Statistical Properties of the Hybrid Radon-Fourier Technique

The hybrid Radon-Fourier technique has been proposed for the discrimination and tracking of deforming and compound targets. The current work investigates the technique’s unique statistical properties which make it inherently robust with respect to performance. The Radon transform is used to generate the geometric signature waveform of the convex hull of the target, this then becomes the input to the Fourier Transform and the Fourier coefficients determine the parameters associated with the shape and motion of the target. Because, in general, relatively few points on the boundary of the object define the convex hull they will follow a Poisson distribution. In addition, for each point in the set of points defining the convex hull, there is a high probability that another neighboring point on or near the boundary may be substituted for that point with no significant effect on the performance of the algorithm. This means that the data may be extremely sparsely sampled without a significant degradation in the performance of the algorithm and with a corresponding reduction in the computational load. The theory is illustrated using 2-D data. The extension of the technique to 3-D data is discussed and is straightforward. 1 Motivation The hybrid Radon-Fourier technique has been proposed as a solution to these machine vision tasks[1]. The current work investigates the technique’s unique statistical properties which make it inherently robust with respect to performance. The remainder of this section deals with previous work concerning tracking and the ways in which it applies to the current work. In the early literature concerning motion tracking the Fourier Transform was proposed as a potential means of tracking single objects in real-time[2]. However, the two suggested methods, global transformation of the image data[3] and segmentation of the object boundary prior to transformation[4], [5] were both seen to have drawbacks, which have precluded their use in any general form. Paul Hough [6] first introduced the Hough transform, a special case of the Radon transform[7] in 1962. Since this time and as a consequence of its potential for grouping image data into perceptually meaningful features[8], the Hough transform has also been used to track rigid objects. For example, in [9]. edge image points are selected at random from two consecutive frames in a sequence of time-varying images. When the sampling produces two pairs of pixels that are similar, i.e. having the same translational motion, the similar translation is the estimated motion. Once detected, moving objects are segmented from the image. This procedure is repeated frame by frame. Other previous methods which use the Radon/Hough transform as part of a rigid object tracking algorithm do not use the image data directly[10]. Optic flow estimates are made using the frame by frame correspondences of each pixel of the image. The transformation process then groups the pixels moving with coherent motions into objects. In addition to the computational expense of the algorithm, the recovery of optic flow is an ill-posed problem and requires additional constraints in order to regularize the problem. In order to track deforming objects, techniques that use Kalman filtering have been developed[11]. One of the most recent of these[12] uses measurements of gradient-based image potential and optical flow along the boundary of the object as the input data to the system. The method relies on the establishment of frame by frame correspondences between points in the image. It detects and rejects spurious measurements which are not consistent with previous estimation of image motion and is thus able to cope with relatively slowly varying deformations of the moving object. There is currently no solution to the problem of tracking an unknown compound target that is relatively rapidly deforming as it moves. It is not possible to extend the available techniques of frame by frame model-based reasoning or inter-frame correspondences between points in the image in order to establish the parameters of motion of such an target. For this reason, the hybrid Fourier-Radon technique has been developed. The remainder of the paper is dedicated to illustrating the theory associated with the technique and its unique statistical properties. 2 The Radon Transform The Radon Transform may be written in the convenient form: