Basis functions and transforms for multiresolution shape analysis and image coding

Basis functions are useful in representing a signal in terms of its expansion coefficients. In this thesis we construct a new family of basis functions for the space {dollar}Lsp2lbrack a, brbrack.{dollar} These basis functions are generated by taking the projections of simple, closed, symmetric 2-d contours on the x and y axis. The projections from elliptic contours generate basis functions which range from almost square wave functions, to the the sine wave, to a periodic delta-like sequence. Integral harmonics of these functions turn out to be a Riesz basis for {dollar}Lsp2lbrack a, brbrack .{dollar} We have shown that the unitary transforms derived from the samples of these functions have good energy compaction properties and these transforms can be parameterized with respect to a variable dependent on the shape of the contour. In the specific case of the ellipse this parameter is the eccentricity of the ellipse.; We have reviewed different families of wavelet basis functions generated from a Multiresolution framework. These basis functions form an orthonomal basis for {dollar}Lsp2(R){dollar}.We have applied the Wavelet multiresolution decomposition scheme to code the different subbands of a still image. We scan the bands along the directions sensitive to which the edge is preserved and code the address and data separately. This technique is promising as it incorporates a progressive coding scheme that contains perceptually meaningful data in the difference (higher) bands. We also apply the multiresolution discrimination algorithm to a set of handwritten characters where block transforms are used on the low-low band and a Fourier-descriptor type discriminator is used on the higher edge sensitive difference bands. This pyramid-like decomposition algorithm has the advantage that the discrimination procedures can work independently starting from a feature space of lower dimensionality (low-low bands) onto the higher dimensional difference bands until a match to a stored template is found.