Soft morphological filtering

Abstract

Abstract. Stack filters are widely used nonlinear filters based on threshold decomposition and positive Boolean functions. They have shown to form a very large class of filters which includes rank-order operations as well as standard morphological operations. The stack filter representation of an order statistic filter provides an efficient tool for the theoretical analysis of the filter.Soft morphological filters form a large subclass of stack filters. They were introduced to improve the behavior of standard morphological filters in noisy conditions. In this paper, different properties of soft morphological filters are analysed and illustrated. Their connection to stack filters is established, and that connection is used in the statistical analysis of soft morphological filters. Soft morphological filters are less sensitive to additive noise than standard morphological, filters. The deterministic properties of soft morphological filters are also analysed and it is shown that soft morphological filters form a class of filters with many desirable properties. For example, they preserve well details of images.

Binary, gray-scale, and vector soft mathematical morphology: Extensions, algorithms, and implementations

“…From Figures 2b-d it is obvious that, as in gray-scale soft morphology, the greater the value of the order index, the better the detail preservation. Comparing Figures 2b-d, it can be also observed that the smaller the value of k, the closer the behavior of a vector soft morphological transform is to that of the corresponding vector standard morphological transform, just as in the case of soft gray-scale morphology (Kuosmanen and Astola, 1995). This is one more similarity of vector soft and gray-scale soft morphological transforms.…”

“…Soft morphological operations are based on weighted order statistics and, therefore, algorithms such as mergesort and quicksort, which were developed for the computation of weighted order statistics, can be used for the computation of soft morphological filters (Kuosmanen and Astola, 1995). The average complexity of the quicksort algorithm is O (N log N), where N is the number of elements to be sorted (Pitas and Venetsanopoulos, 1990).…”

“…Therefore, in this case the nature of soft morphological operations is not preserved (Kuosmanen and Astola, 1995;Pu and Shih, 1995). For this reason the constraint k ≤ min{Card(B)/2, Card(B 2 )} is used.…”

Fuzzy soft mathematical morphology

“…[9,10] In soft morphological operations the maximum or the minimum operations, used in standard morphology, are replaced by weighted order statistics.…”

“…The weights depend on the structuring element, which is divided into two parts : the core, the pixels that participate with weights greater than one, and the soft boundary, the pixels that participate with weights equal to one. It has been shown that soft morphological operations are less sensitive to additive noise and to small variations in object shape [10][11][12].…”