Construction of isothetic covers of a digital object: A combinatorial approach

Abstract

An isothetic cover of a digital object not only specifies a simple representation of the object but also provides an approximate information about its structural content and geometric characteristics. When the cover “tightly” encloses the object, it is said to be an outer isothetic cover; and when the cover tightly inscribes the object, it is an inner isothetic cover. If a set of horizontal and vertical grid lines is imposed on the object plane, then the outer (inner) isothetic cover is defined by a set of isothetic polygons, having their edges lying on the grid lines, such that the effective area corresponding to the object is minimized (maximized). Increasing or decreasing the grid size, therefore, decreases or increases the complexity or the accuracy of the isothetic cover corresponding to a given object, which, in turn, extracts the object information at different resolutions. In this paper, a combinatorial algorithm is presented for varying grid sizes, which is free of any backtracking and can produce the isothetic cover of a connected component in a time linear in the length of the perimeter of the cover. The algorithm has also been extended for finding the isothetic covers of real-world digital objects having multiple components with or without holes. Experimental results that demonstrate applications of an isothetic cover to diverse problems of pattern analysis and computer vision, are reported for various data sets.

Introduction

Determination of the (minimum-)maximum-area (outer) inner isothetic cover corresponding to a 2D digital object is a problem of practical relevance to various fields. Given a set of isothetic grid lines under the object plane, an isothetic cover corresponds to a collection of isothetic polygons, which bears a structural and geometric relation with the concerned object, and hence can be useful to many interesting applications, such as VLSI layout design, robot grasping and navigation, rough sets, inner and outer approximations of polytopes [3], and document image analysis. In VLSI layout design, computation of a minimum-area safety region, referred as the classical safety zone problem [1], may be necessary to ensure the correctness of design rules before fabricating an integrated circuit. The trade-off lies between the minimization of total area of the fabricated parts (an obvious economic constraint) and the necessary electrical relationship (insulation or contact) in the presence of possible production fluctuations [2]. In robotics, identification of free configuration space (path-planner) is a pertinent problem in robot navigation. For example, a real-time robot motion planner often uses standard graphics hardware to rasterize the configuration space into a series of bitmap slices, and then applies a dynamic programming technique to calculate paths in this rasterized space [4]. The motion paths produced by the planner should be minimal with respect to the Manhattan distance $(L_1)$ metric. Similarly, for grasping a 3D object, its outer isothetic cover may be helpful, as the mechanical fingers of a robot may be constrained by only axis-parallel movements [5], [6], [7]. In many applications of rough sets, computation of the lower and upper approximations of a rough set is required [8], [9], [10], [11]. For example, in image mining [12], [13], a challenging problem is to discover valid, novel, potentially useful, and ultimately understandable knowledge from a large image database, in order to overcome the curse of dimensionality. Solutions, using rough-set concepts, mainly include several partitioning and dimension-reduction algorithms, where the (possibly not equi-spaced) demarcating lines (analogous to the background grid lines used while finding isothetic covers) are specified by the corresponding feature values (low, medium, high, etc.). Subsequently, a tight isothetic cover of the region of interest can be obtained following these demarcating lines. Deriving the electronic version of a paper document for the purpose of storage, retrieval, and interpretation, requires an efficient representation scheme. A document representation involves the steps of skew detection, page segmentation, geometric layout analysis, and logical layout analysis, for which isothetic polygons can be used [14], [15], [16], [17]. For example, in the page segmentation method [18], a document page image is cut into polygonal blocks using the inter-column and the inter-paragraph gaps as horizontal and vertical lines, followed by the construction of simple isothetic polygonal blocks using an intersection table. Isothetic polygons can also be used for ground truthing of complex documents [19], [20].

The problem addressed in this paper is stated as follows. Given a 2D digital object S (Definition 3) registered on a set of horizontal and vertical grid lines $\mathcal{G}$ (Definition 5), the problem is to find the tight (outer and inner) isothetic covers (Definition 9, Definition 10) of S. Clearly, the shape of an isothetic cover depends on the resolution of the grid or the grid size (Definition 5) and on registration of the object with the underlying grid. The major difference of our work with the existing ones, therefore, lies in its ability to find an inner/outer isothetic cover that, for a lower grid size, almost grazes the contour of a digital object without touching it, and in its ability to reduce the output complexity (i.e., number of vertices) by producing a rough cover when a higher grid size is specified. Fig. 1 shows the original “bear” image (left), its outer isothetic cover (OIC) and the inner isothetic cover (IIC) in the middle, and the superimposed outer and inner isothetic covers (right). It is seen that the boundary of the object lies in the region of difference given by the OIC minus the interior of IIC.