Proximal Methods for Hierarchical Sparse Coding

Sparse coding consists in representing signals as sparse linear combinations of atoms selected from a dictionary. We consider an extension of this framework where the atoms are further assumed to be embedded in a tree. This is achieved using a recently introduced tree-structured sparse regularization norm, which has proven useful in several applications. This norm leads to regularized problems that are difficult to optimize, and we propose in this paper efficient algorithms for solving them. More precisely, we show that the proximal operator associated with this norm is computable exactly via a dual approach that can be viewed as the composition of elementary proximal operators. Our procedure has a complexity linear, or close to linear, in the number of atoms, and allows the use of accelerated gradient techniques to solve the tree-structured sparse approximation problem at the same computational cost as traditional ones using the L1-norm. Our method is efficient and scales gracefully to millions of variables, which we illustrate in two types of applications: first, we consider fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we apply our optimization tools in the context of dictionary learning, where learned dictionary elements naturally organize in a prespecified arborescent structure, leading to a better performance in reconstruction of natural image patches. When applied to text documents, our method learns hierarchies of topics, thus providing a competitive alternative to probabilistic topic models

Wavelet Theory: for Economic & Financial Cycles

Cycles - their nature in existence, their implications on human-kind and the study thereof have sparked some important philosophical debates since the very pre-historic days. Notable contributions by famous, genius philosophers, mathematicians, historians and economists such as Pareto, Deulofeu, Danielewski, Kuznets, Kondratiev, Elliot and many others in itself shows how cycles and their study have been deemed important, through the history and process of scientific and philosophical inquiry. Particularly, the explication of Business, Economic and Financial cycles have seen some significant research and policy attention. Nevertheless, most of the methodologies employed in this space are either purely empirical in nature, time series based or the so-called Regime-Switching Markov model popularized in Economics by James Hamilton. In this work, we develop a Statistical, non-linear model fit based on circle geometry which is applicable for the dating of cycles. This study proposes a scalable, smooth and differentiable quarter-circular wavelet basis for the smoothing and dating of business, economic and financial cycles. The dating then necessitates the forecasting of the cyclical patterns in the evolution of business, economic and financial time series. The practical significance of dating and forecasting business and financial cycles cannot be over-emphasized. The use of wavelet decomposition in explaining cycles can be seen as an critical contribution of spectral methods of statistical modelling to finance and economic policy at large. Being a relatively new method, wavelet analysis has seen some great contribution in geophysical modelling. This study endeavours to widen the use and application of frequency-time decomposition to the economic and financial space. Wavelets are localized in both time and frequency, such that there is no loss of the time resolution. The importance of time resolution in dating of cycles is another motivation behind using wavelets. Moreover, the preservation of time resolution in wavelet analysis is a fundamental strength employed in the dating of cycles.Thesis (DPhil) -- Faculty of Science, Mathematical Statistics, 201

Wavelet Theory: for Economic & Financial Cycles

Cycles - their nature in existence, their implications on human-kind and the study thereof have sparked some important philosophical debates since the very pre-historic days. Notable contributions by famous, genius philosophers, mathematicians, historians and economists such as Pareto, Deulofeu, Danielewski, Kuznets, Kondratiev, Elliot and many others in itself shows how cycles and their study have been deemed important, through the history and process of scientific and philosophical inquiry. Particularly, the explication of Business, Economic and Financial cycles have seen some significant research and policy attention. Nevertheless, most of the methodologies employed in this space are either purely empirical in nature, time series based or the so-called Regime-Switching Markov model popularized in Economics by James Hamilton. In this work, we develop a Statistical, non-linear model fit based on circle geometry which is applicable for the dating of cycles. This study proposes a scalable, smooth and differentiable quarter-circular wavelet basis for the smoothing and dating of business, economic and financial cycles. The dating then necessitates the forecasting of the cyclical patterns in the evolution of business, economic and financial time series. The practical significance of dating and forecasting business and financial cycles cannot be over-emphasized. The use of wavelet decomposition in explaining cycles can be seen as an critical contribution of spectral methods of statistical modelling to finance and economic policy at large. Being a relatively new method, wavelet analysis has seen some great contribution in geophysical modelling. This study endeavours to widen the use and application of frequency-time decomposition to the economic and financial space. Wavelets are localized in both time and frequency, such that there is no loss of the time resolution. The importance of time resolution in dating of cycles is another motivation behind using wavelets. Moreover, the preservation of time resolution in wavelet analysis is a fundamental strength employed in the dating of cycles.Thesis (DPhil) -- Faculty of Science, Mathematical Statistics, 201

