Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Wavelet and Multiscale Analysis of Network Traffic

The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behaviour in tele-traffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in non-stationary environments. In this paper we describe the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unravelling the mysteries of traffic statistics and dynamics

Study of Lipschitz regularity - Feature extraction on regular and irregular grids

In this paper we study the pointwise Lipschitz regularity, covering several aspects: theoretical and practical, methods for its estimation on regular and irregular grids. The relevance of this value of regularity lies in its invariance properties to several transformations, and its fast computation thanks to wavelets. We study the influence of scale on wavelets transforms and show invariance properties this value of regularity. We also put forward an original technique for its estimation on regular grids. We also address the issue of irregular grids, based on the behavior of smoothing kernels with respect to scale. The obtained results emphasize the usefulness of such features for the applications, and motivate further work on this topic. Keywords: Lipschitz regularity, wavelets, smoothing kernels, robust feature extraction, regular and irregular gridsJRC.G.2-Global security and crisis managemen

Conditionally Strongly Log-Concave Generative Models

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the $\varphi^4$ model and weak lensing convergence maps with higher resolution than in previous works.Comment: 28 pages, 12 figures, accepted at ICML 202

Image Utility Assessment and a Relationship with Image Quality Assessment

International audiencePresent quality assessment (QA) algorithms aim to generate scores for natural images consistent with subjective scores for the quality assessment task. For the quality assessment task, human observers evaluate a natural image based on its perceptual resemblance to a reference. Natural images communicate useful information to humans, and this paper investigates the utility assessment task, where human observers evaluate the usefulness of a natural image as a surrogate for a reference. Current QA algorithms implicitly assess utility insofar as an image that exhibits strong perceptual resemblance to a reference is also of high utility. However, a perceived quality score is not a proxy for a perceived utility score: a decrease in perceived quality may not affect the perceived utility. Two experiments are conducted to investigate the relationship between the quality assessment and utility assessment tasks. The results from these experiments provide evidence that any algorithm optimized to predict perceived quality scores cannot immediately predict perceived utility scores. Several QA algorithms are evaluated in terms of their ability to predict subjective scores for the quality and utility assessment tasks. Among the QA algorithms evaluated, the visual information fidelity (VIF) criterion, which is frequently reported to provide the highest correlation with perceived quality, predicted both perceived quality and utility scores reasonably. The consistent performance of VIF for both the tasks raised suspicions in light of the evidence from the psychophysical experiments. A thorough analysis of VIF revealed that it artificially emphasizes evaluations at finer image scales (i.e., higher spatial frequencies) over those at coarser image scales (i.e., lower spatial frequencies). A modified implementation of VIF, denoted VIF*, is presented that provides statistically significant improvement over VIF for the quality assessment task and statistically worse performance for the utility assessment task. A novel utility assessment algorithm, referred to as the natural image contour evaluation (NICE), is introduced that conducts a comparison of the contours of a test image to those of a reference image across multiple image scales to score the test image. NICE demonstrates a viable departure from traditional QA algorithms that incorporate energy-based approaches and is capable of predicting perceived utility scores