It is widely accepted that the design of morphological filters which are optimal in some sense is a difficult task. In this paper we want to propose a new method for the design of morphological filters. This method is known in Artificial Intelligence as the Genetic Algorithm and is applied to difficult combinatorial problems. Here we want to show the possibility of using this method in the field of mathematical morphology.

In this work we present a new approach to robust image modeling. the proposed method is based on M-estimation algorithms. However, unlike in other M-estimator based image processing algorithms, the new algorithm takes into consideration spatial relations between picture elements. The contribution of the sample to the model depends not only on the current residual of that sample, but also on the neighboring residuals. In order to test the proposed algorithm we apply it to an image filtering problem, where images are modeled as piecewise polynomials. We show that the filter based on our algorithm has excellent detail preserving properties while suppressing additive Gaussian and impulsive noise very efficiently.

Meyer watershed algorithm, which creates watershed lines to the output image, and its extended split-and-merge version are further analyzed in this paper based on our previous work. Specifically, it is shown that no permanently isolated areas exist in the output of the Meyer watershed algorithm when applied to an eight-connected square grid. The scanning order dependency of watershed algorithms is also studied. First, the different scanning order operations which are implementation dependent are identified; then their effects on the output are analyzed. Finally, an alternative implementation of the split-and-merge watershed algorithm is presented which detects isolated areas using a new technique; the goal is to illustrate and analyze the operations of the split-and-merge watershed algorithm.

In this paper we describe our recent work developing automated methods for generation of kernels or structuring elements for use in the hit-or-miss transform. We show how a neural network algorithm (Fuzzy Adaptive Resonance Theory) generates hit and miss structuring elements that can be used with a fuzzy morphology to detect a class of objects and we illustrate with computer simulations.

Optimal openings are considered for extraction of signal from noise in the random binary union-noise model. Disjointness of signal and noise is not assumed, nor are grains within the signal or within the noise assumed to be disjoint. There is a constraint on the overlapping, but this reflects the manner in which binary granular images are derived from gray-scale images of touching objects. The method assumes that the degraded image is segmented by the binary watershed algorithm and that an optimal opening by reconstruction must be found to remove segmented noise grains while passing segmented signal grains.

Given a set of cost coefficients, obtained from a "representative" training data set and some desired set, we have previously shown that the optimal Boolean filter, based on a defined error criterion, is obtained by simple compare/assign operations. If, on the other hand, the desired solution is a stack filter, three steps must be added to the above procedure. Following the compare/assign step above, we check if the resulting solution is of the desired type. If not, we compute the maximal positive Boolean function contained in the resulting Boolean function. Finally, we check if adding other minterms to the positive Boolean function obtained in the previous step will improve the criterion value. The first step requires a very low computational effort. For the following three steps, matrix based procedures using the stacking matrix, are derived. First, we derive a fast procedure for checking the positivity of a Boolean function. This procedure can be written in a single line using Matlab® language. The following step consists of finding the maximal positive Boolean function embedded in a given Boolean function. Again, a fast procedure is derived for this task, which can also be written in one line using Matlab® language. The final step checks for improvement, in the cost criterion, when adding other minterms to the positive Boolean function, resulting from the previous step. We will use again the stacking matrix to accomplish this task, resulting in a three-line Matlab®code. Some examples are provided to illustrate each step in the above procedure.

Let f denote an image and F a morphological operator depending on a set of parameters {m_j}. The purpose of this paper is to find a set of parameters which satisfies the equation F(f,m) equals f where F is one of the two basic morphological operator: erosion or dilation. It is obvious that there exists a trivial solution m of f equals F(f,m), which is given by m(x) equals 0 if x equals 0 and m(x) equals -(infinity) otherwise. Our approach is to consider this problem as an optimization problem: `find the best set of parameters which minimizes the error between the desired image f and the output filtered image F(f,m)'. Among various possible approaches, we have chosen a specific one introduced by Ph. Salembier for adaptative structuring elements. In our problem, we prove that Salembier algorithms always converge toward a solution distinct from m.

Multiresolution image decomposition based on nonlinear filtering has received a lot of attention recently. In this research, we investigate the coding issue for one class of nonlinear multiresolution image decomposition based on mathematical morphology. We consider the use of opening and closing operations with a flat structure element to achieve image decomposition. The entropy and histogram of the difference images in the image pyramid are then examined. We give a numerical example to demonstrate potential advantages of the morphological filtering approach over the conventional linear filtering approach in the context of image coding. However, we also point out difficulties encountered in our study that have to be overcome before the method can be practically used.