Blending techniques for underwater photomosaics

The creation of consistent underwater photomosaics is typically hampered by local misalignments and inhomogeneous illumination of the image frames, which introduce visible seams that complicate post processing of the mosaics for object recognition and shape extraction. In this thesis, methods are proposed to improve blending techniques for underwater photomosaics and the results are compared with traditional methods. Five specific techniques drawn from various areas of image processing, computer vision, and computer graphics have been tested: illumination correction based on the median mosaic, thin plate spline warping, perspective warping, graph-cut applied in the gradient domain and in the wavelet domain. A combination of the first two methods yields globally homogeneous underwater photomosaics with preserved continuous features. Further improvements are obtained with the graph-cut technique applied in the spatial domain

Denoising of Natural Images Using the Wavelet Transform

A new denoising algorithm based on the Haar wavelet transform is proposed. The methodology is based on an algorithm initially developed for image compression using the Tetrolet transform. The Tetrolet transform is an adaptive Haar wavelet transform whose support is tetrominoes, that is, shapes made by connecting four equal sized squares. The proposed algorithm improves denoising performance measured in peak signal-to-noise ratio (PSNR) by 1-2.5 dB over the Haar wavelet transform for images corrupted by additive white Gaussian noise (AWGN) assuming universal hard thresholding. The algorithm is local and works independently on each 4x4 block of the image. It performs equally well when compared with other published Haar wavelet transform-based methods (achieves up to 2 dB better PSNR). The local nature of the algorithm and the simplicity of the Haar wavelet transform computations make the proposed algorithm well suited for efficient hardware implementation

Video Super Resolution

Note: appendices for this title available here. Advances in digital signal processing technology have created a wide variety of video rendering devices from mobile phones and portable digital assistants to desktop computers and high definition television. This has resulted in wide diversity of video content with spatial and temporal properties fitting into their intended rendering devices. However the sheer ubiquity of video content creation and distribution mechanisms has effectively blurred the classification line resulting in the need for interchangeable rendering of video content across devices of varying spatio-temporal properties. This results in a need for efficient and effective conversion techniques; mostly to increase the resolution (referred to as super resolution) in-order to enhance quality of perception, user satisfaction and overall the utility of the video content

Wavelet-based Image Denoising

Käesolev töö uurib laineteisenduste kasutust piltide kvaliteedi parandamise eesmärgil, neilt müra eemaldades. See annab ülevaate erinevates müra tüüpidest ning müraeemaldusmeetoditest. Edasi keskendub töö laineteisenduspõhistele müraeemaldusskeemidele. Samuti uurib töö laineteisenduspõhise müraeemaldusmeetodi ning kokkupakkimise kombineerimise kasulikkust ja pakub välja uue lävendamise tüübi ning muudatuse eksisteerivale BayesShrink meetodile. Pakutud meetod implementeeritakse C# keeles ning selle implementatsiooni tulemusi, jõudlust ning optimaalseid parameetreid analüüsitakse eksperimentaalsete tulemuste abil.This thesis studies the use of wavelets for the purpose of improving the quality of images by removing noise from them. It presents an overview of different types of noise and denoising methods. The thesis then focuses on the wavelet transform based denoising schemes. It explores the potential of combining wavelet-based denoising and compression, and presents a new thresholding type and a modification to the existing BayesShrink method. The proposed method is implemented in the C# language and its performance and optimal parameters are analyzed through experimental results