Two different formal definitions of gray-scale reconstruction are presented. The use of gray-scale reconstruction in various image processing applications discussed to illustrate the usefulness of this transformation for image filtering and segmentation tasks. The standard parallel and sequential approaches to reconstruction are reviewed. It is shown that their common drawback is their inefficiency on conventional computers. To improve this situation, an algorithm that is based on the notion of regional maxima and makes use of breadth-first image scannings implemented using a queue of pixels is introduced. Its combination with the sequential technique results in a hybrid gray-scale reconstruction algorithm which is an order of magnitude faster than any previously known algorithm. >

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Morphological grayscale reconstruction in image analysis: applications and efficient algorithms

The results of a study on multiscale shape description, smoothing and representation are reported. Multiscale nonlinear smoothing filters are first developed, using morphological opening and closings. G. Matheron (1975) used openings and closings to obtain probabilistic size distributions of Euclidean-space sets (continuous binary images). These distributions are used to develop a concept of pattern spectrum (a shape-size descriptor). A pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals, as well as for discrete images, by developing a discrete-size family of patterns. Large jumps in the pattern spectrum at a certain scale indicate the existence of major (protruding or intruding) substructures of the signal at the scale. An entropy-like shape-size complexity measure is also developed based on the pattern spectrum. For shape representation, a reduced morphological skeleton transform is introduced for discrete binary and graytone images. This transform is a sequence of skeleton components (sparse images) which represent the original shape at various scales. It is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components; in addition, two-way correspondences are established among the degree of shape smoothing via multiscale openings or closings, the pattern spectrum zero values, and the elimination or nonexistence of skeleton components at certain scales. >

/pdf/pattern-spectrum-and-multiscale-shape-representation-p6m9mqq83k.pdf

Pattern spectrum and multiscale shape representation

This paper examines the set-theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations, which involve supremum/infimum operations and additions. Binary signals are classified as sets, and multilevel signals as functions. Two set-theoretic representations of signals are reviewed. Filters are classified as set-processing (SP) or function-processing (FP). Conditions are provided for certain FP filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as SP filters. The basic morphological operations of set erosion, dilation, opening, and closing are related to Minkowski set operations and are used to construct FP morphological filters. Emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions; the latter case allows the definition of morphological convolutions and correlations. Toward this goal, various properties of FP morphological filters are also examined. Linear shift-invariant filters (due to their translation-invariance) are uniquely characterized by their kernel, which is a special collection of input signals. Increasing linear filters are represented as the supremum of erosions by their kernel functions. If the filters are also discrete and have a finite-extent impulse response, they can be represented as the supremum of erosions only by their minimal (with respect to a signal ordering) kernel functions. Stable linear filters can be represented as the sum of (at most) two weighted suprema of erosions. These results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal-shaping strategies.

/pdf/morphological-filters-part-i-their-set-theoretic-analysis-4077mmj6zm.pdf

Morphological filters--Part I: Their set-theoretic analysis and relations to linear shift-invariant filters

Weighted Median (WM) filters have the robustness and edge preserving capability of the classical median filter and resemble linear FIR filters in certain properties. Furthermore, WM filters belong to the broad class of nonlinear filters called stack filters. This enables the use of the tools developed for the latter class in characterizing and analyzing the behavior and properties of WM filters, e.g. noise attenuation capability. The fact that WM filters are threshold functions allows the use of neural network training methods to obtain adaptive WM filters. In this tutorial paper we trace the development of the theory of WM filtering from its beginnings in the median filter to the recently developed theory of optimal weighted median filtering. Applications discussed include: idempotent weighted median filters for speech processing, adaptive weighted median and optimal weighted median filters for image and image sequence restoration, weighted medians as robust predictors in DPCM coding and Quincunx coding, and weighted median filters in scan rate conversion in normal TV and HDTV systems.

Weighted median filters: a tutorial

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described. >

Adaptive polynomial filters

This paper extends the theory of median, order-statistic (OS), and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions, dilations, openings, closings, and open-closings that commute with thresholding. The max-min representation of OS filters is introduced by showing that any median or other OS filter is equal to a maximum of erosions (moving local minima) and also to a minimum of dilations (moving local maxima). Thus, OS filters can be computed by a closed formula that involves a max-min on prespecified sets of numbers and no sorting. Stack filters are established as the class of filters that are composed exactly of a finite number of max-min operations. The kernels of median, OS, and stack filters are collections of input signals that uniquely represent these filters due to their translation-invariance. The max-min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal (with respect to a signal ordering) kernel elements, and also to a minimum of dilations by minimal kernel elements of dual filters. The representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum-of-products expressions of Boolean functions. It is also shown that median filtering (and its iterations) of any signal by convex 1-D windows is bounded below by openings and above by closings; a signal is a root (fixed point) of the median iff it is a root of both an opening and a closing; the open-closing and clos-opening yield median roots in one pass, suppress impulse noise similarly to the median, can discriminate between positive and negative noise impulses, and are computationally less complex than the median. Some similar results are obtained for 2-D median filtering.

R. Schafer

Papers

Morphological filters--Part II: Their relations to median, order-statistic, and stack filters