Number-theoretic interpretation and construction of a digital circle

Bhowmick, Partha; Bhattacharya, Bhargab B.

doi:10.1016/j.dam.2007.10.022

Cited by 36 publications

(30 citation statements)

References 43 publications

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“…2), since for k = 4, we have r = 4 for which v (r ) [4,4] in which lies the square number 4 = i 2 and u (r ) r − j = r 2 − j 2 − j = 10 2 − 9 2 − 9 = 10, v (r ) r − j = r 2 − j 2 + j = 10 2 − 9 2 + 9 = 28, thus giving I On the contrary, (3,9,4) is not an absentee-voxel, as for k = 4, there is no such r for which J (r ) r −4 contains 3 2 ; in fact, for k = 4, we get the interval I To characterize the absentees as a whole, we use Lemma 3 for the expanded form of (the lower and the upper limits of) J (r ) r −k . We replace r by k + h and r + 1 by k + (h + 1), where the h(≥ 0)th run of pixels in C 1 (r ) drawn on zx-plane has z = k [7]. Thus,…”

Section: Theorem 5 a Voxel P(i J K) Is An Absentee If And Only Ifmentioning

confidence: 97%

“…Lemma 2 (circle pixel [7]) The squares of abscissae of the pixels with z = k in C 1 (r ) drawn on zx-plane lie in the interval I (r ) …”

Section: Characterizing the Absentee Familymentioning

confidence: 99%

“…We use the following lemmas from [6,7] for deriving the necessary and sufficient conditions to decide whether a given voxel is an absentee or not.…”

Section: Characterizing the Absentee Familymentioning

confidence: 99%

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On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

2016

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We show that the construction of a digital sphere by circularly sweeping a digital semi-circle (generatrix) around its diameter results in the appearance of some holes (absentee-voxels) in its spherical surface of revolution. This incompleteness calls for a proper characterization of the absentee-voxels whose restoration in the surface of revolution can ensure the required completeness. In this paper, we present a characterization of the absentee-voxels using certain techniques of digital geometry and show that their count varies quadratically with the radius of the semi-circular generatrix. Next, we design an algorithm to fill up the absentee-voxels so as to generate a spherical surface of revolution, which is complete and realistic from the viewpoint of visual perception. We also show how the proposed technique for absentee-filling can be used to generate a variety of digital surfaces of revolution by choosing an arbitrary curve as the generatrix. We further show that covering a solid sphere by a set of complete spheres also results to an asymptotically larger count of absentees, which is cubic in the radius of the sphere. A complete characterization of the absenteevoxels that aids the subsequent generation of a solid digital A preliminary version of this work appeared in ICAA'14 [5].

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Section: Theorem 5 a Voxel P(i J K) Is An Absentee If And Only Ifmentioning

confidence: 97%

“…Lemma 2 (circle pixel [7]) The squares of abscissae of the pixels with z = k in C 1 (r ) drawn on zx-plane lie in the interval I (r ) …”

Section: Characterizing the Absentee Familymentioning

confidence: 99%

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On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

2016

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“…First let us show that in the first octant for any given point p 0 = (x 0 , y 0 ) on the circle at the point p 1 = (x 0 + 1, y 0 − 1) must also be in the circle as |p 1 | < |p 0 |. When p 0 is in the circle and in the first octant segment such that x < y it follows that the magnitude of p 1 is less than p 0 as |p 0 | − |p 1…”

Section: Geometrical Properties Of Discrete Circlesmentioning

confidence: 99%

“…The concept of neighbourhood sequences is of importance in a number of practical applications and was originally applied for measuring distances in a digital world [13]. Initially two classical digital motions (cityblock and chessboard) 1 were introduced. Based on these two types of motions periodic neighbourhood sequences were defined in [3] by allowing arbitrary mixture of cityblock and chessboard motions.…”

Section: Introductionmentioning

confidence: 99%

Broadcasting Automata and Patterns on ℤ2

Nickson

Potapov

2015

Automata, Universality, Computation

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The Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood sequences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for broadcasting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to automata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a variety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An exploration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and generation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and approximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions.

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A digital geometric approach for discrete Voronoi diagram construction using GPU

Dhar,

Pal,

Bhattacharjee

et al. 2024

Concurrency and Computation

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SummaryVoronoi diagram is a versatile geometric structure used in solving many computational geometry problems in the field of image analysis, computer vision, engineering, medicine, molecular biology and various other disciplines. As a result it is a popular research domain. The literature boasts of many elegant and efficient methods for the construction of the Voronoi diagram in the real space. But these algorithms often fail to perform well when applied to the discrete domain. We present in this paper a novel algorithm for the construction of discrete Voronoi diagram using digital geometric concepts that takes care of the geometric perspective of the Voronoi diagram in the discrete domain. Although our technique is more suited for the CPU but by careful use of some cut‐off heuristics we are able to design a parallel programming approach for the construction of the Voronoi diagram making use of the GPU computational power. The speedup of our algorithm is compatible with the popular GPU technique for Voronoi diagram construction, namely, Jump flood algorithm. But since our algorithm uses the geometric perspective for the construction of Voronoi diagram it gives more accurate output and hence requires no correction step as is the case for the Jump flood algorithm. Furthermore, our approach focuses on extensibility, allowing for easy amenability for construction of higher dimensional Voronoi diagram. This ensures that the Voronoi diagram generation process can be adapted and expanded upon to meet specific requirements or incorporate future enhancements. We present an implementation of the proposed algorithm on a GPU using TAICHI language and describe its performance benefits on multiple input sites.

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Number-theoretic interpretation and construction of a digital circle

Cited by 36 publications

References 43 publications

On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

Broadcasting Automata and Patterns on ℤ2

A digital geometric approach for discrete Voronoi diagram construction using GPU

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