## Doctoral Theses

### Polygonal Approximation and Scale-Space Analysis of Closed Digital Curves.

2-28-1994

2-28-1995

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Electronics and Communication Sciences Unit (ECSU-Kolkata)

#### Supervisor

Ray, Kumar Sankar (ECSU-Kolkata; ISI)

#### Abstract (Summary of the Work)

This thesis presents a series of algorithms for polygonal approximation of closed digital curves followed by scale-space analysis with its application to corner detection.Approximation of a closed curve by plece straight line segments is known as polygonal approximation. Any curve can be approximated by a polygon with any desired degree of accuracy.Polygonal approximation is useful in reducing the number of points required to represent a curve and to smooth data. Such representation facilitates extraction of numerical features for description and classification of curves. Basically there are two approaches to the problem. One is to subdivide the points into groups each of which satisfies a specific criterion function measuring the collinearity of the points. The collinearity is measured either by integral square error or by absolute error or by some other criterion function such as area deviation. This approach treats polygonal approximation as a side detection problem. Another approach to polygonal approximation is to detect the significant points and join the adjacent significant points by straight line segments. This approach treats polygonal approximation as an angle detection problem. In this thesis we treat polygonal approximation as a side detection as well as an angle detection problem.A number of algorithms for polygonal approximation of digital curves are already existing. Ramer [38] and Duda and Hart (13] propose a splitting technique which iteratively splits a curve into smaller and smaller curve segments until the maximumof the perpendicular distances of the points of the curve segment from the line joining the initium and the terminus of the curve segment falls within a specified tolerance. The curve segments are split at the point most distant from the line segment. The polygon is obtained joining the adjacent break-points. The procedure need multiple passes through data. Pavlidis and Horowitz (33] use split-and-merge technique which fits lines to an initial segmentation of the boundary points and computes the least squares error. The procedure then iteratively splits a curve if the error is too large and merges two lines if the error is too small. Pavlidis (32] develops another procedure which is based on the concept of 'almoet collinearity' of a set of points. To check whether a set of points are collinear / almost collinear the procedure computes an error of fit which is a function of two variables C and T. T is the maximum of the perpendicular distances of the points being tested for collinearity from the yet-to-be obtained line segment. And C is a normalized variable (0 0 then the line segment ia rejected, T, being the acceptable error and W. is the weighting factor of C. The procedure need multiple passes through data. The fundamental problem in the splitting technique and in the split-and-merge process is the initial segnentation. For open curves one can start with the two end points of a curve as the initial break-points.

#### Comments

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:28843021

ISILib-TH262

#### Creative Commons License

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

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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