Design of statistically optimal stack filters

Any gray-scale image can be represented as a "stack" of a decreasing sequence of binary images, obtained by thresholding the gray-scale image at each level. Stack filters are a special class of gray-scale image operators whose filtered images can be represented as the stack of binary images resulting from applying an increasing binary operator for each image of the stack. A classical example of stack filter is the median filter: the median of a gray-scale image is the same as the sum of the binary median computed for each of the binary images in the stack. Thus, the design of stack filters can be reduced to the design of the binary operators that characterize them. The paper reviews stack filters in the context of mathematical morphology on discrete images, and shows that the problem of designing optimal mean-absolute error (MAE) stack filters is an optimization problem equivalent to the design of optimal MAE binary increasing operators. A new combinatorial algorithm for designing binary increasing operators is applied on the design of stack filters for impulse and speckle noise reduction.