Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Advance Computing and Microelectronics Unit (ACMU-Kolkata)


Bhattacharaya, Bhargab Bikram (ACMU-Kolkata; ISI)

Abstract (Summary of the Work)

In this thesis, we have reported some new theoretical findings, empirical formulations, useful heuristics, and efficient algorithms related to digital circle, digital disc, and digital sphere, along with their practical applications to the analysis of geometric information embedded in a digital image. Detecting digital circles and circular arcs from a digital image is very important in shape recognition. Several image processing techniques were proposed over the years to extract circles and circular arc from a digital image and to interpret related issues. We have proposed a novel technique for the segmentation of a digital circle, which is based on a variant of well known chord property and sagitta property of an euclidean circle. We show that the radius of a digital circle estimated by this technique is more accurate and thus, this method is very effective in segmenting circles and circular arcs from digitized engineering drawings. Next, we use a set of consecutive and concentric digital circles to construct a digital disc. Such a construction raises a new problem of absentee-pixel characterization. We present a novel characterization of the absentee-pixels that appear in the cover of a digital disc with concentric digital circles. The characterization is based on several number-theoretic and geometric properties of a digital circle. The notion of infimum parabola and supremum parabola has been used to derive the count of these absentees. Using this parabolic characterization, we derive a necessary and sufficient condition for a pixel to be a disc absentee, and obtain the underlying geometric properties of the absentees. An algorithm for identifying the absentee-pixels is also presented. Later, we have generalized this idea to 3D and show that the construction of a digital sphere of revolution obtained by circularly sweeping a digital semicircle (generatrix) around its diameter, results in the appearance of some holes (absentee-voxels) in its spherical surface of revolution. We present a characterization of these absentee-voxels using certain properties of digital geometry and show that their count varies quadratically with the radius of the semicircular generatrix. Also, we design an algorithm to fill the holes of the absentee-voxels so as to generate a spherical surface of revolution, which is complete and realistic from theviewpoint of visual perception. We further show that covering a solid sphere by a set of complete spheres also results to an asymptotically larger count of absentee-voxels, which is cubic in the radius of the sphere. Necessary characterization and generation of a complete solid sphere have also been worked out in the final stage.The segmentation of objects embedded in a digital image is an important task in image processing with numerous applications. In this thesis, we study two specific engineering problems: (i) characterization of micro-pores on a porous silicon (PS) chip by image analysis, and (ii) granular object segmentation for application to agriculture. Whilst regular structures like micro-test tubes and micro-beakers fabricated on a PS chip offer potential platforms for implementing various biosensors, controlling the uniformity of pores during electrochemical etching is a challenging problem. One important objective of such fabrication procedure is to ensure the circularity of pore boundaries. Thus, to tune up and standardize the etching process, a fast image analysis technique is needed to evaluate and quantify the geometry of these nano-scale PS structures. We present an automated approach to pore image analysis: given a top-view image of a PS chip captured by a scanning electron microscope (SEM), the porous regions are segmented and each of the pore boundaries is approximated by a circle. Granular object segmentation is another important task of image processing with several fields of applications including agriculture. A simple algorithm for automated analysis of granulometric images consisting of touching or overlapping convex objects such as coffee bean, food grain, is presented. The algorithm is based on certain underlying digital-geometric features embedded in their snapshots. Using the concept of an outer isothetic cover and geometric convexity, the separator of two overlapping objects is identified. The objects can then be isolated by removing the isothetic covers and the separator. The technique needs only integer computation and its termination time can be controlled by choosing a resolution parameter. The thesis ends with future research directions and a few interesting problems related with digital circle, digital disc, and digital sphere.


ProQuest Collection ID:

Control Number


Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Included in

Mathematics Commons