SPLINES : A PERFECT FIT FOR SIGNAL / IMAGE PROCESSING

1. INTRODUCTION Finding a general mechanism for switching between the continuous and discrete signal domains is one of the fundamental issues in signal processing. It is a question that arises naturally during the acquisition process where an analog signal or an image is to be converted into a sequence of numbers (discrete representation). Conversely, the need for a continuous signal representation comes up every time one wishes to implement numerically an operator that is initially defined in the continuous domain. Typical examples in image processing are the detection of edges through the computation of gradients (spatial derivatives), and geometric transformations such as rotations and scaling (interpolation). The textbook approach to those problems is provided by Shannon's sampling theory which describes an equivalence between a bandlimited function and its equidistant samples taken at a frequency that is superior or equal to the Nyquist rate [76]. Even though this theory has had an enormous impact on the field, it has a number of problems associated with it. First, it relies on the use of ideal filters which are devices not commonly found in nature. Second, the bandlimited hypothesis is in contradiction with the idea of a finite (or finite duration) signal. Third, the bandlimiting operation tends to generate Gibbs oscillations which can be visually disturbing in images. Lastly, the underlying cardinal basis function (sinc(x)) has a very slow decay which makes computations in the signal domain very inefficient. While the first two problems can be dealt with by using approximations and introducing concepts such as an essential bandwidth and an essential time duration [78], there is no way to address the last two issues other than changing basis functions. Our purpose here will be to provide arguments in favor of an alternative approach that uses splines, which is equally justifiable on a theoretical basis, and which offers many practical advantages. To reassure the reader who may be afraid to enter new territory, we must emphasize that we are not loosing anything because we will retain the traditional theory as a particular case (i.e., a spline of infinite degree). The basic computational tools will also be familiar to a signal processing audience (filters and recursive algorithms), even though-2-their use in the present context is less conventional. In the course of the presentation, we will also bring out the connection with the multiresolution theory of the wavelet transform. Interestingly, splines are slightly older than …