Ranita Biswas

On different topological classes of spherical geodesic paths and circles in Z 3

A discrete spherical geodesic path (DSGP) between two voxels s and t lying on a discrete sphere is a/the shortest path from s to t, comprising voxels of the discrete sphere intersected by the discrete geodesic plane passing through s, t, and the center of the sphere. We consider two classes of discretization, namely naive and standard, for both the sphere and the geodesic plane, which gives rise to four distinct topological classes of DSGP. We show that the naive-naive class does not guarantee the existence of a DSGP, whereas the other three classes do. We derive the upper bounds of the distance of a DSGP belonging to each class, from the real sphere and the real plane, for different neighborhood conditions. We propose an efficient integer-based algorithm to compute the DSGP for any class-and-neighborhood combination. Novel number-theoretic characterization of discrete sphere has been used for searching the voxels comprising a DSGP. The algorithm is output-sensitive, having its time and space complexities both linear in the length of the DSGP. It can also be extended for constructing discrete 3D circles of arbitrary orientations, specified by a few appropriate input parameters. Experimental results and related analysis demonstrate its efficiency and versatility.

From prima quadraginta octant to lattice sphere through primitive integer operations

We present here the first integer-based algorithm for constructing a well-defined lattice sphere specified by integer radius and integer center. The algorithm evolves from a unique correspondence between the lattice points comprising the sphere and the distribution of sum of three square numbers in integer intervals. We characterize these intervals to derive a useful set of recurrences, which, in turn, aids in efficient computation. Each point of the lattice sphere is determined by resorting to only a few primitive operations in the integer domain. The symmetry of its quadraginta octants provides an added advantage by confining the computation to its prima quadraginta octant. Detailed theoretical analysis and experimental results have been furnished to demonstrate its simplicity and elegance.

On the Connectivity and Smoothness of Discrete Spherical Circles

A discrete spherical circle is a topologically well-connected 3D circle in the integer space, which belongs to a discrete sphere as well as a discrete plane. It is one of the most important 3D geometric primitives, but has not possibly yet been studied up to its merit. This paper is a maiden exposition of some of its elementary properties, which indicates a sense of its profound theoretical prospects in the framework of digital geometry. We have shown how different types of discretization can lead to forbidden and admissible classes, when one attempts to define the discretization of a spherical circle in terms of intersection between a discrete sphere and a discrete plane. Several fundamental theoretical results have been presented, the algorithm for construction of discrete spherical circles has been discussed, and some test results have been furnished to demonstrate its practicality and usefulness.

On Finding Spherical Geodesic Paths and Circles in ℤ3

A discrete spherical geodesic path between two voxels s and t lying on a discrete sphere is a/the 1-connected shortest path from s to t, comprising voxels of the discrete sphere intersected by the real plane passing through s, t, and the center of the sphere. We show that the set of sphere voxels intersected by the aforesaid real plane always contains a 1-connected cycle passing through s and t, and each voxel in this set lies within an isothetic distance of \(\frac32\) from the concerned plane. Hence, to compute the path, the algorithm starts from s, and iteratively computes each voxel p of the path from the predecessor of p. A novel number-theoretic property and the 48-symmetry of discrete sphere are used for searching the 1-connected voxels comprising the path. The algorithm is output-sensitive, having its time and space complexities both linear in the length of the path. It can be extended for constructing 1-connected discrete 3D circles of arbitrary orientations, specified by a few appropriate input parameters. Experimental results and related analysis demonstrate its efficiency and versatility.

On the polyhedra of graceful spheres and circular geodesics

We construct a polyhedral surface called a graceful surface, which provides best possible approximation to a given sphere regarding certain criteria. In digital geometry terms, the graceful surface is uniquely characterized by its minimality while guaranteeing the connectivity of certain discrete (polyhedral) curves defined on it. The notion of “gracefulness” was first proposed in Brimkov and Barneva (1999) and shown to be useful for triangular mesh discretization through graceful planes and graceful lines. In this paper we extend the considerations to a nonlinear object such as a sphere. In particular, we investigate the properties of a discrete geodesic path between two voxels and show that discrete 3D circles, circular arcs, and Mobius triangles are all constructible on a graceful sphere, with guaranteed minimum thickness and the desired connectivity in the discrete topological space.

Layer the sphere

Voxelation today is not only limited to discretization and representation of 3D objects, but has also been gaining tremendous importance in rapid prototyping through 3D printing. In this paper, we introduce a novel technique for discretization of a sphere in the integer space, which gives rise to a set of mathematically accurate, 3D-printable physical voxels up to the desired level of precision. The proposed technique is based on an interesting correspondence between the voxel set forming a discrete sphere and certain easy-to-compute integer intervals defined by voxel position and sphere dimension. It gives us several algorithmic leverages, such as computational sufficiency with simple integer operations and amenability to layer-by-layer additive fabrication. Theoretical analysis, prototype characteristics, and experimental results demonstrate its efficiency, versatility, and further prospects.

Digital Primitives Defined by Weighted Focal Set

This papers introduces a definition of digital primitives based on focal points and weighted distances (with positive weights). The proposed definition is applicable to general dimensions and covers in its gamut various regular curves and surfaces like circles, ellipses, digital spheres and hyperspheres, ellipsoids and k-ellipsoids, Cartesian k-ovals, etc. Several interesting properties are presented for this class of digital primitives such as space partitioning, topological separation, and connectivity properties. To demonstrate further the potential of this new way of defining digital primitives, we propose, as extension, another class of digital conics defined by focus-directrix combination.

Construction of Persistent Voronoi Diagram on 3D Digital Plane

Different distance metrics produce Voronoi diagrams with different properties. It is a well-known that on the (real) 2D plane or even on any 3D plane, a Voronoi diagram (VD) based on the Euclidean distance metric produces convex Voronoi regions. In this paper, we first show that this metric produces a persistent VD on the 2D digital plane, as it comprises digitally convex Voronoi regions and hence correctly approximates the corresponding VD on the 2D real plane. Next, we show that on a 3D digital plane D, the Euclidean metric spanning over its voxel set does not guarantee a digital VD which is persistent with the real-space VD. As a solution, we introduce a novel concept of functional-plane-convexity, which is ensured by the Euclidean metric spanning over the pedal set of D. Necessary proofs and some visual result have been provided to adjudge the merit and usefulness of the proposed concept.

Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space

Space filling circles and spheres have various applications in mathematical imaging and physical modeling. In this paper, we first show how the thinnest (i.e., 2-minimal) model of digital sphere can be augmented to a space filling model by fixing certain “simple voxels” and “filler voxels” associated with it. Based on elementary number-theoretic properties of such voxels, we design an efficient incremental algorithm for generation of these space filling spheres with successively increasing radius. The novelty of the proposed technique is established further through circular space filling on 3D digital plane. As evident from a preliminary set of experimental result, this can particularly be useful for parallel computing of 3D Voronoi diagrams in the digital space.

On Some Local Topological Properties of Naive Discrete Sphere

Discretization of sphere in the integer space follows a particular discretization scheme, which, ini¾?principle, conforms to some topological model. This eventually gives rise to interesting topological properties of a discrete spherical surface, which need to be investigated for its analytical characterization. This paper presents some novel results on the local topological properties of the naive model of discrete sphere. They follow from the bijection of each quadraginta octant of naive sphere with its projection map called f-map on the corresponding functional plane and from the characterization of certain jumps in the f-map. As an application, we have shown how these properties can be used in designing an efficient reconstruction algorithm for a naive spherical surface from an input voxel set when it is sparse or noisy.