Mathematical Modeling and Analysis Directional Multiresolutional Image Analysis

Efcient representations of signals require that coefcients of functions that represent the regions of interest are sparse. This also means that we can reconstruct the original function with a smaller set of basis functions with better accuracy. Wavelets have had great success in signal pro-cessing because of their good approximation properties [1] and ability to pick up disconti-nuities efciently in one dimensional piecewise smooth functions, the discontinuities here are zero-dimensional or point discontinuities. In two dimensional functions, however, discontinuities are one-dimensional, like discontinuities occur-ring over edges or curves. Intuitively, wavelets in 2-D obtained by a tensor product of one-dimensional wavelets will be good at isolating the discontinuity at an edge point, but will not see the smoothness along a contour. For example in Fig-ure 1 the wavelet transform of the vertical edge is represented by coefcients in only one direc-tional orientation, whereas the diagonal edge is represented by signicant coefcients in all the directional orientations. Numerous methods have been proposed inde-pendently to overcome the problem. A major part of the work done this summer was to study and comprehend the rapidly growing literature on directional multiresolutional image analysis and classify the different methods based on some common approach they use. The other part of the work was to study the different properties of these methods and choose appropriate transforms for either image compression or denoising. Classication of Techniques We have classied the different directional multiresolutional image analysis techniques a