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: highquality 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 Bspline 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. 1! Splines and Continuous/Discrete Signal Processing What follows is a brief synopsis of the presentation, with some pointers to the relevant literature. Splines provide a unifying framework for linking the continuous and discrete domains. They are well understood theoretically, and are ideally suited for performing numerical computations [6]. This makes them the perfect tool for solving a whole variety of signal and image processing (or pattern recognition) problems that are best formulated in the continuous domain but call for a discrete solution [21]. This leads to a class of computational techniques that we refer to as “continuous/discrete signal processing”, and which may be best summarized by the motto “think analog, act digital”. The cardinal splines, which are the type of splines considering here, were invented by Schoenberg almost 60 years ago [14]. These polynomial splines are 1D functions that Splines and Wavelets: New Perspectives for Pattern Recognition 245 are defined on a uniform grid with knots at the integers when the degree is odd, or at the mid-integers when the degree is even. For each segment defined by two successive knots, the spline is a polynomial of degree n; the polynomial pieces are patched together at the knots in a way that guarantees the continuity of the function and of all its derivative up to order (n-1). Schoenberg in his landmark paper showed that these splines could be represented most conveniently in terms of basis functions that are integer translates of a generating function: the B-spline of degree n. These Bsplines have a number of very desirable properties. They have a simple analytical form (piecewise polynomial of degree n) that facilitates their manipulation [6, 23]. In our earlier work, we have shown that most B-spline computations can be made using digital filters [22, 23], provided that the grid is regular, which is always the case in image/signal processing. The B-splines satisfy a two-scale relation which makes them prime candidates for constructing wavelet bases [24]. In this respect, B-splines form a category apart since the scaling relation holds for any positive integer m—and not just powers of two; this property can be used advantageously for designing fast multi-scale filtering algorithms [26] . Splines have excellent approximation properties, mainly because the underlying Bspline basis functions are very regular [19, 4, 3]. They are also optimal in the sense that they provide the signal interpolant with the least oscillating energy [13]. Finally, the spline framework allows for a progressive transition between the two extreme signal representations: the piecewise-constant model (spline of degree zero) that uses the most localized, but least regular, basis functions, and the bandlimited model that corresponds to a spline of infinite degree [1]. Also note that there are recent extensions of Shannon’s sampling theory that consider spline-like representations of functions instead of the traditional bandlimited model [20]. 2! Splines: Applications in Pattern Recognition The primary applications of splines in pattern recognition are sampling and interpolation, feature extraction, image matching, and motion analysis. In the presentation, we briefly discuss the following topics: 2.1 High-Quality Image Interpolation When compared to other interpolation algorithms, splines provide the best tradeoff in terms of quality and of computational cost. In other words, if you want to improve interpolation quality, your best choice over any other method is to increase the order of the spline. This is a finding that has been confirmed independently by several research teams [16, 11, 10, 9]. 2.2! Feature Extraction After fitting the image with a spline, it is straightforward to compute exact image derivatives to derive gradients or Hessians for the detection of various image features such as contours or ridges [22]