Summary form only given. Two approaches have been used in the past to design rank-order-based nonlinear filters to enhance or restore images: the structural approach and the estimation approach. The first approach requires structural descriptions of the image and the process which has altered it, whereas the second required statistical descriptions. The many different classes of rank-order-based filters that have been developed over the last few decades have been reviewed in the context of these two approaches. One of these filter classes, stack filters, has been investigated. These filters, which are defined by a weak superposition property and an ordering property, contain all compositions of 2D rank-order operations. The recently developed theory of minimum-mean-absolute-error (MMAE) stack filtering has been extended to two dimensions. A theory of optimal stack filtering under structural constraints and goals has been developed for the structural approach to image processing. These two optimal stack filtering theories have been combined into a single design theory for rank-order-based filters. >

Optimal stack filtering and the estimation and structural approaches to image processing

Within the last two decades a small group of researchers has built a useful, nontrivial theory of nonlinear signal processing around the median-related filters known as rank-order filters, order-statistic filters, weighted median filters, and stack filters. This required significant effort to overcome the bias, both in education and research, toward linear theory, which has been dominant since the days of Fourier, Laplace, and “Convolute.” We trace the development of this theory of nonlinear filtering from its beginnings in the study of noise-removal properties and structural behavior of the median filter to the recently developed theory of optimal stack filtering. The theory of stack filtering provides a point of view which unifies many different filter classes, including morphological filters, so it is discussed in detail. Of particular importance is the way this theory has brought together, in a single analytical framework, both the estimation-based and the structural-based approaches to the design of these filters. Some recent applications of median and stack filters are provided to demonstrate the effectiveness of this approach to nonlinear filtering. They include: the design of an optimal stack filter for image restoration; the use of vector median filters to attenuate impulsive noise in color images and to eliminate cross luminance and cross color in TV images; and the use of median-based filters for image sequence coding, reconstruction, and scan rate conversion in normal TV and HDTV systems.

An overview of median and stack filtering

A theory for the structural behavior of stack filters is developed. This theory provides a test which can determine if a given stack filter has any root signals; a method for classifying the root signal behavior of any stack filter found to have roots; and, perhaps most important, a method for designing stack filters with specific root signals or other structural behavior. This theory of root signals for stack filters is then combined with the theory of minimum mean absolute error stack filtering. This unified theory allows the designer to pick a filter which minimizes noise subject to constraints on its structural behavior. >

Minimum mean absolute error stack filtering with structural constraint and goals

A new joint region-merging criterion for watershed-based image segmentation is proposed. The proposed method combines, in a weighted fashion, criteria based on region homogeneity and edge integrity, which have each been applied separately in image and video segmentation with some success. However, each criterion has shown some significant drawbacks when applied independently. The new criterion takes joint advantage of these two methods, giving a final segmentation that is more visually appropriate. Additionally, a nonlinear pre- and postfiltering procedure is proposed for reducing oversegmentation in the watershed algorithm. This procedure preserves the location of the watershed boundaries and yields a merging procedure that requires fewer steps to produce a final segmentation. Experimental results are presented showing the advantages of the new merging criterion and the pre- and postfiltering method.

Region merging using homogeneity and edge integrity for watershed-based image segmentation

The notion of the on-set of a positive Boolean function is used to classify stack filters into three different types, called decreasing, increasing, and mixed. The associative memory capability of each of these three types of stack filters is then investigated. The associative memory of a stack filter is defined to be the set of root signals of that filter. In a class of stack filters in which each filter's root set contains a desired set of patterns, those filters whose root sets have the smallest cardinality are said to be minimal among all filters in that class for that set of patterns. Some learning schemes are proposed to find minimal decreasing and increasing stack filters. It is also shown that, for any specified set of patterns, there is always a mixed stack filter which is minimal when one considers all stack filters which preserve those patterns. In this sense, mixed stack filters are always at least as good as decreasing or increasing stack filters. >

The classification and associative memory capability of stack filters

Median filtering is a simple digital technique for smoothing signals A root signal is defined as an invariant signal to the median filtering We describe directed graph representation for the root-signal set of median filters The directed graph representation allows us to obtain a set of roots and the number of roots in a straightforward manner

http://ieeexplore.ieee.org/iel5/5/31379/01458185.pdf

Directed graph representation for root-signal set of median filters

We described spiking filtering of all-pass filter impulse responses. We established a rule about the optimum delay and obtained the optimum spiking filter. We also showed that this result can be successfully applied to additional filtering to increase the resolution even further.

Souguil Ann

Papers

Directed graph representation for root-signal set of median filters

Spiking filtering of all-pass filter impulse responses