A method for adaptation of the basis matrix of the gray-scale function processing (FP) opening and closing under the least mean square (LMS) error criterion is presented. We previously proposed the basis matrix for efficient representation of opening and closing (see IEEE Trans. Signal Processing, vol.43, p.3058-61, Dec. 1995 and IEEE Signal Processing Lett., vol.2, p.7-9, Jan. 1995). With this representation, the opening and closing operations are accomplished by a local matrix operation rather than cascade operation. Moreover, the analysis of the basis matrix shows that the basis matrix is skew symmetric, permitting to derive a simpler matrix representation for opening and closing operators. Furthermore, we propose an adaptation algorithm of the basis matrix for both opening and closing. The LMS and backpropagation algorithms are utilized for adaptation of the basis matrix. At each iteration of the adaptation process, the elements of the basis matrix are updated using the estimation of gradient to decrease the mean square error (MSE) between the desired signal and the actual filter output. Some results of optimal morphological filters applied to two-dimensional (2-D) images are presented.
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