Ridgelets: A promising new wavelet-like transform to represent objects with linear singularities

In the last two decades plenty of research has been carried out in the field of Wavelet theory and it is well known that wavelets can efficiently deal with point-like singularities. Unfortunately, such is not the case for higher dimensions singularities. To overcome this weakness of the Wavelet transform E. Candes and D. Donoho [4] introduced a new wavelet-like transform that can effectively deal with linear singularities in two dimensions, namely the Ridgelet transform. This new representation tool exploits the ability of wavelets to deal with point singularities. In fact, the Ridgelet transform is equivalent to a one-dimensional wavelet transform in the Radon domain. By doing so, a line singularity is transformed into a point singularity (by means of the Radon transform) which can then be efficiently analyzed by the wavelet transform; This thesis presents the Ridgelet transform, its properties and connections to the Radon and Wavelet transform. Also, the reader is presented with practical results that allow us to see how the Ridgelet transform is much better suited than the Wavelet transform for representing images with straight edges (linear singularities)