ADAPTIVE BASIS MATRIX FOR THE MORPHOLOGICAL FUNCTION PROCESSING OPENING AND CLOSING

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 proposed the basis matrix for effici ent representation of opening and closing in [1] and [2]. Withz this r epresentation, 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 skeiy symm etric, permitting to derive a simpler matrix representation for openin g and closing operators. Furthermore, we propose an adaptation algorit hm of the basis matrix for both opening and closing. The LMS and backp ropagation algorithms are utilized for adaptation of the basis matrix. At each iteration of the adaptation process, the elements of the basi s matrix are updated using the estimation of gradient to decrease the mean square error (MSE) between the desired signal and the actual alte r output. Some results of optimal morphological filters applied to two -dimensional (2-D) images are presented.