The Generalised Radon Transform: Sampling and Memory Considerations

The generalised Radon transform is a well-known tool for detecting parameterised shapes in an image. Applying the Radon transform to an image results in a parameter response function (PRF). Curves in the image become peaks in the PRF. The location of a peak corresponds to the parameters of a shape, and the amplitude to the amount of evidence for that shape. In this paper we discuss two important aspects of the Radon transform. The first aspect is discretisation. Using concepts from sampling theory we derive a set of sampling criteria for the Radon transform. The second aspect concerns a projection-based algorithm to reduce memory requirements. 1 The Radon transform The (generalised) Radon transform is a technique for detecting parameterised shapes. Given a model of the shape, it defines a mapping from the image space onto a parameter space. The axes of the parameter space correspond to the parameters of the model. When applied to an image, the Radon transform yields a parameter response function (PRF) defined on the parameter space. A shape in the image becomes a peak in the PRF. The location of the peak corresponds to the parameters of the shape. Shape detection is thus reduced to peak detection. We discuss two aspects of the Radon transform: its discretisation and an algorithm to reduce its storage requirements. We focus on the Radon transform for (hyper-)spheres, but the discussion of the discretisation holds for arbitrary shapes. In its most general form, the Radon transform is