arXiv subjects

Julien Fageot

Publications and source records attributed to Julien Fageot.

42 records · Page 3Linked to original sources

arXiv2017

Angular Accuracy of Steerable Feature Detectors

The detection of landmarks or patterns is of interest for extracting features in biological images. Hence, algorithms for finding these keypoints have been extensively investigated in the literature, and their localization and detection properties are well known. In this paper, we study the complementary topic of local orientation estimation, which has not received similar attention. Simply stated, the problem that we address is the following: estimate the angle of rotation of a pattern with steerable filters centered at the same location, where the image is corrupted by colored isotropic Gaussian noise. For this problem, we use a statistical framework based on the Cramér-Rao lower bound (CRLB) that sets a fundamental limit on the accuracy of the corresponding class of estimators. We propose a scheme to measure the performance of estimators based on steerable filters (as a lower bound), while considering the connection to maximum likelihood estimation. Beyond the general results, we analyze the asymptotic behaviour of the lower bound in terms of the order of steerablility and propose an optimal subset of components that minimizes the bound. We define a mechanism for selecting optimal subspaces of the span of the detectors. These are characterized by the most relevant angular frequencies. Finally, we project our template to a basis of steerable functions and experimentally show that the prediction accuracy achieves the predicted CRLB. As an extension, we also consider steerable wavelet detectors.

cs.IT↗

arXiv2017

Gaussian and Sparse Processes Are Limits of Generalized Poisson Processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by white Lévy noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.

math.PR↗

arXiv2016

Splines are Universal Solutions of Linear Inverse Problems with Generalized-TV regularization

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This leads us to the definition of the generalized Beppo-Levi space M, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV) over M, subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that non-uniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L1, or TV-like regularization constraints. Remarkably, the spline-type is fully determined by the choice of L and does not depend on the actual nature of the measurements.

math.FA↗

arXiv2016

Multidimensional Lévy White Noises in Weighted Besov Spaces

In this paper, we study the Besov regularity of d-dimensional Lévy white noises. More precisely, we describe new sample paths properties of a given white noise in terms of weighted Besov spaces. In particular, the smoothness and integrability properties of Lévy white noises are characterized using the Blumenthal-Getoor indices. Our techniques rely on wavelet methods and generalized moments estimates for Lévy white noises.

math.PR↗

arXiv2015

On the Besov Regularity of Periodic Lévy Noises

In this paper, we study the Besov regularity of Lévy white noises on the $d$-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Lévy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-$α$-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical $σ$-field of the space of generalized functions. These results pave the way to the characterization of the $n$-term wavelet approximation properties of stochastic processes.

math.PR↗

arXiv2014

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos-Bochner theorem. This requires a careful study of the regularity properties, especially the boundedness in Lp-spaces, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for p<1 since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.

math.PR↗