Charles R. Giardina

Discrete black and white object recognition via morphological functions

Two morphological algorithms that attempt to recognize a black and white object directly in its discrete domain are presented. The first algorithm is based on covariance functions, while the second is based on a variant of size distribution functions. In both these algorithms, the scale correction has been automated. Also presented is a complete geometric and algebraic characterization of objects that are identical with respect to the proposed methodologies, and it is shown that the induced equivalent classes over binary images contain objects that are structurally very similar. This has been accomplished by introducing the notions of a strongly attached pixel, discrete structure of an image, and a structure preserving operation. An outcome of the analysis is the insight into the relationship between the discrete structure of an image and the induced equivalence classes. >

Image Processing Using Pointed Fuzzy Sets

Pointed fuzzy sets are introduced and are shown to provide a useful model for describing and manipulating grey values of images. Algebraic operations involving grey values and points in the integral lattice induce corresponding basic operations on images from which important image processing algorithms are described.

Fast dilation and erosion of time-varying grey-valued images with uncertainty

Two of the most important basic morphological operations used in image filtering are erosion and dilation. In this paper the authors consider the case when a finite image is part of a pixel display which changes ate discrete times. Taking advantage of the fact that not all pixels will change from time t to time t + 1, they develop two important algorithms for computing the dilation and erosion of such images in o(n2) less time then with brute force. These results hold also for translation and rotation and can be extended to opening and closing of images by structuring elements. These results are also extended to images which contain multi-uncertain values, that is, the extended fuzzy pointing set. The advantages of these fast operations are obvious in on-the-fly image processing schemes such as real-time filtering of images.

Error bounds for morphologically derived measurements

The morphological analysis of black and white images was originated by G. Matheron for the textural study of porous materials. The method is grounded upon the classical Minkowski algebra as developed by H. Hadwiger. The present paper provides sampling error bounds which result from employing a digital methodology in the computation of morphologically derived measurements. Specifically, the digitization error is examined in the case of size distributions resulting from erosions and linear granulometries.

Representation Theorems In A L-Fuzzy Set Theory Based Algebra For Morphology

Perhaps the most promising area of morphological image processing is that dealing with morphological filters. An image to image mapping is called a morphological filter if it is increasing and translation compatible. In other words, morphological filters preserve the natural set theoretic ordering and are space invariant. Examples of such filters are the convex hull operator, the topological closure, the umbra transform and various other topological algorithms [2-4]. Matheron showed that all such mappings can be characterized by the set theoretic operations erosion and dilation. His representation theorems has been adapted to digital images with great success [4].

Closed-form representation of convolution, dilation, and erosion in the context of image algebra

Using fundamental operators from image algebra, the authors present simple closed-form expressions for dilation, erosion, and convolution. Algebraically, these expressions appear as terms within the algebra. Moreover, the methodology for obtaining the expressions reveals a universal operational structure within image algebra, of which the three aforementioned operations are particular instances. The result is a natural parallel mechanism for computation and a representation of convolution that naturally overcomes the difficulties arising from the variability of image domains in the defining relation.<<ETX>>

MORPHOLOGY ON UMBRA MATRICES

The umbra transform serves as a connection between gray-scale morphology and the classical two-valued morphology of G. Matheron and H. Hadwiger. From a general set-theoretic perspective, the umbra transform of an image (or signal) results in an infinite set, even in the discrete case. By employing bound matrix image representation it is possible to represent the umbra by a finite data structure, the result being an approach that is both intuitive and computational. Moreover, the method is essentially dimensionally independent and thus applies to both morphological image and signal processing.

Application of the extended fuzzy pointing set to coin grading

The application of the extended fuzzy pointing set to coin grading is described and illustrated by examples. The coin grading example considered takes advantage of the fact that the extended fuzzy pointing set S/sub n/ has been equipped with special operators with certain induced properties. In addition, the fact that the valuation space is endowed with a lattice structure is exploited. It is believed that coin grading is a paradigm for more important applications. For example, the technique considered here could be used in grading semiconductor wafers before implantation or for the interpretation of data transmitted over noisy or damaged channels such as in space communications.<<ETX>>

Universal Systolic Architecture For Morphological And Convolutional Image Filters

Systolic arrays are employed in a number of computational settings where a simple computation is repeated a large number of times and where the algorithm exhibits a high degree of concurrency. The present paper demonstrates the relationship between a universal algebraic paradigm appearing in image algebra and a corresponding universal systolic scheme that can be employed upon algorithms that fit the paradigm.

Image processing operations and their digital signal analogs

The authors demonstrate several homomorphic relationships between Minkowskis addition of black and white images and their discrete signal counterparts. In particular, several special relationships involving the Z-transform and discrete convolution of signals are used to demonstrate analagous properties in the image domain. In addition to the work in black and white images, extensions to fuzzy sets and colored images are suggested. The importance of the propositions discussed is twofold; by relating the theories of mathematical morphology in the one area are shown to have analogs in the other. The work contributes to a unified theory of image and signal processing.<<ETX>>