In this thesis, the problem of data denoising is considered and a new data denoising method is developed. This approach is an adaptive, data-driven thresholding method that is based on Minimum Noiseless Description Length (MNDL). MNDL is an approach to subspace selection which estimates bounds on the desired Mean Square Error (MSE). The subspace minimizing these bounds is chosen as the optimum one. In this research, we explore application of MNDL Subspace Selection (MNDL-SS) as a thresholding method. Although the basic idea and desired criterion of MNDL thresholding and MNDL-SS are the same, the challenges in calculation of the desired criterion in MNDL thresholding are very different. In MNDL-SS, the additive noise effects are in the form of samples of a Chi-Square random variable. However, this assumption does not hold for MNDL thresholding anymore. In this research, we developed a new method for calculation of the desired criterion based on characteristics of noise in thresholding. Our simulation results show that MNDL thresholding outperforms the compared methods.In this thesis, we also explore the area of image denoising. In image denoising approaches, some properties of the image are considered. One of the well known image denoising methods, that outperforms other methods, is BayesShrink. We compare our method with BayesShrink. We show that the results of MNDS thresholding are comparable with BayesShrink in our simulations.