| In the last decade, the wavelet transform has become a standard tool in both theoretical and applied signal and image processing. Several reasons may be adduced for the popularity of wavelet analysis, including scalability, energy compaction, simplicity of computation, and suitability to the processing of natural time-varying signals. One aspect of the wavelet transform that has been rarely explored in practical signal and image processing is the ability of such multiresolution transforms to yield information about the local smoothness of signals, quantified mathematically by Hölder regularity.; This dissertation explores several applications of regularity-based signal processing using the wavelet transform. First, we consider the discrimination of edges in images by detecting their non-smooth regions. Next, we consider a regularity-based method for improving the results of wavelet-based image compression. This method produces not only an improved visual quality, but also a lower mean squared error. The next chapter considers the image interpolation problem from the point of view of regularity. The predominant problem with existing image interpolation techniques is that they increase the smoothness of edges, which causes blurring. This artifact can be mitigated by ensuring that the interpolation algorithm preserves the low regularity of edges. This approach yields better-looking images and higher signal to noise ratios than conventional interpolation techniques—often significantly higher.; Several authors have considered methods for removing noise from images using the wavelet transform, but few people have addressed this problem using regularity. Studies have shown that noise is most easily perceptible in smooth areas, so it is of paramount importance to remove noise in these areas while leaving edges intact to avoid blurring. The algorithm we propose addresses this problem by adapting the denoising procedure to take into account local regularity. This approach also shows good mean squared error results which depend on the degree of smoothing performed. Finally, we consider a new algorithm for scalable and progressive image compression. This algorithm depends on the fact that the wavelet transform is inherently able to distinguish perceptually important edges from less important textures and smooth areas. |
