Synthetic aperture radar (SAR) and ultrasound imaging systems are widely used for applications in remote sensing and medical diagnosis, respectively. However, the SAR and ultrasound images get corrupted by speckle noise during the process of image generation. The presence of speckle gives these images a granular appearance, thus hampering the interpretation of the image and reducing the efficiency of the algorithms in performing tasks such as compression, segmentation and classification. Hence, it is crucial to reduce the speckle from the SAR and ultrasound images before performing analysis or processing of these images.  The objective of this thesis is to develop efficient wavelet-based methods for an improved reduction of the speckle from SAR and medical ultrasound images at a reduced computational cost. It is shown that the symmetric normal inverse Gaussian (SNIG) distribution is highly suitable for modelling the wavelet coefficients of the log-transformed reflectivity. Bayesian minimum mean absolute error, minimum mean squared error and maximum a posteriori estimators are developed using the SNIG PDFs. Fast and efficient techniques are introduced to estimate the model parameters from the noisy wavelet coefficients. A fast and efficient technique is presented to calculate the Bayesian minimum mean absolute error and minimum mean squared error estimators, while closed-form expressions are obtained for the Bayesian maximum a posteriori estimators. New methods of reduced complexity are proposed to incorporate the spatial dependencies of the wavelet coefficients with the Bayesian estimation processes. Extensive simulations using synthetically-speckled, SAR and medical ultrasound images are carried out to study the performance of the proposed techniques and the results show that they perform better than several existing techniques in terms of the peak signal-to-noise ratio, speckle statistics, edge preservation index, structural similarity index, ability to suppress the speckle in the homogeneous regions and visual quality, without an undue increase in computational complexity.  Finally, it is shown that the SNIG PDF can also be used to advantage in developing an efficient method for the classical case of reducing the additive white Gaussian noise from natural images.