| This thesis investigates some innovative approaches to inverse problems in image restoration and enhancement. The specific problems addressed include image denoising and image interpolation. Before developing an algorithm, the first step is to find an appropriate image model to use. To achieve this, we identify two successful domains of image processing, namely, image compression and edge analysis, from which ideas can be applied to image denoising and interpolation. The underlying framework for signal analysis and algorithm development is based on wavelets , which conveniently provides a multiresolution, localized space-frequency representation of the signal.; We use a Bayesian model for the distribution of the wavelet coefficients, namely, the Generalized Gaussian distribution which has been widely used for image compression. From this distribution, we propose a near-optimal threshold selection.; One of the first ideas we examine is using lossy compression for removing noise from corrupted images. We make a connection between lossy compression and wavelet thresholding, and develop a systematic lossy compression method to achieve simultaneous compression and denoising.; Next, we develop a spatially adaptive algorithm for image denoising. Images typically consist of edges, textures and smooth regions. In the wavelet transform domain, the first two features are characterized by clusters of high energy transform coefficients and the latter by low energy coefficients.; To conclude the denoising topic, we investigate the best strategy to combine multiple noisy copies of the same image. Typically, multiple sets of noisy observations of the same data are averaged to obtain the best estimate of the noiseless version. Since wavelet thresholding is effective for denoising one set of noisy observations, it is worthwhile to incorporate it with weighted averaging when multiple noisy copies are available. In particular, we investigate which sequential ordering of the averaging and wavelet thresholding operation would yield a final result with the lowest mean squared error. The result shows that, under the assumed Laplacian distribution for the coefficients (a special, simple case of the Generalized Gaussian), the ordering is dependent on the distribution parameter, the noise power, and the number of noisy copies.; Lastly, we develop an edge-preserving image interpolation algorithm. The available image is modeled as a low resolution image of some higher resolution image which we wish to estimate. The additional details needed to obtain the desired image is estimated by extrapolating edge characteristics from the low resolution image. The problem model and the edge analysis can be developed very naturally in the wavelet framework. (Abstract shortened by UMI.)... |