The present paper discusses an approach of efficient coding of the error images obtained using the morphology based multiresolution pyramid decomposition technique proposed by Heijmans and Toel. A commonly used approach to achieve significant compression ratios in pyramid compression techniques is to discard the first error image. However, this may cause degradation of fine edges, texture information, and thin features. In the present paper we have proposed an estimator for the first error image. Directional filtering and sampling is used to decompose the error image into two components.

This paper deals with the robust extraction of line segments in noisy binary images. The principle is to find maximal geodesic arcs lying in the shape. A rigorous digital framework is first presented, then an efficient algorithm is described and results on real data presented. Some hints for filling the gaps between disconnected arcs are finally exposed.

In radiographic inspection used by Electricite De France (EDF) for pipe control in nuclear power plants, pipe radiographs are digitized according to a well-defined protocol. The aim of EDF consists of developing a non-destructive testing system for identifying defects. In this paper, we propose an approach based on a multiscale analysis allowing to achieve simultaneously defect-enhancement and noise-filtering and leading to accurate defects segmentation using mathematical morphology. The principle consists of defining a one- parameter family of operators which are successively applied to the digitized images, after drift-removal. Equivalently, the multiscale analysis can be expressed as the solution of an evolution equation, whose analytical form is conditioned by the specific properties of the operators (e.g., mean curvature motion, affine scale-space). This results in a family of images with increasing smoothness and consistent edge preservation.

Several works have recently underlined the relevance of morphological-filtering-based techniques for accurately segmenting gray-scale images of corneal endothelial tissue. Nonetheless, applied to low-quality and/or highly pathological images, methods exploiting standard morphological filters fail to provide correct segmentations. Moreover, quantification issues remain essentially an open problem. In this paper, we present a robust and accurate method for automatically segmenting and quantifying corneal endothelial images.

This paper presents a neural network application to target classification using a new type of neural network called the Fuzzy Image Algebra Neural Network (FIANN). The FIANN is used in a heterogenous network structure. The first layer of the net performs feature extraction, while the remaining layers are used for classification. Generalized image algebra operations are used to obtain fuzzy morphological or linear operation. The parameters for the generalized operations are learned in a fashion similar to standard backpropagation, but with training rules based on a combination of stochastic learning and gradient descent techniques. The type of data used is the range data part of tank LADAR data. The objective is to classify the tanks by type. The range data is first converted to elevation data, which is input to the net for feature extraction and classification. A two tiered approach is used for training. The first layer learns image features, while the top layers perform classification.

A neural network structure that learns feature extraction and classification operations simultaneously is described. The feature extraction operations are represented using generalized image algebra operations. Learning rules are described. Linear operations and nonlinear, hit-or-miss operations are used to perform handwritten digit recognition.

New criteria for shape preservation are presented. These criteria are applied in optimizing soft morphological filters. The filters are optimized by simulated annealing and genetic algorithms which are briefly reviewed. A situation where the given criteria give better results compared to the traditional MAE and MSE criteria is illustrated.

Image Algebra (IA) was developed to provide a standard mathematical means of describing image processing algorithms. The goal of IA was to reduce the amount of programming code required in implementing an image processing algorithm. IA has been successful in expressing many linear and nonlinear image processing algorithms in a short and concise manner using a basic set of operators. When placed in a programming environment, IA enables the programmer to write image processing algorithms at a high level of abstraction and with a high degree of readability. IA functions have been developed for several programming languages such as ADA, FORTRAN, and C either as an external library or as a preprocessor.

This paper explores the template concept as defined in the University of Florida image algebra with special attention to computer representation of template operands and the implementation of image-template operations. We discuss some of the basic properties that support template decomposition and relate these properties to computer implementation structures. We also discuss the practical considerations that have led to provide users of the image algebra C++ class library (iac++) with neighborhoods and neighborhood operations and provide formal definitions for these concepts. We argue that neighborhoods represent an important concept which should not be ignored in image algebra. We then exploit properties of point set representation to yield a technique for automatic decomposition of neighborhoods employed in the iac++ class library and extend this technique to provide some initial techniques for template decomposition as well.

