On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence

We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-$r$ modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115

Non-Iterative Tone Mapping With High Efficiency and Robustness

This paper proposes an efficient approach for tone mapping, which provides a high perceptual image quality for diverse scenes. Most existing methods, optimizing images for the perceptual model, use an iterative process and this process is time consuming. To solve this problem, we proposed a new layer-based non-iterative approach to finding an optimal detail layer for generating a tone-mapped image. The proposed method consists of the following three steps. First, an image is decomposed into a base layer and a detail layer to separate the illumination and detail components. Next, the base layer is globally compressed by applying the statistical naturalness model based on the statistics of the luminance and contrast in the natural scenes. The detail layer is locally optimized based on the structure fidelity measure, representing the degree of local structural detail preservation. Finally, the proposed method constructs the final tone-mapped image by combining the resultant layers. The performance evaluation reveals that the proposed method outperforms the benchmarking methods for almost all the benchmarking test images. Specifically, the proposed method improves an average tone mapping quality index-II (TMQI-II), a feature similarity index for tone-mapped images (FSITM), and a high-dynamic range-visible difference predictor (HDR-VDP)-2.2 by up to 0.651 (223.4%), 0.088 (11.5%), and 10.371 (25.2%), respectively, compared with the benchmarking methods, whereas it improves the processing speed by over 2611 times. Furthermore, the proposed method decreases the standard deviations of TMQI-II, FSITM, and HDR-VDP-2.2, and processing time by up to 81.4%, 18.9%, 12.6%, and 99.9%, respectively, when compared with the benchmarking methods.11Ysciescopu

Streaming Algorithms for Submodular Function Maximization

We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $\Omega(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes value oracle access to $f$ and membership oracles for the matroids defining the $p$-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201

Exact Penalty Method for Knot Selection of B-Spline Regression

This paper presents a new approach to selecting knots at the same time as estimating the B-spline regression model. Such simultaneous selection of knots and model is not trivial, but our strategy can make it possible by employing a nonconvex regularization on the least square method that is usually applied. More specifically, motivated by the constraint that directly designates (the upper bound of) the number of knots to be used, we present an (unconstrained) regularized least square reformulation, which is later shown to be equivalent to the motivating cardinality-constrained formulation. The obtained formulation is further modified so that we can employ a proximal gradient-type algorithm, known as GIST, for a class of non-smooth non-convex optimization problems. We show that under a mild technical assumption, the algorithm is shown to reach a local minimum of the problem. Since it is shown that any local minimum of the problem satisfies the cardinality constraint, the proposed algorithm can be used to obtain a spline regression model that depends only on a designated number of knots at most. Numerical experiments demonstrate how our approach performs on synthetic and real data sets

Põhjalik uuring ülisuure dünaamilise ulatusega piltide toonivastendamisest koos subjektiivsete testidega

A high dynamic range (HDR) image has a very wide range of luminance levels that traditional low dynamic range (LDR) displays cannot visualize. For this reason, HDR images are usually transformed to 8-bit representations, so that the alpha channel for each pixel is used as an exponent value, sometimes referred to as exponential notation [43]. Tone mapping operators (TMOs) are used to transform high dynamic range to low dynamic range domain by compressing pixels so that traditional LDR display can visualize them. The purpose of this thesis is to identify and analyse differences and similarities between the wide range of tone mapping operators that are available in the literature. Each TMO has been analyzed using subjective studies considering different conditions, which include environment, luminance, and colour. Also, several inverse tone mapping operators, HDR mappings with exposure fusion, histogram adjustment, and retinex have been analysed in this study. 19 different TMOs have been examined using a variety of HDR images. Mean opinion score (MOS) is calculated on those selected TMOs by asking the opinion of 25 independent people considering candidates’ age, vision, and colour blindness

2-D Prony-Huang Transform: A New Tool for 2-D Spectral Analysis

This work proposes an extension of the 1-D Hilbert Huang transform for the analysis of images. The proposed method consists in (i) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure, and (ii) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2-D mode decompositions based on non-smooth convex optimization: a "Genuine 2-D" approach, that constrains the local extrema of the IMFs, and a "Pseudo 2-D" approach, which constrains separately the extrema of lines, columns, and diagonals. The spectral analysis step is based on Prony annihilation property that is applied on small square patches of the IMFs. The resulting 2-D Prony-Huang transform is validated on simulated and real data.Comment: 24 pages, 7 figure

On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression

In this paper, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures, namely, minimizers of the least squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for the degrees of freedom in many nonparametric regression problems, e.g., bounded isotonic regression, multivariate (penalized) convex regression, and additive total variation regularization. Our theory also yields, as special cases, known results on the degrees of freedom of many well-studied estimators in the statistics literature, such as ridge regression, Lasso and generalized Lasso. Our results can be readily used to choose the tuning parameter(s) involved in the estimation procedure by minimizing the Stein's unbiased risk estimate. As a by-product of our analysis we derive an interesting connection between bounded isotonic regression and isotonic regression on a general partially ordered set, which is of independent interest.Comment: 72 pages, 7 figures, Journal of the American Statistical Association (Theory and Methods), 201