Title: Efficient implementation with FIR filters of operators based on B-splines to represent and classify signals of one and two dimensions
Abstract: Digital signals have a lot of advantages: facility in storage, compression, easy to operate numerically, but they have disadvantages, too. For example, operations relative to differential calculus like derivatives, integration, etc., are not well defined. In this thesis a way to solve this problem is presented. First, digital signals are converted into a continuous signal using polynomial spline interpolation, and then, the derivatives and others operators are calculated in the same way as in the continuous signals. Another main objective is to find the most efficient way to implement these operators with digital FIR filters. The splines are curves or functions defined piecewise by polynomials of different degrees. The splines of order 1 are straight lines that connect the different samples. The junction points of the polynomial are called knots and at this point the splines have the characteristic of having continuity in the curve as well as in derivatives up to one order less than the spline. For example with cubic splines the continuity in the knots of polynomials is insured up to the second derivative. In Fig.1 it’s shown an example of interpolation with linear splines. Fig. 2 shows the same points but interpolated with cubic splines. As it can be seen with cubic splines a curve much smoother than with linear splines is achieved. In the case of cubic splines, to find each coefficient of each spline polynomial is necessary to solve a system of three equations per point to interpolate. In this thesis we proposed to solve this by applying a digital FIR filter. One of the first reference works on the spline is [1]. At first, it was applied in the field of graphic design and basically to define continuous curves, interpolating or approximating specific points, without needing that these points be evenly spaced. It was in early 1990's when Michael Unser, professor and director of research Biomedical Imaging group of the Federal Polytechnic School of Lausanne, who developed much of the mathematical theory to apply B-splines in signal processing [2], [3]. This imply to have equidistant samples and normalized period (T=1). The B-spline function of order zero is a rectangle defined in the real domain between [-0.5 and +0.5]. The B-spline of order n is a function that has compact support and can be generated by convolving the Bspline of order zero n+1 times with itself. Any polynomial spline can be represented by a linear combination of Bspline functions displaced. In figures 3 and 4 the Bsplines functions of order 0, 1, 2 and 3 respectively are shown.
Publication Year: 2012
Publication Date: 2012-01-01
Language: en
Type: article
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Cited By Count: 2
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