Morphological Dilation Research Articles

Mutational equations of the morphological dilation tubes

The present paper provides some differential results dealing with the morphological dilation of a compact set in the nonregular case. Indeed the evolution of dilated sets with respect to time is characterized through mutational equations which are new mathematical tools extending the concept of differential equations to the metric space of all nonempty compact sets of R n Using the generalized gradient notion and viability theorems, we prove that a mutation of the dilation tube corresponds to the normal cone which is a generalization of the classical notion of normal

Algebraic Models for Feature Displacement in the Generalization of Digital Map Data Using Morphological Techniques

Feature displacement is one of the many operations (or operators) in map-data generalization. It has become a priority item on the research agenda. This paper describes some algebraic models (or mathematical models) for this operation. These models are based on the operators developed in mathematical morphology, which is a science dealing with form, shape, and structure of objects. Feature displacements can be classified into two groups, i.e., feature translation and feature modification. For the first type, a set of structuring elements (key elements in morphological operators) are developed. Using this set of structuring elements, the two basic morphological operators, i.e., dilation and erosion, can be used for translating features in any of the eight directions, freely. Translation of features in other directions can be achieved by a combination of these eight directions. For the second type of feature displacement, a number of models have been developed to suit different cases, i.e., an area feature and a linear feature, two linear features, etc. These models have also been tested using various examples. The results show that these techniques are very promising.

Slope transforms: theory and application to nonlinear signal processing

Fourier transforms are among the most useful linear signal transformations for quantifying the frequency content of signals and for analyzing their processing by linear time-invariant systems. Some nonlinear signal transforms are developed that can provide information about the slope content of signals and are useful analytic tools for large classes of nonlinear systems. Many of their theoretical properties are examined, showing a striking conceptual resemblance to Fourier transforms and their application to linear systems. These novel transforms, called slope transforms, are originally derived from the eigenvalues of morphological dilation and erosion systems, where the corresponding eigenfunctions are lines /spl alpha/t+b parameterized by their slope /spl alpha/. They obey a nonlinear superposition principle of the supremum- or infimum-of-sums type. Applied to the impulse response of dilation or erosion systems, the slope transforms provide a slope response function for these systems, which allows their analysis and design in a transform domain, the slope domain. Applied to arbitrary signals, the slope transforms provide information about upper or lower tangents to the signal's graph at varying slopes. The upper or lower envelopes of the signal can be obtained from the inverse transforms. Overall, the slope transforms provide a new transform domain for signals and morphological systems where time lines become slope impulses, time cones become slope bandpass filters, and time dilation/erosion transform into addition of slope transforms. Their application to the design of slope-selective filters is also presented. >

Fully automatic determination of soil bacterium numbers, cell volumes, and frequencies of dividing cells by confocal laser scanning microscopy and image analysis

We describe a fully automatic image analysis system capable of measuring cell numbers, volumes, lengths, and widths of bacteria in soil smears. The system also determines the number of cells in agglomerates and thus provides the frequency of dividing cells (FDC). Images are acquired from a confocal laser scanning microscope. The grey images are smoothed by convolution and by morphological erosion and dilation to remove noise. The background is equalized by flooding holes in the image and is then subtracted by two top hat transforms. Finally, the grey image is sharpened by delineation, and all particles above a fixed threshold are detected. The number of cells in each detected particle is determined by counting the number of local grey-level maxima in the particle. Thus, up to 1,500 cells in 10 fields of view in a soil smear are analyzed in 30 min without human intervention. Automatic counts of cell numbers and FDC were similar to visual counts in field samples. In microcosms, automatic measurements showed significant increases in cell numbers, FDC, mean cell volume, and length-to-width ratio after amendment of the soil. Volumes of fluorescent microspheres were measured with good approximation, but the absolute values obtained were strongly affected by the settings of the detector sensitivity. Independent measurements of bacterial cell numbers and volumes by image analysis and of cell carbon by a total organic carbon analyzer yielded an average specific carbon content of 200 fg of C (mu)m(sup-3), which indicates that our volume estimates are reasonable.

