Showing papers on "Minkowski addition published in 1991"

An algebra of polygons through the notion of negative shapes

Abstract: 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.

On the vector sum of two convex sets in space

Abstract: In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C 20/3 (this result is sharp). When A and B have C ∞ boundaries, then the smoothness of the sum set breaks down at the level C 5 (see [KIS2] for the various pathologies that arise).

Method for calculating minkowski's sum of general polyhedron

Abstract: PURPOSE: To reduce the complication of the Minkowski sum of a general polyhedron by generating a geometrical object or an internal computer expression expressed by the Minkowski sym. CONSTITUTION: Boolean operations and the Minkowski sum of a simple polyhedron are used to generate the Minkowski sum of a general polyhedron. The image of the geometrical object or the internal computer expression expressed by this Minkowski sum is generated and is displayed in a parallel displacement free space 9.

Decomposing morphological structure element into neighborhood configurations

Abstract: Decomposing morphological structure element into Minkowski sum of several small ones is very useful for fast implementation of morphological operations and important for multiscale systems. This paper presents some theoretical results on decomposition of digital structure element, such as geometrical constraints, singularity, compatibility, and decomposability, etc., which is very different from that in continuous space. Based on those, methodology for decomposition will be proposed, including approximate decomposition, correction of singularity, and etc. These results will be used in decomposition into four and eight neighborhood configurations and series decomposition of multiscale structure sequence in the paper.

Regularity of constraints in the Minkowski space Yang-Mills theory

Abstract: For Yang-Mills theory in the Minkowski space it is proved that the constraint set is a smooth submanifold of the phase space consisting of square integrable Cauchy data.

Quadtree decomposition of binary structuring elements

Abstract: In order to efficiently perform morphological binary operations by relatively large structuring elements, we propose to decompose each structuring element into squares with 2 X 2 pixels by the quadtree approach. There are two types of decomposition--the dilation decomposition and the union decomposition. The first type decomposition is very efficient, but it is not necessarily always possible. The decomposition of second type is available for any structuring element, but the time cost of computation is proportional to the area of the structuring element. The quadtree decomposition proposed here is the combination of these two types of decomposition, and exists for any structuring element. When the Minkowski addition A (direct sum) B or the Minkowski subtraction A - B is computed, the number of times of the union/intersection of translations of the binary image is about the number of leaves of the quadtree representation of the structuring element B, which is roughly proportional to the square root of the area of B. In this paper, an algorithm for quadtree decomposition is described, and experimental results of this decomposition for some structuring elements are shown.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Regularity results for harmonic maps of Minkowski space

Abstract: We establish regularity results for harmonic maps of (m + 1)-dimensional Minkowski-space (m = 2, 3) into any given compact Riemannian manifold. Our results are sharp if m = 3.

Curves and lattices of points in the minkowski plane

Abstract: The main goal of this work is to derive an integral formula refering to bounded convex sets (section 3), in order to obtain some results involving lattices of points in the Minkowski plane (seccion 4). To prove such a formula it is necessary to develop some tools of differential geometry in the large. This is done in section 2, where the turning tangents theorem for Euclidean curves is carried over to Minkowski plane.