Sample and rank selection probabilities are important in the analysis of weighted median filters as well as in comparing linear and nonlinear filters. We propose a new, efficient algorithm to calculate these probabilities for weighted median filters. The computational savings of this new algorithm are substantial and compare favorably with other known algorithms.

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.

A real-time implementation method for the most general morphological system, the so-called grayscale function processing (FP) system is presented. The proposed method is an extension of our previous works (1993, 1995) using the matrix representation of the FP system with a basis matrix (BM) and a block basis matrix (BBM) composed of grayscale structuring elements (GSE). In order to further improve the computational efficiency of the basis matrix representation, we propose recursive algorithms based on the observation of the BM and BBM. The efficiency of the proposed algorithms is gained by avoiding redundant steps in computing overlapping local maximum or minimum operations. It is shown that, with the proposed scheme, both opening and closing can be determined in real time by 2N-2 additions and 2N-2 comparisons, and OC and CO by 4N-4 additions and 4N-4 comparisons, when the size of the GSE is equal to N. It is also shown that the proposed recursive opening and closing require only 3N-3 memory elements.

We propose a method for the real-time implementation of function processing (FP), open-closing (OC), and close-opening (CO) morphological operations. The proposed method is based on the block basis matrix (BBM), which is an extension of the basis matrix of Ko and Shridar (1990). A procedure to obtain the BBM is proposed. This matrix is skew symmetric, and some related properties lead to simplify the FP morphological operations. It is shown that the real-time calculation of open-closing and close-opening is accomplished by local matrix operations rather than cascade operations (opening/closing followed by closing/opening), thus eliminating delays and requiring less memory storage.>

The aim of this paper is to find a relationship between alternating sequential filters (ASF) and the morphological sampling theorem (MST) developed by Haralick et al. (1987). The motivation behind this approach is to take advantage of the computational efficiency offered by the MST to implement morphological operations. First, we show alternative proofs for opening and closing in the sampled and unsampled domain using the basis functions. These proofs are important because they show that it possible to obtain any level of a morphological pyramid in one step rather than the traditional two-step procedure. This decomposition is then used to show the relationship of the open-closing in the sampled and unsampled domain. An upper and a lower bound, for the above relationships, are presented. Under certain circumstances, an equivalence is shown for open-closing between the sampled and the unsampled domain. An extension to more complicated algorithms using a union of openings and an intersection of closings is also proposed. Using the Hausdorff metric, it is shown that a morphologically reconstructed image cannot have a better accuracy than twice the radius of the reconstruction structuring element. Binary and gray scale examples are presented.

A multiscale filtering scheme based on the three Matheron axioms for morphological openings is developed. It is shown that opening a signal with a gray scale operator does not introduce additional zero-crossings as one moves to coarser scales. Within this framework, the problem of choosing an appropriate structuring element is studied. In order to obtain a measure of the performance of different structuring elements, the statistical properties of gray scale opening are studied, using a powerful tool in mathematical morphology, namely, basis functions.>

The aim of this research is to obtain a morphological pyramid with the aid of alternating sequential filters. The authors establish a relationship between alternating sequential filters and the morphological sampling theorem developed by R. Haralick et al. (1987). An alternative proof for opening and closing in the sampled and unsampled domain using the basis functions is shown. This decomposition is then used to show the relationship of the compound mapping opening-closing in the sampled and unsampled domain. An upper and a lower bound for the above relationships are presented. Under certain circumstances, an equivalence is shown for opening-closing between the sampled and the unsampled domain. An extension to more complicated algorithms using union of openings and intersection of closings is also proposed.>