Digital Image Processing

G. Ramesh Chandra ¹, G. Sathya ², E. G. Rajan ³

Affiliations
1 Department of Computer Science, VNR Vignana Jyothi Institute of Engineering and Technology, Bachupally, Hyderabad-500090, IN
2 New York Institute of Technology, Westbury Campus, New York, US
3 Pentagram Research Centre, Jubilee Hills, Hyderabad, A.P., IN

Abstract

Three dimensional (3-D) structuring elements plays a vital role in the processing of three dimensional volumetric images such as 3-D medical images. These structuring elements are used in 3-D mathematical morphological operations such as erosion, dilation etc. There is no algebraic framework as such to systematically generate the 3-D structuring elements. The work discussed in this paper is systematically generating the 3-D structuring elements by using geometric filters (G-Filters).
This paper initially gives a novel concept of what we call as geometric filters (G-Filters) defined over a 3-D rectangular grid of pixels. By using these G-Filters, 256 convex Polyhedrons are generated. The 256 different structuring elements could be created by using the 256 convex polyhedrons. These G-Filters have potential applications to processing of volumetric images. In addition, the paper describes in brief the algebra of G-Filters by formulating a lattice of convex polyhedrons constructed in a 3×3×3 grid of pixels.
This paper proposes a new algorithm for automatic generation of 255 convex polyhedrons from its initial set named A containing all eight corners of the 3×3×3 rectangular grid. This paper also proposes an algorithm for hierarchal path enumeration used in visualizing the relationships between convex polyhedron sets and their corresponding subsets of other convex polyhedrons. Finally, an algorithm is proposed to generate 3D structuring element from a selected convex polyhedron.

Keywords

Geometric Filters, 3-D Structuring Elements, Convex Polyhedrons, Hierarchal Path Enumeration, 3-D Image Processing.

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Towheed Sultana ¹, E. G. Rajan ²

Affiliations
1 Pentagram Research Centre, IN
2 Pentagram Research Centre, Jubilee Hills, Hyderabad, A.P., IN

Abstract

Coding of n dimensional image is a critical task in image compression. There exist a number of image coding methods based on symbolic or number coding. This paper discusses shape coding or pextral coding, in which the coding is done based on morphological analysis of shapes. This shape based coding method uses the permutation and cyclic groups of polygons, polyhedrons and hyper polyhedrons for image coding called pextral coding. This paper discusses the generation of permutation groups for 2-D, 3-D and multidimensional digital images. There are 16 convex polygons in a 3×3 two dimensional array of cells and 256 convex polyhedrons in a 3×3×3 three dimensional array of cells. These two arrays of cells are sub lattices of discrete infinite lattices Z2 and Z3 respectively. Similarly, one can construct 216 convex hyper polyhedrons in a sub lattice of the discrete infinite lattice Z4. In the abstract sense, convex hyper polyhedrons could be constructed in a sub lattice of the discrete infinite lattice Zn. Thus, given an n-dimensional discrete infinite lattice, one can build a finite or a potentially infinite set of convex geometric shapes which could be treated as the ground set of a permutation group. This paper describes the results of a study carried out on permutations groups of convex geometric shapes defined over n-dimensional discrete lattices Zn, where n varies from 2 to infinity. Also an application of pextral coding using permutation and cyclic groups, for 2-D image is explained.

Keywords

Geometric Filters, Permutation Invariant Groups, Pextral Coding.

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