Polynomial fitting and total variation based techniques on 1-D and 2-D signal denoising

New techniques are developed for signal denoising and texture recovery. Geometrical theory of total variation (TV) is explored, and an algorithm that uses quadratic programming is introduced for total variation reduction. To minimize the staircase effect associated with commonly used total variation based techniques, robust algorithms are proposed for accurate localization of transition boundaries. For this boundary detection problem, three techniques are proposed. In the first method, the 1−D total variation is applied in first derivative domain. This technique is based on the fact that total variation forms piecewise constant parts and the constant parts in the derivative domain corresponds to lines in time domain. The boundaries of these constant parts are used as the transition boundaries for the line fitting. In the second technique proposed for boundary detection, a wavelet based technique is proposed. Since the mother wavelet can be used to detect local abrupt changes, the Haar wavelet function is used for the purpose of boundary detection. Convolution of a signal or its derivative family with this Haar mother wavelet gives responses at the edge locations, attaining local maxima. A basic local maximization technique is used to find the boundary locations. The last technique proposed for boundary detection is the well known Particle Swarm Optimization (PSO). The locations of the boundaries are randomly perturbed yielding an error for each set of boundaries. Pursuing the personal and global best positions, the boundary locations converge to a set of boundaries. In all of the techniques, polynomial fitting is applied to the part of the signal between the edges. A more complicated scenario for 1−D signal denoising is texture recovery. In the technique proposed in this thesis, the periodicity of the texture is exploited. Periodic and non-periodic parts are distinguished by examining total variation of the autocorrelation of the signal. In the periodic parts, the period size was found by PSO evolution. All the periods were averaged to remove the noise, and the final signal was synthesized. For the purpose of image denoising, optimum one dimensional total variation minimization is carried to two dimensions by Radon transform and slicing method. In the proposed techniques, the stopping criterion for the procedures is chosen as the error norm. The processes are stopped when the residual norm is comparable to noise standard deviation. 1−D and 2−D noise statistics estimation methods based on Maximum Likelihood Estimation (MLE) are presented. The proposed denoising techniques are compared with principal curve projection technique, total variation by Rudin et al, total variation by Willsky et al, and curvelets. The simulations show that our techniques outperform these widely used techniques in the literature.