Spatial domain filter design

In this thesis, we present both a methodology and a rationale for designing filters based on spatial smoothness and accuracy criteria. Frequency based methods have been used to design function and derivative reconstruction filters especially in image and signal processing. Desired filter characteristics are specified in terms of spectral properties. However, for visualization and image synthesis algorithms it is more natural to specify a filter in terms of the smoothness of the resulting reconstructed function and the spatial reconstruction error. Most currently known filter choices are limited to a specific set of functions, and the visualization practitioner has, so far, no way to state his preferences in a convenient fashion. First we present a Taylor series expansion of the convolution sum, which leads to four evaluation criteria of general reconstruction filters. These evaluation criteria help us state our design criteria. The filter performance in our framework is specified by listing the accuracy and smoothness (of the reconstructed function) through the use of three criteria. These criteria help us design reconstruction filters for arbitrary derivatives of arbitrary (asymptotic) accuracy and arbitrary smoothness. We provide rigorous proof for the existence of a filter using results from approximation theory. Also, we provide a frequency domain perspective of our three design criteria. In addition, we show that the filters designed using our framework have desirable spectral properties by measuring the deviation from the ideal function or derivative filter frequency response. We then provide a cookbook for designing filters and illustrate this process with examples. Finally, we present applications from volume rendering and pattern matching to highlight our design methodology.