Christian Ronse

The algebraic basis of mathematical morphology. I. dilations and erosions

Abstract Mathematical morphology is a theory of image transformations and functionals deriving its tools from set theory and integral geometry. This paper deals with a general algebraic approach which both reveals the mathematical structure of morphological operations and unifies several examples into one framework. The main assumption is that the object space is a complete lattice and that the transformations of interest are invariant under a given abelian group of automorphisms on that lattice. It turns out that the basic operations called dilation and erosion are adjoints of each other in a very specific lattice sense and can be completely characterized if the automorphism group is assumed to be transitive on a sup-generating subset of the complete lattice. The abstract theory is illustrated by a large variety of examples and applications.

Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images

Abstract In successive deletion stages of parallel thinning algorithms for binary digital images, one usually checks the preservation of connectivity by verifying that: (a) every removed pixel is individually deletable without modifying connectivity (well-known criteria, such as those of Rosenfeld and Yokoi, exist for that purpose); (b) every pair of 8-adjacent removed pixels is deletable without connectivity modification. In the case of the 8-connectivity for the figure (and the 4-connectivity for the background), two more patterns must be tested for connectivity preservation: an isolated triple or quadruple of mutually 8-adjacent pixels. In this paper we give a formal characterization of these patterns for testing connectivity preservation by what we call minimal non-x-deletable sets (x-MND sets), where x=4, 8 or {4,8} (the type of connectivity considered for the figure). A parallel thinning algorithm whose deletion stage cannot remove an x-MND set is guaranteed to preserve the connectivity properties of any figure. We show that an x-MND set consists in either (1) a single pixel; or (2) a pair of 8-adjacent pixels; or (3) an isolated triple or quadruple of mutually 8-adjacent pixels (for x=8 only).

A topological characterization of thinning

Abstract A large number of skeletonization algorithms for binary images use the method of thinning: successive layers of pixels are deleted from the figure until it becomes one pixel thick. In this paper we analyze the topological properties of the set D of pixels to be deleted from a figure F in order to get a skeleton. We characterize them by the concept of strong k -deletability ( k = 4 or 8). For individual pixels, strong k -deletability is equivalent to a more general property that we call k -deletability, which is well-known connectivity requirement assumed—at least implicity—in all existing thinning algorithms. We show then that a strongly k -deletable subset D of a figure F can be deleted by a succession of deletions of individual pixels p 1 ,..., p t , where each p i is k -deletable from F{ p j ¶ j i }. This justifies our definition of strong deletability and shows that any topologically valid skeleton can be obtained by some thinning process.

A bibliography on digital and computational convexity (1961-1988)

A bibliography of 370 references of books, papers in serial journals, and conference papers, on convexity in relation to computer science is presented. The subject is divided into five topics: (1) convexity and straightness in digital images; (2) convex hull algorithms and their complexity; (3) other computational problems related to convexity; (4) miscellaneous applications; and (5) general mathematical sources. These references range in time from 1961 to September 1988. >

A simple proof of Rosenfeld's characterization of digital straight line segments

A digital straight line segment is defined as the grid-intersect quantization of a straight line segment in the plane. Let S be a set of pixels on a square grid. Rosenfeld [8] showed that S is a digital straight line segment if and only if it is a Digital arc having the chord property. Then Kim and Rosenfeld [3,6] showed that S has the chord properly if and if for every p, q@eS there is a digital straight line segment C @? S such that p and q are the extremities of C. We give a simple proof of these two results based on the Transversal Theorem of Santalo. We show how the underlying methodology can be generalized to the case of (infinite) digital straight lines and to the quantization of hyperplanes in an n-dimensional space for n >= 3.

On idempotence and related requirements in edge detection

The energy feature detectors described by R. Owens et al. (1989) are good candidates for idempotent edge detectors. However, some of them (in particular, the Gabor energy feature detector) suffer from a serious defect that is absent in gradient-type operators: their sensitivity to gray-level shift in the original image. This leads to errors in the localization of step edges. The Fourier phase and amplitude conditions outlined by M.C. Morrone and D.C. Burr (1988) for the class of energy feature detectors guarantee a zero DC level when the convolution masks are taken in L/sup 1/; therefore, the resulting energy feature detector is invariant under grey-level shift in the original image. Also, the properties of the underlying edge model are invariant under a smoothing of the image by a Gaussian or any function in L/sup 1/ having zero Fourier phase. In particular, such a smoothing does not deteriorate the idempotence of the edge detector. Some concrete examples of energy feature detectors satisfying the Morrone conditions are described. >

Order-configuration functions: Mathematical characterizations and applications to digital signal and image processing

We call a function f in n variables an order-configuration function if for any x1,…, xn such that xi1 ⩽ … ⩽ xin we have f(x1,…, xn) = xt, where t is determined by the n-tuple (i1,…, in) corresponding to that ordering. Equivalently, it is a function built as a minimum of maxima, or a maximum of minima. Well-known examples are the minimum, the maximum, the median, and more generally rank functions, or the composition of rank functions. Such types of functions are often used in nonlinear processing of digital signals or images (for example in the median or separable median filter, min-max filters, rank filters, etc.). In this paper we study the mathematical properties of order-configuration functions and of a wider class of functions that we call order-subconfiguration functions. We give several characterization theorems for them. We show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order-configuration functions.

A strong chord property for 4-connected convex digital sets

The chord property was introduced by Rosenfeld in the study of digital straight line segments. It is known that a digital set has that property if and only if it is 8-connected and convex. We define a stronger property, which we call the strong chord property , and show that a digital set satisfies it if and only if it is 4-connected and convex. We give also another expression for the chord and strong chord properties, and some consequences of our results.

Criteria for approximation of linear and affine functions

Generalisation du theoreme de Santalo aux fonctions lineaires et affines. Application au traitement numerique des signaux

An isomorphism for digital images

Abstract In usual topology, a homeomorphism is a one to one mapping between two topological spaces which induces a one to one mapping between their open subsets and so establishes an equivalence between their topologies. In digital images, as well as in several discrete structures (e.g., planar graphs), one encounters concepts and features analogous to those of topology, for example connectedness, holes, surrounding relations, but it is impossible to define on these structures an isomorphism in the classical sense, if one excepts a trivial one, and this only between images having the same number of points for each colour. It is thus necessary to define in a new way a corresponding concept for digital images. In this paper, an isomorphism between two digital images as a relation, not a map, which satisfies several requirements related to the equivalence of the two digital structures is defined. Such an isomorphism wil then play the same role as the homeomorphism in classical topology. The requirements for this isomorphism are found by a study of the special case of binary images on a rectangular grid, on which we can construct such an isomorphism from a Euclidean plane homeomorphism thanks to a correspondence that we establish between the digital rectangular grid structure and the Euclidean plane topology. It is shown then how this new type of isomorphism preserves certain digital features related to topology (connected components, surrounding relations, etc.). These properties, together with the correspondence with the Euclidean topology in the case of the rectangular grid, validate our definition of the digital isomorphism.