , j + n denote the intensity values of pixel ( i , j ) and ( i + m, j + n ) , respectively; β is a constant parameter. One can easily figure out that the smaller contrast between the two pixels is, the closer ( m,n )

w( i , j ) reaches to 1, which indicates the more likely the

two pixels belong to the same scattering mechanism. The reason we adopt the expression in eq.(1) instead of simple difference of the intensity value rises from the multiplicative noise of SAR images. The multiplicative noise model of SAR images can be described as follows: zi , j = xi , j vi , j , (2) Fig. 1 Eight edge-aligned windows. Depending on the edge direction, one of the eight windows is to be selected. Pixels in white are used in the filtering computation.

where zi , j denotes the observed pixel value at pixel

The fuzzy window is shown in Fig. 2, where each pixel in the window is assigned a membership function value ranging from 0 to 1. One can easily see that the edge-aligned-window can be regarded as a special form of the fuzzy window, where the pixels inside the window take the membership function value as 1, the pixels outside the window take as 0.

estimated and vi , j denotes the noise term. From eq.(2),

Fig. 2 Fuzzy window (right) v.s. edge-aligned-window (left), different grayscales stand for different membership function values.

2.2 Definition of fuzzy membership function

( i, j )

, xi , j denotes the noise-free value to be

it can be noticed that, within a homogeneous area, the larger the mean power, the larger the intensity variation. So, if we use the simple difference of the intensity value to calculate the membership function, the membership function value would vary significantly in the high intensity region even the region may be homogeneous. Instead, it has been generally accepted that by adopting the function form in eq.(1), the variation of the membership function values is able to remain to the same extent, regardless the high intensity homogeneous region or low intensity homogeneous region.

2.3 The Fuzzy filtering algorithm Given the membership function between the central pixel and the neighborhood pixel defined in the

above section, we then present the proposed filtering algorithm based on the fuzzy-weighed MMSE rule.

( var ( z ) ) i, j

∑ w(( w

=

m,n

∑

2.3.1 Lee's MMSE Filter

2

zi + m , j + n

− ( zi , j ) w . 2

(m,n) w( i , j )

(6)

m,n

Based on the multiplicative noise model in eq.(2), the linear minimum mean-square filter was proposed by Lee et al. [1], which has shown good performance for SAR speckle filtering. Lee's filter needs to calculate the local statistics in the local neighborhood window of the filtered pixel. The filter is (3) xˆi , j = zi , j + k i , j ( zi , j − zi , j ) , where xˆi , j is the filtered value of pixel ( i , j ) , zi , j is the local mean, and ki , j is the weighting function having a value between 0 and 1. The parameter ki , j adjusts the tradeoff between the observed value zi , j and the local mean zi , j , which is computed by var ( zi , j ) − zi , jσ v 2

ki , j =

m,n)

i, j)

2

(1 + σ ) var ( z ) 2

v

,

(4)

i, j

where var ( zi , j ) is the local variance and σ v is the

Eq.(5) and eq.(6) are then substituted to eq.(4) yielding w

the coefficient ki , j : w

ki , j

( var ( z ) ) − ( z ) σ = (1 + σ ) ( var ( z ) ) i, j

w

i, j

2

2

w

v

2

v

i, j

.

(7)

w

w

By replacing k i , j of ki , j in eq.(3), we then get our fuzzy filter. Due to the introduction of membership function between the central pixel and the neighborhood pixel, more contribution would be provided by the highmembership value pixel when calculating the local statistics. The introduction of membership function between pixels helps to make more concentration on similar scattering mechanism pixels when calculating the local statistics. Therefore, it is promising that more reasonable filtering results would be obtained as well as good preservation for edge and structure during filtering process.

