A New Model for the Evolution of Open Ended Curves in a Two Dimensional Plane (A Special Case of Evolving Manifolds of Arbitrary Codimension with Boundaries).

Date of Submission

December 2005

Date of Award

Winter 12-12-2006

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Master's Dissertation

Degree Name

Master of Technology

Subject Name

Computer Science


Electronics and Communication Sciences Unit (ECSU-Kolkata)


Mukherjee, Dipti Prasad (ECSU-Kolkata; ISI)

Abstract (Summary of the Work)

Most of the existing mathematical theory for implicit models of manifold evolution like level set methods, focus on the evolution of hypersurfaces, i.e., codimension one motion. For example, level set methods are consistently used in the segmentation and tracking of object boundaries in medical or aerial images. This family of curve evolution works on instances of codimension one motion of a closed curve in a 2D image space. The limitation of implicit methods for evolving smooth manifolds of arbitrary codimension is a serious difficulty in segmenting shapes like blood vessels in volumetric angiogram images or roads and rivers from satellite images. The objects in previous examples are essentially one dimensional in nature and requires evolution of a curve in 3D space or that of an open-ended curve in 2D space, both of which are instances of codimension two motion.The goal of this thesis is to build a theoretical and numerical framework to represent and evolvr smooth open ended curves in a two dimensional plane, which is a special case of a manifold wit, boundaries and having codimension greater than one. Original ideas of the level set formulation of active contours need to be adapted to images that exhibit one-dimensional filament like structures, since, segmentation of one-dimensional filament like structures are more suitable using open ended curves rather than closed ones (classical theory of geometric active contoui + work only for closed curves).We have introduced two new approaches to the problem of evolving an open-ended curve in a two-dimensional image plane. The first approach is an extension to the existing level set theory and is suitable for tracking the contours of objects in images with open curves. However, the first method essentially works for object contours that can be tracked with closed curves too. The second approach is a completely new visualization of the implicit representation of open curves in a 2D plane, and is particularly suitable for segmenting one-dimensional filament like structures in 2D images. The second approach solves a segmentation problem (of ID filaments) that was hitherto inefficiently catered to by the traditional level set method.We have tested the new models on both synthetic and real images that contain one-dimensional filament like structures. Results show that our models are successful in segmenting complex topological filaments with minimal distortion, and the theory is robust enough to withstand effects of shape distortions like kinks, bending, circularity and inconsistent edges and correctly approximate the shapes of the filaments. We hope to have built a novel approach to one dimensional shape segmentation, and future work on the improvement of the models a expected to give rise to hugely efficient filament tracking algorithms encompassing a wide range of image types.


ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843308

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Creative Commons License

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



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