Analysing Arbitrary Curves from the Line Hough Transform

Area of Science:

• Computer Vision
• Image Processing
• Pattern Recognition

Background:

• The Hough transform is a standard technique for detecting linear features in digital images.
• Lines are represented as peaks in a parameter space, simplifying detection.
• Extending this to arbitrary curves requires analyzing the shape of these peaks, known as the peak locus.

Purpose of the Study:

• To demonstrate that the Hough transform can detect and analyze arbitrary (non-parametric) curves by examining the peak locus in parameter space.
• To establish a one-to-one relationship between curves in image space and their peak loci in parameter space.
• To explore simplifications and direct measurements achievable from the Hough transform representation.

Main Methods:

• Analyzing the patterns of the peak locus in parameter space for various curve types.
• Investigating the reconstruction of curves from their peak loci.
• Demonstrating simplification techniques by selectively ignoring parts of the peak locus.
• Applying the method to derive convex hulls and measure parameters of elliptical blobs.

Main Results:

• A one-to-one correspondence between image space curves and parameter space peak loci was established.
• Specific peak locus patterns were identified for closed curves, linear segments, inflection points, and corners.
• Curve simplification by peak locus manipulation was demonstrated, including direct convex hull derivation.
• Parameters of elliptical blobs were directly measurable from the Hough transform representation.

Conclusions:

• The Hough transform, through peak locus analysis, offers a powerful method for detecting and analyzing arbitrary curves, not just lines.
• This approach enables complete curve reconstruction and facilitates shape simplification and feature extraction.
• The method provides direct measurement capabilities for specific shapes like elliptical blobs within the Hough parameter space.