2

prior estimation of the noise's variation. It has been

proved that the larger var ( zi , j ) is, the more extent

ki , j reaches to 1 so xˆi , j is dominated by the local mean zi , j ; on the other hand, the less var ( zi , j ) is, the more ki , j reaches to 0, in this case, xˆi , j is mainly determined by the observed pixel value zi , j . 2.3.2 The Fuzzy Filter Lee's filter needs to estimate the local statistics from each filtered pixel's local window, which hints that the choice of this window is important for the performance of the filter. This point is also mentioned in the above section, where we propose the fuzzy window to solve this problem. Given the fuzzy membership function between the neighborhood pixel and the central pixel, the local statistics of Lee's filter is modified accordingly as

(z ) i, j

∑ w((

m,n)

zi + m , j + n

i, j)

w

=

m,n

∑ w((

m,n)

i, j)

m,n

,

(5)

3. Experimental Results The proposed algorithm is evaluated using two sets of data, collected by the National Aeronautics and Space Administration Jet Propulsion Laboratory airborne L-band SAR. The power images are used for filtering. For better visualization, only the selected areas of the two images are displayed. In the first image, we compare our method with Lee's filter [1] and the revised Lee's filter [6]. In the revised Lee's filter, 8 edge-aligned-windows are used (as shown in Fig. 1 and the one with the least variation is selected for filtering. For comparison, the three filters' parameters are set to the same value: the moving average window sizes are all set to 9 by 9 and σ v is 2

set to 0.1. Experimental results show the fuzzy filter's advance in both de-noising and edge/structure preserving. It can be seen that Lee's filter got the smoothest result (see Fig. 3 (b)) but lost most of the detailed features; the revised Lee's filter (see Fig. 3 (c)) tries to obtain an edge-preserving image by the averaging in the edge-aligned-window, however, the result is inconsistent with the original image, for some edges actually do not exist in the original image. The result from the new method is shown in Fig. 3 (d), the edges and the linear structures (such as roads in the blocked region) are well preserved; in addition, the

filtered image is the most consistent with the original one.

Fig. 3 (a) The filtered image by the fuzzy filter, (b)the filtered image by “box-car” MMSE filter, (c) the filtered image by revised Lee’s filter using the edge-aligned window, (d) the filtered image by the fuzzy filter.

To further show the details of the proposed algorithm, some intermediate results are plotted. In Fig. 4, three pixels are selected, which belong to homogeneous area (a), road area (b) and edge area (c), respectively. Their corresponding fuzzy windows are plotted on the right side, where the lighter the square is, the larger the membership value is. These plots demonstrate that the fuzzy filter is capable of adaptive selecting neighborhood window, which is essential for edge/structure preserving filtering.

4. Conclusion In this paper, a novel fuzzy filter for SAR image denoising is proposed. Unlike the traditional filters which use box-car window or edge-aligned-windows, the proposed filter use fuzzy window to measure the uniform properties of SAR data of local pixels, where more concentration is paid to high membership-valued pixels. The fuzzy filter provides a more effective and

flexible way for local window shape selection, which is essential for preserving edge/structure during the filtering. Experimental results with detail intermediate results have shown the effectiveness of the proposed filter. The new method is proposed for SAR speckle noise reduction. However, our filtering framework is also capable of filtering for polarimetric SAR data. For the polarimetric SAR data, much more information can be obtained, which hints that more effective method for calculating the membership function could be devised. This would be included in our future work.

Fig. 4 Fuzzy windows of typical pixels: (a) homogeneous area, (b) road area, (c) edge area.

Acknowledgement This work was supported by the National Important Fundamental Research Plan of China (2001CB309401) and by the Fundamental Research Foundation of Tsinghua University.

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[4] I. R. Joughin, D. P. Winebrenner, and D. B. Percival, “Maximum likelihood estimation of K-distributed parameters for SAR data,” IEEE Trans. Geosci. Remote Sensing, vol. 31, pp. 989-999, Sept. 1993. [5] A. Lopes, E. Nezry, R. Touzi, and H. Laur, “MAP speckle filtering and first order texture models in SAR images,” in Proc. IGARSS, Washington, DC, May 1990, pp. 2409-2412. [6] J S. Lee，”Refined filtering of image noise using local statistics,” Comput. Graph. Image Proc., 1981, 15(3): 380-389. [7] J. Gu, J. Yang, H. Zhang and et al, “Speckle filtering in polarimetric SAR data based on the subspace decomposition”, IEEE Trans. on Geoscience and Remote Sensing, vol. 42, Aug. 2004.