Many scientific data sets are contaminated by noise, either because of data acquisition process, or because of naturally occurring phenomena. A first step in analyzing such data sets is denoising, i.e., removing additive noise from a noisy image. For images, noise suppression is a delicate and a difficult task. A trade of between noise reduction and the preservation of actual image features has to be made in a way that enhances the relevant image content. The beginning chapter in this thesis is introductory in nature and discusses the Popular denoising techniques in spatial and frequency domains. Wavelet transform has wide applications in image processing especially in denoising of images. Wavelet systems are a set of building blocks that represent a signal in an expansion set involving indices for time and scale. These systems allow the multi-resolution representation of signals. Several well known denoising algorithms exist in wavelet domain which penalize the noisy coefficients by threshold them. We discuss the wavelet transform based denoising of images using bit planes. This approach preserves the edges in an image. The proposed approach relies on the fact that wavelet transform allows the denoising strategy to adapt itself according to directional features of coefficients in respective sub-bands. Further, issues related to low complexity implementation of this algorithm are discussed. The proposed approach has been tested on different sets images under different noise intensities. Studies have shown that this approach provides a significant reduction in normalized mean square error (NMSE). The denoised images are visually pleasing. Many of the image compression techniques still use the redundancy reduction property of the discrete cosine transform (DCT). So, the development of a denoising algorithm in DCT domain has a practical significance. In chapter 3, a DCT based denoising algorithm is presented. In general, the design of filters largely depends on the a-priori knowledge about the type of noise corrupting the image and image features. This makes the standard filters to be application and image specific. The most popular filters such as average, Gaussian and Wiener reduce noisy artifacts by smoothing. However, this operation normally results in smoothing of the edges as well. On the other hand, sharpening filters enhance the high frequency details making the image non-smooth. An integrated approach to design filters based on DCT is proposed in chapter 3. This algorithm reorganizes DCT coefficients in a wavelet transform manner to get the better energy clustering at desired spatial locations. An adaptive threshold is chosen because such adaptively can improve the wavelet threshold performance as it allows additional local information of the image to be incorporated in the algorithm. Evaluation results show that the proposed filter is robust under various noise distributions and does not require any a-priori Knowledge about the image. Inpainting is another application that comes under the category of image processing. In painting provides a way for reconstruction of small damaged portions of an image. Filling-in missing data in digital images has a number of applications such as, image coding and wireless image transmission for recovering lost blocks, special effects (e.g., removal of objects) and image restoration (e.g., removal of solid lines, scratches and noise removal). In chapter 4, a wavelet based in painting algorithm is presented for reconstruction of small missing and damaged portion of an image while preserving the overall image quality. This approach exploits the directional features that exist in wavelet coefficients in respective sub-bands. The concluding chapter presents a brief review of the three new approaches: wavelet and DCT based denoising schemes and wavelet based inpainting method.