In this paper, we develop a new formulation and methodology for including invariance in a general form of the Hough transform. Essentially, the transformations that control a shape's appearance are extracted using invariance, for arbitrary shapes with a continuous description. We first develop a formal definition of the Hough transform mapping for arbitrary shapes and general transformations. We then include an invariant characterisation of shapes and develop and apply our new technique to extract shapes under similarity and affine transformations. Our formulation and implementation is based directly on parametric curves and so avoids the use of indexed look-up tables. This confers the attributes of a continuous shape description avoiding discretisation problems inherent in earlier formulations. To obtain an invariant characterisation, each point in the model is related to a collection of other points defining a geometric arrangement. This characterisation does not require the computation of properties for lines or other primitives that compose the model, but is based solely on the local geometry of the points on shapes. The transformation is obtained by solving for the parameters of the curve according to an arrangement of points defined for a point in the image and a corresponding arrangement of points for a point in the model with the same invariant proper-ties. The location parameters can be gathered in a 2D accumulator space independent of the transformation and of a shape's complexity. Experimental results show that the new technique is capable of extracting arbitrary shapes under occlusion and when the image contains significant noise. (C) 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. [References: 35]
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