The dynamic generalized Hough transform: Its relationship to the probabilistic Hough transforms and an application to the concurrent detection of circles and ellipses

Abstract Parametric transformation is a powerful tool in shape analysis. Major shortcomings of the technique are excessive storage requirements and computational complexity. Using a standard Hough transform (SHT) algorithm, each image point votes independently for all instances of the shape under detection on which it may lie. In this way a great redundancy of evidence concerning the image is generated. A new type of Hough-like techniques have evolved, the probabilistic Hough transforms (PHTs). These attempt to reduce the generation of redundant information by sampling the image data in various ways. The dynamic generalized Hough transform (DGHT) belongs to this family. It differs from other SHT and PHT algorithms in two fundamental ways. The first difference is that the algorithm selects a single connected point, ( x c , y c ), and uses this point to seed the transformation. The n parameters associated with the shape under detection are calculated using ( x c , y c ) together with sets of ( n − 1) randomly sampled image points. In this way voting is constrained to be on the hypersurface in transform space which would be generated by the standard transformation of ( x c , y c ). The algorithm maps each set of n image points to a single point on this surface. Voting is further restricted by appropriate segmentation of the image data. The second fundamental difference exploits the production of a sparse transform space by projecting the results of the transformation onto the axes of the n -dimensional transform space. Hence if T is the resolution in transform space and n is the number of parameters under detection then use of the DGHT reduces memory requirements from T n to nT and introduces the opportunity for parallel calculation and accumulation of parameters. Essential to the efficient use of the probabilistic techniques are stopping criteria which ensure adequate sampling with respect to the desired detection result and which also give optimum computational savings. A robust stopping criterion is deduced for the DGHT. This is applied to the concurrent detection of circles and ellipses using real image data over a range and variety of noise conditions. It is shown that the DGHT copes well with both occlusion and the effects of correlated noise. In addition, the method provides an efficient feedback mechanism linking the accumulated boundary point evidence and the contributing boundary point data. It achieves this goal automatically with an intelligent monitoring of the transformation.