The most powerful tools of image algebra in terms of image processing are image-template operations. An image-template operation can be classified as a local operation or a nonlocal operation. For local image-template operations, we have an efficient parallel implementation. However, there seems to be no efficient way to map general nonlocal image-template operations onto parallel architectures, since the communication patterns involved with nonlocal operations are very complex in general. A practical way is to classify nonlocal image-template operations into several commonly used classes and develop efficient implementation for each class. In this paper, we define a special class of image-template operations with nonlocal templates and develop a general efficient algorithm for this class of image-template operations. We then demonstrate how to use this class of image-template operations to compute geometric properties of image components such as area, perimeter, compactness, height, width, diameter, moments, and centroid.

In this paper, we discuss methods for multispectral ATR (Automated Target Recognition) of small targets that are sensed under suboptimal conditions, such as haze, smoke, and low light levels. In particular, we discuss our ongoing development of algorithms and software that effect intelligent object recognition by selecting ATR filter parameters according to ambient conditions. Our algorithms are expressed in terms of IA (image algebra), a concise, rigorous notation that unifies linear and nonlinear mathematics in the image processing domain. IA has been implemented on a variety of parallel computers, with preprocessors available for the Ada and FORTRAN languages. An image algebra C++ class library has recently been made available. Thus, our algorithms are both feasible implementationally and portable to numerous machines. Analyses emphasize the aspects of image algebra that aid the design of multispectral vision algorithms, such as parameterized templates that facilitate the flexible specification of ATR filters.

An important aspect of Mathematical Morphology is the description of set operators by a formal language, the Binary Morphological Language, whose vocabulary are dilations, erosions, antidilations, antierosions, union and intersection. This language is complete (i.e. it can represent any set operator) and expressive (i.e. many useful operators can be represented as phrases with relatively few words). Since the sixties special machines, the Binary Morphological Machines (BMMach's), have been built to implement the BML with increasing efficiency. However, designing useful BMMach programs is not an elementary task. Recently, much research effort has been addressed to automating the programming of BMMach's. The goal is to find suitable knowledge representation formalisms to describe operations over geometric structures and to translate them into BMMach programs.

Mathematical Morphology is a general theory that studies the decompositions of mappings between complete lattices in terms of some families of simple mappings: dilations, erosions, anti-dilations and anti-erosions. Nowadays, this theory is largely used in Image Processing and Computer Vision to extract information from images. The KHOROS system is an open and general environment for Image Processing and Visualization that has become very popular. One of the main characteristics of KHOROS is its flexibility, since it runs out on standard machines, supports several standard data formats, uses a visual programming language, and has tools to help the user to build and install his own programs. A set of new programs can be organized as a subsystem, called Toolbox. This paper presents a fast and comprehensive Mathematical Morphology Toolbox for the KHOROS system, that deals with binary, gray- scale and multiple band images. Each program has specialized algorithms for binary and gray- scale images, that are chosen automatically according to the input data. These implemented algorithms running on current general purpose workstations are as fast as the equivalent ones running on specialized hardware with 1986 technology.

In this paper, we propose a novel fuzzy morphology (FM) which is induced by threshold decomposition. Its operators are the measures on a (sigma) _ring in range space of the membership functions of fuzzy sets, which depend upon the ordinary binary morphological operators of threshold sets of fuzzy sets. Some of the characteristic of these FM are similar to those of the traditional morphology. All the operators of these FM prove to be the morphological operators in the sense of complete lattice. It is shown how one can use binary morphological operators, thresholding techniques and stacking properties to implement these FM's operators. The VLSI implementation is simple and fast. The concept of fuzzification of set_intersection is introduced. This paper also presents a general algebraic approach to analysis fuzzy morphological operators on the space of fuzzy sets.

In this paper, we give some results of threshold decomposition of soft morphological (SM's) filters. Some of new properties of SM have been developed by means of these results. We give the necessary and sufficient condition for the SM's operators to be the morphological operators in the sense of complete lattice. We propose the binary SM, which simplify the analysis of SM. An algebraic approach to bounded signal processing has been given. The VLSI implementation of SM's filters have been discussed.

In this paper, we present a comprehensive theory of spatially-variant (SV) mathematical morphology. A kernel representation of increasing operators in terms of the union (resp., intersection) of SV erosions (resp., SV dilations) is provided. A representation of algebraic openings (resp., algebraic closings) in terms of the union (resp., intersection) of SV openings (resp., SV closings) is also provided. The SV morphological skeleton representation is finally presented, some of its properties investigated, and conditions for its invertibility derived.

In this paper, we present an investigation of the generalized morphological center. The generalized morphological center is first introduced. Conditions are subsequently derived for the self-duality of the generalized morphological center. Several important examples of self- dual generalized morphological centers are finally presented.

