In most signal processing applications, a given range of data is best described by a set of local characteristics as opposed to a single global characteristic. In image processing, for example, a region of an image that contains numerous edges is best described as a region whose pixel color values change abruptly, i.e., they are not continuous values of color. A region of constant color, or gradually changing color, is best described as a region whose pixel values are constant, or whose values increase linearly by some factor. It is advantageous to represent this data with signals capable of adapting to these types of local characteristics, as opposed to choosing the best global characteristic. Here, the B-spline wavelet recurrence relation is presented. The B-spline wavelet recurrence relation allows a wavelet of order n +1 to be constructed from a wavelet of order n. This recurrence relation provides a mathematical tool capable of locally varying its degree of continuity. The order of differentiability of a B-spline wavelet increases as the order of the B-spline wavelet increases, and the values range from discontinuous to an arbitrary degree of continuity. A brief discussion of interpolation for splines and B-spline wavelets is introduced as a step toward a future application.
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