A digit-serial architecture for gray-scale morphological filtering

We present a digit-serial architecture for gray-scale morphological operations that operate on radix-2 redundant numbers. We present new implementations for a redundant number adder and a maximum unit that are used in the morphological dilation unit. These new designs have areas comparable to 2's complement implementations but have significantly smaller latencies.

Second-order statistics of morphological dilation and erosion of a memoryless source

The paper shows that morphological dilation and erosion preserve strict sense stationarity, derives the 2-D probability distribution and density functions of dilated and eroded stationary memoryless sources, and presents numerical results for a memoryless uniform source, which shows identical closed-form autocovariance functions of dilated and eroded sources. >

Morphological systems: Slope transforms and max-min difference and differential equations

Linear time-invariant systems are well understood in the time domain either as convolutions with their impulse response or by describing their dynamics via linear differential equations. Their analysis in the frequency domain using their exponential eigenfunctions and related frequency response is also greatly facilitated via Fourier transforms. Attempting to extend such ideas to nonlinear systems, we present in this paper a theory for a broad class of nonlinear systems and a collection of related analytic tools, which parallel the functionality of and have many conceptual similarities with ideas and tools used in linear systems. These nonlinear systems are time-invariant dilations or erosions, in continuous and discrete time, and obey a supremum- or infimum-of-sums superposition. In the time domain, their equivalence with morphological dilation or erosion by their impulse response is established, and their causality and stability are examined. A class of nonlinear difference did differential equations based on max-min operations is also introduced to describe their dynamics. After finding that the affine signals αt + b are eigenfunctions of morphological systems, their slope response is introduced as a function of the slope α, and related slope transforms for arbitrary signals are developed. These ideas provide a transform (slope) domain for morphological systems, where dilation and erosion in time corresponds to addition of slope transforms. Recursive morphological systems, described by max-min difference equations, are also investigated and shown to be equivalent to dilation or erosion by infinite-support structuring elements. Their analysis is significantly aided by using slope transforms. These recursive morphological systems are applied to the design of ideal-cutoff slope-selective filters which are useful for signal envelope estimation.

Optical cellular fuzzy logic gray-scale image processor

A new architecture of cellular fuzzy logic array for gray-scale image processing is proposed, which consists of a parallel fuzzy logic processor and a morphological gray-tone dilation processor. All possible image processing functions based on the gray-scale morphological operations can thus be realized. The architecture can be optically implemented on the basis of spatial coding and thresholding of gray intensity. A primary experimental result is given.

Gibbs random fields, cooccurrences, and texture modeling

Gibbs random field (GRF) models and features from cooccurrence matrices are typically considered as separate but useful tools for texture discrimination. The authors show an explicit relationship between cooccurrences and a large class of GRF's. This result comes from a new framework based on a set-theoretic concept called the and on measures of this set, measures. This framework is also shown to be useful for relating different texture analysis tools. The authors show how the aura set can be constructed with morphological dilation, how its measure yields cooccurrences, and how it can be applied to characterizing the behavior of the Gibbs model for texture. In particular, they show how the aura measure generalizes, to any number of gray levels and neighborhood order, some properties previously known for just the binary, nearest-neighbor GRF. Finally, the authors illustrate how these properties can guide one's intuition about the types of GRF patterns which are most likely to form. >

Decomposition of morphological structuring elements

Mathematical morphology has become an important tool for machine vision since the influential work by Serra (1982). It is a branch of image analysis based on set-theoretic descriptions of images and their transformations. As is well known, the chain rule for basic morphological operations, i.e., dilation and erosion, lends itself well to pipelining. Specialized pipeline architecture hardware built in the past decade is capable of efficiently performing morphological operations. The nature of specialized hardware depends on the strategy for morphologically decomposing a structuring element. The two-pixel decomposition technique and the cellular decomposition technique are the two main techniques for morphological structuring-element decomposition. The former was used by the image flow computer and the latter by the cytocomputer.

