Image denoising is often used for pre-processing images so that subsequent image analyses are more reliable. Many existing methods can not preserve complicated edge-structures well, but those structures contain useful information about the image objects. So, besides noise removal, a good denoising method should preserve important edge-structures. The major goal of this dissertation is to develop image denoising techniques so that complicated edge-structures are preserved efficiently. The developed methods are based on nonparametric estimation of discontinuous surfaces, because a monochrome image can be regarded as a surface of the image intensity function and its discontinuities are usually at the outlines of the objects. The first part of this dissertation introduces some existing methods and related literature. Next, an edge-structure preserving 2-D image denoising technique is proposed, and it is shown that it performs well in many applications. The next part considers 3-D images. Because of emerging popularity of 3-D MRI images, 3-D image denoising becomes an important research area. The edge-surfaces in 3-D images can have much more complicated structures, compared to the edge-curves in 2-D images. So, direct generalizations of 2-D methods would not be sufficient. This part handles the challenging task of mathematically describing different possible structures of the edge-surfaces in 3-D images. The proposed procedures are shown to outperform many popular methods. The next part deals with the well-known bias issue in denoising MRI images that is corrupted with rician noise, and provides an efficient method to remove that bias. The final part of this dissertation discusses the future research directions along the line of previous parts. One of them is image denoising by appropriate multilevel local smoothing techniques so that the fine details of the images are well preserved.
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