Bias Error Analysis of the Generalised Hough Transform

Abstract

The generalised Hough transform (GHT) extends the Hough transform (HT) to the extraction of arbitrary shapes. In practice, the performance of both techniques differs considerably. The literature suggests that, whilst the HT can provide accurate results with significant levels of noise and occlusion, the performance of the GHT is in fact much more sensitive to noise. In this paper we extend previous error analyses by considering the possible causes of bias errors of the GHT. Our analysis considers both formulation and implementation issues. First, we compare the formulation of the GHT against the general formulation of the standard HT. This shows that, in fact, the GHT definition increases the robustness of the standard HT formulation. Then, in order to explain this paradoxical situation we consider four possible sources of errors that are introduced due to the implementation of the GHT: (i) errors in the computation of gradient directions (ii) errors due to false evidence attributed to the range of values defined by the point spread functions (iii) errors due to the contribution of false evidence by background pointss and (iv) errors due to the non-analytic (i.e., tabular) representation used to store the properties of the model. After considering the effects of each source of error we conclude that: (i) in theory, the GHT is actually more robust than the standard HTs (ii) that clutter and occlusion have a reduced effect in the GHT with respect to the HTs and (iii) that a significant source of error can be due to the use of a non-analytic representation. A non-analytic representation defines a discrete point spread function that is mapped into a discrete accumulator array. The discrete point spread function is scaled and rotated in the gathering process, increasing the amount of inaccurate evidence. Experimental results demonstrate that the analysis of errors is congruent with practical implementation issues. Our results demonstrate that the GHT is more robust than the HT when the non-analytic representation is replaced by an analytic representation and when evidence is gathered using a suitable range of values in gradient direction. As such, we show that errors in the GHT are due to implementation issues and that the technique actually provides a more powerful model-based shape extraction approach than has previously been acknowledged.