Moment-preserving piecewise linear approximations of signals and images

Approximation techniques are an important aspect of digital signal and image processing. Many lossy signal compression procedures such as the Fourier transform and discrete cosine transform are based on the idea that a signal can be represented by a small number of transformed coefficients which are an approximation of the original. Existing approximation techniques approach this problem in either a time/spatial domain or transform domain, but not both. This paper briefly reviews various existing approximation techniques. Subsequently, we present a new strategy to obtain an approximation fˆ(x) of f(x) in such a way that it is reasonably close to the original function in the domain of the variable x, and exactly preserves some properties of the transformed domain. In this particular case, the properties of the transformed values that are preserved are geometric moments of the original function. The proposed technique has been applied to single-variable functions, two-dimensional planar curves, and two-dimensional images, and the results obtained are demonstrative