Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients

We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods

Adaptive Image Restoration: Perception Based Neural Nework Models and Algorithms.

Abstract This thesis describes research into the field of image restoration. Restoration is a process by which an image suffering some form of distortion or degradation can be recovered to its original form. Two primary concepts within this field have been investigated. The first concept is the use of a Hopfield neural network to implement the constrained least square error method of image restoration. In this thesis, the author reviews previous neural network restoration algorithms in the literature and builds on these algorithms to develop a new faster version of the Hopfield neural network algorithm for image restoration. The versatility of the neural network approach is then extended by the author to deal with the cases of spatially variant distortion and adaptive regularisation. It is found that using the Hopfield-based neural network approach, an image suffering spatially variant degradation can be accurately restored without a substantial penalty in restoration time. In addition, the adaptive regularisation restoration technique presented in this thesis is shown to produce superior results when compared to non-adaptive techniques and is particularly effective when applied to the difficult, yet important, problem of semi-blind deconvolution. The second concept investigated in this thesis, is the difficult problem of incorporating concepts involved in human visual perception into image restoration techniques. In this thesis, the author develops a novel image error measure which compares two images based on the differences between local regional statistics rather than pixel level differences. This measure more closely corresponds to the way humans perceive the differences between two images. Two restoration algorithms are developed by the author based on versions of the novel image error measure. It is shown that the algorithms which utilise this error measure have improved performance and produce visually more pleasing images in the cases of colour and grayscale images under high noise conditions. Most importantly, the perception based algorithms are shown to be extremely tolerant of faults in the restoration algorithm and hence are very robust. A number of experiments have been performed to demonstrate the performance of the various algorithms presented

Induced seismicity analysis for reservoir characterization at a petroleum field in Oman

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 2005.Includes bibliographical references.This thesis presents the analysis and interpretation of passive seismic data collected in a 20-month monitoring period. The investigation is divided into four studies, each focusing on a different aspect of the seismic data to infer the reservoir properties. First, I applied three different methods (the iterative linearized, nonlinear grid-search, and double-difference methods) to relocate 405 microearthquakes that occurred between October 1999 and June 2001 in a producing field in Oman. A numerical technique is applied to "collapse" the relocated hypocenters and to find the simplest structural interpretation consistent with the data. Comparing the methods, the applicability of waveform correlation methods such as the double-difference in this case is limited by the relatively large number of events with dissimilar waveforms. Unlike the iterative linearized method, the nonlinear grid-search method gives the best results with the smallest average rms error of the absolute locations because it avoids the local minimum problem.(cont.) The relocated hypocenters clearly delineate nearly vertical, northeast-southwest striking faults near the crest of the field, which is consistent with the graben fault system mapped by surface geologic surveys and reflection seismic interpretations. I also performed statistical tests to estimate location errors, and found that the station geometry is the major factor that limits the accuracy of focal depths. Secondly, this thesis presents a non-linear wavelet-based approach to linear waveform inversion of high-frequency seismograms for the estimation of a point source mechanism and its time function. For earthquake mechanism inversions, it is important to stabilize the problem by reducing the number of parameters to be determined. Commonly, overlapping isosceles triangles or boxcar functions are used for the parameterization of the moment tensor rate functions (MTRFs). Here, I develop a wavelet-based strategy that allows us to construct an adaptive, problem-dependent parameterization for the MTRFs employing fractional spline wavelets. Synthetic results demonstrate that the adaptive parameterization improves the numerical approximation to the model space and therefore, allows more accurate estimations of the MTRFs.(cont.) The waveform inversion is performed in the wavelet domain and leads to a multiresolution sparse matrix representation of the inverse problem. At each resolution level a regularized least-squares solution is obtained using the conjugate gradient method. The wavelet-based waveform inversion method has been applied successfully in three real- data examples: the April 22, 2002 Au Sable Forks, New York earthquake, the September 3, 2002 Yorba Linda, California earthquakes, and 11 M>1 microearthquakes in a producing field in Oman. In the Oman field, the dominant styles of focal mechanism are left-lateral strike-slip for events with focal depths less than 1.5 km, and dip-slip along an obliquely trending fault for those with focal depths greater than 2.0 km. Thirdly, the covariance matrix method of shear-wave splitting analysis is presented. Different from conventional methods that usually analyze only two horizontal components, this method processes all three components of the seismogram simultaneously, allowing not only orientation but also dip information of fractures to be resolved. Synthetic test results show that this method is stable even for high noise level.(cont.) The method is applied to the Oman microearthquake records that display distinctive shear-wave splitting and polarization directions. From the polarizations, I estimate the predominant subsurface fracture directions and dipping angles. From the time delays of the split wave I determine the fracture density distributions in the reservoir. Finally, I examine the spatio-temporal characteristics of the microseismicity in the producing reservoir. The frequency-magnitude distribution measured by the b-value is determined using the maximum likelihood method. I found that b-values are higher for events below the deeper Shuaiba oil reservoir than those above. Also, the feasibility of monitoring the temporal change of b-values is demonstrated. The analysis of production and injection well data shows that seismicity event rates in the field all strongly correlated with gas production from the shallower Natih Formation. Microseismicity, focal mechanisms, GPS analysis, and production / injection well data all suggest the NE- SW bounding graben fault system responds elastically to the gas-production-induced stresses. Normal faulting is enhanced in the reservoirs by the compaction related stresses acting on the graben fault system.by Edmond Kin-Man Sze.Ph.D