In order to be able to optimally design morphological shape extraction algorithms operating on binary digital images, there needs to be a probability theory for finite random sets and probability relations that show how the probability changes as a finite random set is propagated through a morphological operation. In this paper, we develop such a theory for finite random sets. We then demonstrate how to apply this theory for calculating the probability that a set S perturbed by min or max noise N and dilated or eroded by a structuring element K is a subset, superset, or hits a given set R. In some cases we obtain exact results and in some cases we obtain bounds for the desired probability.

It is known that if a binary random image is composed of a disjoint union of translates of i.i.d. randomly scaled homothetics of an arbitrary compact set, then, for any convex, compact granulometric generator, the granulometric moments are asymptotically normal and there exist asymptotic representations of the moments of the granulometric moments. The present paper extends the asymptotic theory to a random image composed of a disjoint union of translates of scaled homothetics of a finite collection of compact primitives (shapes) under the condition that the mixture proportions of the shapes are known and fixed. Grain sizing is independent, with the sizings for each primitive being identically distributed. Based on this new granulometric structure theorem for mixed grain images, an estimation method is proposed that estimates the mixture proportions from estimates of the granulometric-moment means derived from running the granulometry on realizations of the mixed process. The granulometric mixture estimation is compared to maximum-likelihood estimation.

In this paper the symmetry properties of the cascaded order statistics filters are studied. The relationship between the mask and the Boolean function expression of the filter is considered. It is shown that a symmetric transformation of the mask of a cascaded filter leads to an isomorphic Boolean function expression of the filter. Using this connection certain statistical symmetry properties of the filters are derived.

The Hopfield network model associates an input pattern with trained patterns and is generally considered to be a pattern recognition system that completes missing pieces of the input image. In this paper the Morphological Hopfield Net associates segments in input patterns with trained pattern segments and is used to reconstruct known patterns degraded by noise by reconstructing the individual segments. A very simple Hopfield model is defined over an image space and consists of a large number of identical Hopfield networks, one about each pixel site, each with a local connectivity to a neighborhood of pixels. The weights are all 1 and the thresholds are adjusted to extreme values (max or min). It is shown that this Hopfield model is equivalent to a union of openings. Convergence occurs in only one iteration since the union of openings is idempotent.

A method of based on genetic algorithms for finding an optima soft morphological filter for specific situation is presented. The behavior of soft morphological filters is illustrated by empirical results. Some optimal parameter will also be proposed for MAE and MSE error criteria.

The digital Mathematical Morphology and the Distance Transform (DT) have many points of intersection. The DT combines numerical features and objects shapes. The properties of digital distance functions (metrics, asymmetries and quasi-metrics) and DT based on these functions are studied. Some extensions of the transform and the interpretation of the grey scale DT by application of the binary DT in the n-dimensional digital space are given. The morphological erosion and dilation may be performed by the DT for binary and grey scale images.

This paper states and proves a number of properties of the tophat and the tophat spectrum. These include: the tophat is antiextensive and idempotent (but not increasing); each image in the tophat spectrum is size-limited and open; the structuring element family need not be mutually open to generate a tophat spectrum; if the SE family is mutually open, and the original image is binary, each image in the tophat spectrum includes the open part of the corresponding image from the opening spectrum; and, the tophat spectrum is identical to the opening spectrum created with a family of flat, 1D structuring elements.

In this paper, a comprehensive set of fast algorithms for computing granulometries in binary images is first proposed: linear granulometries (i.e., granulometries based on openings with line segments) constitute the easiest case, and are computed using image `run-length'. The 2D case (granulometries with square or `diamond'-shaped structuring elements, or granulometries with unions of line-segments at different orientations) involves the determination of opening functions or granulometry functions. The grayscale case is then addressed, and a new algorithm for computing grayscale linear granulometries is introduced. This algorithm is orders of magnitude faster than any previously available technique. The techniques introduced in this paper open up a new range of applications for granulometries, examples of which are described in the paper.

Euclidean granulometries are used to decompose a binary image into a disjoint union based on interaction between image shape and the structuring elements generating the granulometry. Each subset of the resulting granulometric spectral bands composing the union defines a filter by passing precisely the bands in the subset. Given an observed image and an ideal image to be estimated, an optimal filter must minimize the expected symmetric-difference error between the ideal image and filtered observed image. For the signal-union-noise model, and for both discrete and Euclidean images, given a granulometry, a procedure is developed for finding a filter that optimally passes bands of the observed noisy image.