TY - CONF
AB - We propose a new non-orthogonal basis to express the 3D Euclidean space in terms of a regular grid. Every grid point, each represented by integer 3-coordinates, corresponds to rhombic dodecahedron centroid. Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. A characterization of a 3D digital sphere with relevant topological features is proposed as well with the help of a 48 symmetry that comes with the new coordinate system.
AU - Biswas, Ranita
AU - Largeteau-Skapin, Gaëlle
AU - Zrour, Rita
AU - Andres, Eric
ID - 6163
SN - 0302-9743
T2 - Lecture Notes in Computer Science
TI - Rhombic dodecahedron grid—coordinate system and 3D digital object definitions
VL - 11414
ER -
TY - CONF
AB - In this paper, we propose an algorithm to build discrete spherical shell having integer center and real-valued inner and outer radii on the face-centered cubic (FCC) grid. We address the problem by mapping it to a 2D scenario and building the shell layer by layer on hexagonal grids with additive manufacturing in mind. The layered hexagonal grids get shifted according to need as we move from one layer to another and forms the FCC grid in 3D. However, we restrict our computation strictly to 2D in order to utilize symmetry and simplicity.
AU - Koshti, Girish
AU - Biswas, Ranita
AU - Largeteau-Skapin, Gaëlle
AU - Zrour, Rita
AU - Andres, Eric
AU - Bhowmick, Partha
ID - 6164
SN - 0302-9743
T2 - Lecture Notes in Computer Science
TI - Sphere construction on the FCC grid interpreted as layered hexagonal grids in 3D
VL - 11255
ER -
TY - JOUR
AB - 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.
AU - Biswas, Ranita
AU - Bhowmick, Partha
AU - Brimkov, Valentin E.
ID - 5799
JF - Discrete Applied Mathematics
SN - 0166-218X
TI - On the polyhedra of graceful spheres and circular geodesics
VL - 216
ER -
TY - JOUR
AB - This paper presents a novel study on the functional gradation of coordinate planes in connection with the thinnest and tunnel-free (i.e., naive) discretization of sphere in the integer space. For each of the 48-symmetric quadraginta octants of naive sphere with integer radius and integer center, we show that the corresponding voxel set forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as its functional plane. We use this fundamental property to prove several other theoretical results for naive sphere. First, the quadraginta octants form symmetry groups and subgroups with certain equivalent topological properties. Second, a naive sphere is always unique and consists of fewest voxels. Third, it is efficiently constructible from its functional-plane projection. And finally, a special class of 4-symmetric discrete 3D circles can be constructed on a naive sphere based on back projection from the functional plane.
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5800
IS - 1
JF - Journal of Mathematical Imaging and Vision
SN - 09249907
TI - On the functionality and usefulness of Quadraginta octants of naive sphere
VL - 59
ER -
TY - CONF
AB - 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.
AU - Dwivedi, Shivam
AU - Gupta, Aniket
AU - Roy, Siddhant
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5801
SN - 0302-9743
TI - Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space
VL - 10502
ER -
TY - CONF
AB - 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.
AU - Andres, Eric
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5802
SN - 0302-9743
TI - Digital Primitives Defined by Weighted Focal Set
VL - 10502
ER -
TY - CHAP
AB - 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.
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5803
SN - 0302-9743
T2 - Combinatorial image analysis
TI - Construction of Persistent Voronoi Diagram on 3D Digital Plane
VL - 10256
ER -
TY - CHAP
AU - Sen, Nabhasmita
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5805
SN - 0302-9743
T2 - Computational Topology in Image Context
TI - On Some Local Topological Properties of Naive Discrete Sphere
VL - 9667
ER -
TY - CONF
AB - Although the concept of functional plane for naive plane is studied and reported in the literature in great detail, no similar study is yet found for naive sphere. This article exposes the first study in this line, opening up further prospects of analyzing the topological properties of sphere in the discrete space. We show that each quadraginta octant Q of a naive sphere forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as the functional plane of Q, and hence gives rise to merely mono-jumps during back projection. The other two coordinate planes serve as para-functional and dia-functional planes for Q, as the former is ‘mono-jumping’ but not bijective, whereas the latter holds neither of the two. Owing to this, the quadraginta octants form symmetry groups and subgroups with equivalent jump conditions. We also show a potential application in generating a special class of discrete 3D circles based on back projection and jump bridging by Steiner voxels. A circle in this class possesses 4-symmetry, uniqueness, and bounded distance from the underlying real sphere and real plane.
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5806
SN - 0302-9743
T2 - Discrete Geometry for Computer Imagery
TI - On Functionality of Quadraginta Octants of Naive Sphere with Application to Circle Drawing
VL - 9647
ER -
TY - CHAP
AB - 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.
AU - Biswas, Ranita
AU - Bhowmick, Partha
AU - Brimkov, Valentin E.
ID - 5809
SN - 0302-9743
T2 - Combinatorial image analysis
TI - On the Connectivity and Smoothness of Discrete Spherical Circles
VL - 9448
ER -
TY - JOUR
AB - 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.
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5804
IS - 4
JF - Theoretical Computer Science
SN - 0304-3975
TI - From prima quadraginta octant to lattice sphere through primitive integer operations
VL - 624
ER -
TY - JOUR
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5807
IS - 11
JF - Theoretical Computer Science
SN - 0304-3975
TI - On different topological classes of spherical geodesic paths and circles inZ3
VL - 605
ER -
TY - JOUR
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5808
IS - 6-8
JF - The Visual Computer
SN - 0178-2789
TI - Layer the sphere
VL - 31
ER -
TY - CONF
AB - 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 32 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.
AU - Biswas, Ranita
AU - Bhowmick, Partha
ID - 5810
SN - 0302-9743
TI - On Finding Spherical Geodesic Paths and Circles in ℤ3
VL - 8668
ER -
TY - JOUR
AB - Canny's edge detection algorithm is a classical and robust method for edge detection in gray-scale images. The two
significant features of this method are introduction of NMS (Non-Maximum Suppression) and double thresholding of
the gradient image. Due to poor illumination, the region boundaries in an image may become vague, creating
uncertainties in the gradient image. In this paper, we have proposed an algorithm based on the concept of type-2 fuzzy sets to handle uncertainties that automatically selects the threshold values needed to segment the gradient image using classical Canny’s edge detection algorithm. The results show that our algorithm works significantly well on different benchmark images as well as medical images (hand radiography images).
AU - Biswas, Ranita
AU - Sil, Jaya
ID - 5839
JF - Procedia Technology
SN - 2212-0173
TI - An Improved Canny Edge Detection Algorithm Based on Type-2 Fuzzy Sets
VL - 4
ER -