Bias Error Analysis of the Generalised Hough Transform

• Authors: Alberto S. Aguado;Eugenia Montiel;Mark S. Nixon
• Venue: Journal of Mathematical Imaging and Vision
• Year: 2000

The generalised Hough transform (GHT) extends the Houghtransform (HT) to the extraction of arbitrary shapes. In practice,the performance of both techniques differs considerably. Theliterature suggests that, whilst the HT can provide accurate resultswith significant levels of noise and occlusion, the performance ofthe GHT is in fact much more sensitive to noise. In this paper weextend previous error analyses by considering the possible causes ofbias errors of the GHT. Our analysis considers both formulation andimplementation issues. First, we compare the formulation of the GHTagainst the general formulation of the standard HT. This shows that,in fact, the GHT definition increases the robustness of the standardHT formulation. Then, in order to explain this paradoxical situationwe consider four possible sources of errors that are introduced dueto the implementation of the GHT: (i) errors in the computation ofgradient direction; (ii) errors due to false evidence attributed tothe range of values defined by the point spread function; (iii)errors due to the contribution of false evidence by backgroundpoints; and (iv) errors due to the non-analytic (i.e., tabular)representation used to store the properties of the model. Afterconsidering the effects of each source of error we conclude that: (i)in theory, the GHT is actually more robust than the standard HT; (ii)that clutter and occlusion have a reduced effect in the GHT withrespect to the HT; and (iii) that a significant source of error canbe due to the use of a non-analytic representation. A non-analyticrepresentation defines a discrete point spread function that ismapped into a discrete accumulator array. The discrete point spreadfunction is scaled and rotated in the gathering process, increasingthe amount of inaccurate evidence. Experimental results demonstratethat the analysis of errors is congruent with practicalimplementation issues. Our results demonstrate that the GHT is morerobust than the HT when the non-analytic representation is replacedby an analytic representation and when evidence is gathered using asuitable range of values in gradient direction. As such, we show thaterrors in the GHT are due to implementation issues and that thetechnique actually provides a more powerful model-based shapeextraction approach than has previously been acknowledged.