Evolution equations for continuous-scale morphological filtering

Multiscale signal analysis has emerged as a useful framework for many computer vision and signal processing tasks. Morphological filters can be used to develop nonlinear multiscale operations that have certain advantages over linear multiscale approaches in that they preserve important signal features such as edges. The authors discuss several nonlinear partial differential equations that model the scale evolution associated with continuous-space multiscale morphological erosions, dilations, openings, and closings. These equations relate the rate of change of the multiscale signal ensemble as scale increases to a nonlinear operator acting on the space of signals. The nonlinear operator is characterized by the shape and dimensionality of the structuring element used by the morphological operators, generally taking the form of a nonlinear function of certain partial differential operators. >

Decomposition of geometric-shaped structuring elements using morphological transformations on binary images

Mathematical morphology which is based on geometric shape, provides an approach to the processing and analysis of digital images. Several widely-used geometric-shaped structuring elements can be used to explore the shape characteristics of an object. A unified technique is presented to simplify the decomposition of various types of big geometric-shaped structuring elements into dilations of smaller structuring components by the use of a mathematical transformation. Hence, the desired morphological erosion and dilation are equivalent to a simple inverse transformation over the result of operations on the transformed decomposable structuring elements. A strategy is also presented to decompose a large cyclic cosine structuring element. The technique of decomposing a two-dimensional (2D) convex structuring element into one-dimensional (1D) ones is also developed.

An algebra of polygons through the notion of negative shapes

A new notion of negative shape is introduced in order to develop an algebraic system of geometric shapes within which one can add and subtract shapes exactly as one adds and subtracts within the integer number system. Concentrating on polygonal shapes in 2 dimensions, we show that this simple extension of our commonsense concept of geometric shapes opens up many new areas with a great potential for understanding and developing 2-dimensional geometry and geometric algorithms. In the course of this pursuit the concept of a new equivalence relation on convex polygons evolves that also appears to be significant in understanding convex polygons, particularly various symmetries in them. In constructing the algebraic system of shapes we use the Minkowski addition operation (in mathematical morphology dilation) as the composition Operation.

A representation theory for morphological image and signal processing

A unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented. The unification requires a set-theoretic methodology, where signals are modeled as sets, systems (signal transformations) are viewed as set mappings, and translational-invariant systems are uniquely characterized by special collections of input signals. This approach leads to a general representation theory, in which any translation-invariant, increasing, upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations. The theory is used to analyze some special cases of image/signal analysis systems, such as morphological filters, median and order-statistic filters, linear filters, and shape recognition transforms. Although the developed theory is algebraic, its prototype operations are well suited for shape analysis; hence, the results also apply to systems that extract information about the geometrical structure of signals.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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

Tutorial On Advances In Morphological Image Processing And Analysis

This paper reviews some recent advances in the theory and applications of morphological image analysis. Regarding applications, we show how the morphological filters can be used to provide simple and systematic algorithms for image processing and analysis tasks as diverse as nonlinear image filtering, noise suppression, edge detection, region filling, skeletonization, coding, shape representation, smoothing, and recognition. Regarding theory, we summarize the representation of a large class of translation-invariant nonlinear filters (including morphological, median, order-statistic, and shape recognition filters) as a minimal combination of morphological erosions or dilations; these results provide new realizations of these filters and lead to a unified image algebra.

Morphological structuring element decomposition

To efficiently perform morphological operations with specialized pipeline hardware which is not able to utilize all the points in the domain of the structuring element in one pipeline stage requires the capability of decomposing the structuring element into a morphological dilation of smaller structuring elements each of which is utilized in a successive stage of the pipeline. In this paper, we give the theory and algorithm for such optimal structuring element decomposition.