The first part of this thesis is on contour stencils, a new method for edge adaptive image processing. We focus particularly on image zooming, which is the problem of increasing the resolution of a given image. An important aspect of zooming is accurate estimation of edge orientations. Contour stencils is a new method for estimating the image contours based on total variation along curves. Contour stencils are applied in designing several edge-adaptive color zooming methods. These zooming methods fall at different points in the balance between speed and quality. One of these zooming methods, contour stencil windowed zooming, is particular successful. Although most zooming methods require either solving a large linear system or running many iterations, this method has linear complexity in the number of pixels and can be computed in a single pass through the image. The zoomed image is constructed as a function that may be sampled anywhere, enabling arbitrary resampling operations. Comparisons show that contour stencil zooming methods are competitive with existing methods. Applications of contour stencils to corner detection and image enhancement are also illustrated.

The second part of this thesis is on topics in variational image processing. First, we apply variational techniques to formulate a total variation optimal prediction in Harten multiresolution schemes. We show that this prediction is well-defined, construct a Harten multiresolution using this prediction, and show that a modified encoding strategy is possible for approximation using the scheme. We also investigate the efficient numerical solution of the prediction and compare several different algorithms. Examples show that image approximation with this scheme is competitive with the CDF 9/7 wavelet.

Next, we investigate nonconvex potentials in variational image problems. For the approximate solution of these nonconvex problems, we develop a particle swarm optimization like algorithm that avoids becoming trapped in shallow local minima. Examples in denoising and image zooming show that the method can outperform gradient descent.

Finally, the last topic is on image restoration with Rician noise. Total variation regularization is usually applied with L² data fidelity assuming an additive white Gaussian noise model. However, better results are possible when the noise model accurately describes the noise in the given image. Total variation denoising has already been developed with the Laplace noise model (L¹ data fidelity) and the Poisson noise model. A challenge with Rician noise is that the resulting objective function is nonconvex. We develop a convex variational problem that closely approximates the Rician noise model. The problem is efficiently solved using the split Bregman method.