M. A. Schulze and Q. X. Wu. "Noise reduction in synthetic aperture radar imagery using a morphology-based nonlinear filter." Proceedings of DICTA95, Digital Image Computing: Techniques and Applications, pp. 661-666. (Brisbane, Australia, December 6-8, 1995.)
Imaging techniques using coherent illumination, such as laser imaging and synthetic aperture radar (SAR), are subject to the phenomenon of speckle noise. Speckle noise is usually modeled successfully as a purely multiplicative noise process. An interesting property of the SAR speckle noise model is that the ratio of the standard deviation to the signal value is theoretically constant at every point in a SAR image. This ratio is called the coefficient of variation. Estimates of the coefficient of variation made over local windows are usually close to the theoretical value in areas where the signal is constant, but windows that extend over a region where the signal changes significantly give estimates of the coefficient that are larger than the theoretical value. This observation can be used as part of a nonlinear filter structure to ensure that smoothing is performed over constant signal areas and not across edges and other areas where the signal changes rapidly.
The value-and-criterion filter structure is a new structure for designing nonlinear filters based on mathematical morphology. Value-and-criterion filters have a "value" function (V) and a "criterion" function (C), which operate independently on the original image, and a "selection" operator (S) which acts on the output of C. The selection operator chooses a location from the output of C, and the output of V at that point is the output of the overall filter. This structure is similar to the morphological opening and closing operators, except that the value function and criterion function operate in parallel as the first step in the filtering process. The value-and-criterion structure is also much more flexible than morphological structure because it allows the use of different linear and nonlinear elements in a single filter. The shape control provided by the morphological structure is retained by value-and-criterion filters, however.
We have used the value-and-criterion filter structure to design a filter to reduce speckle noise in SAR imagery. As noted above, the sample coefficient of variation reflects the degree of homogeneity of the signal over the sample window; a high coefficient indicates a rapidly changing signal, whereas an area of constant signal returns a low coefficient of variation. Using the sample coefficient of variation (that is, the sample standard deviation divided by the sample mean) as the criterion function, the sample mean as the value function, and the minimum operator as the selection function yields a value-and-criterion filter suitable for speckle reduction in SAR images. The sample mean (a simple low-pass filter) provides the signal estimation, and the filter structure gives the output of this estimator in the region with the smallest coefficient of variation. These regions are less likely to contain significant features or edges which would be distorted by low-pass filtering. We call this new filter the "Minimum Coefficient of Variation" (MCV) filter and demonstrate its ability to reduce speckle noise and enhance edges simultaneously.
The operation of the MCV filter can be visualized in a different way that helps explain its behavior. If the basic functions of the MCV filter are performed over a 3x3 window, the overall window from which the MCV filter draws its values is 5x5. Within this overall window, there are nine 3x3 subwindows. The MCV filter effectively finds which of these nine subwindows has the smallest coefficient of variation, and returns the mean over that 3x3 window. The operation of the filter described earlier is much more efficient, but functionally the two descriptions are the same. The number of subwindows increases dramatically with window size: there are 25 subwindows of size 5x5 in an overall window of size 9x9.
We illustrate the use of the MCV filter on phantom images with simulated SAR speckle noise and on real SAR images. The MCV filter is shown to be a significant improvement over established SAR speckle removal techniques both in reducing noise and enhancing edges. The MCV filter is an excellent noise reduction technique for SAR imagery that is particularly useful as part of edge detection and segmentation algorithms.
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