Splines and Wavelets: New Perspectives for Pattern Recognition

We provide an overview of spline and wavelet techniques with an emphasis on applications in pattern recognition. The presentation is divided in three parts. In the first one, we argue that the spline representation is ideally suited for all processing tasks that require a continuous model of the underlying signals or images. We show that most forms of spline fitting (interpolation, least-squares approximation, smoothing splines) can be performed most efficiently using recursive digital filtering. We also discuss the connection between splines and Shannon's sampling theory. In the second part, we illustrate their use in pattern recognition with the help of a few examples: high-quality interpolation of medical images, computation of image differentials for feature extraction, B-spline snakes, image registration, and estimation of optical flow. In the third and last part, we discuss the fundamental role of splines in wavelet theory. After a brief review of some key wavelet concepts, we show that every wavelet can be expressed as a convolution product between a B-spline and a distribution. The B-spline constitutes the regular part of the wavelet and is entirely responsible for its key mathematical properties. We also describe fractional B-spline wavelet bases, which have the unique property of being continuously adjustable. As the order of the spline increases, these wavelets converge to modulated Gaussians which are optimally localized in time (or space) and frequency.