139 research outputs found
Projected gradient descent for non-convex sparse spike estimation
We propose a new algorithm for sparse spike estimation from Fourier
measurements. Based on theoretical results on non-convex optimization
techniques for off-the-grid sparse spike estimation, we present a projected
gradient descent algorithm coupled with a spectral initialization procedure.
Our algorithm permits to estimate the positions of large numbers of Diracs in
2d from random Fourier measurements. We present, along with the algorithm,
theoretical qualitative insights explaining the success of our algorithm. This
opens a new direction for practical off-the-grid spike estimation with
theoretical guarantees in imaging applications
PROJECTED GRADIENT DESCENT FOR NON-CONVEX SPARSE SPIKE ESTIMATION
We propose an algorithm to perform sparse spike estimation from Fourier measurements. Based on theoretical results on non-convex optimization techniques for off-the-grid sparse spike estimation, we present a simple projected descent algorithm coupled with an initialization procedure. Our algorithm permits to estimate the positions of large numbers of Diracs in 2d from random Fourier measurements. This opens the way for practical estimation of such signals for imaging applications as the algorithm scales well with respect to the dimensions of the problem. We present, along with the algorithm, theoretical qualitative insights explaining the success of our algorithm
The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension
Non-convex methods for linear inverse problems with low-dimensional models
have emerged as an alternative to convex techniques. We propose a theoretical
framework where both finite dimensional and infinite dimensional linear inverse
problems can be studied. We show how the size of the the basins of attraction
of the minimizers of such problems is linked with the number of available
measurements. This framework recovers known results about low-rank matrix
estimation and off-the-grid sparse spike estimation, and it provides new
results for Gaussian mixture estimation from linear measurements. keywords:
low-dimensional models, non-convex methods, low-rank matrix recovery,
off-the-grid sparse recovery, Gaussian mixture model estimation from linear
measurements
Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations
In this paper, we first review some properties of the signed distance function. In particular, we examine the skeleton of a curve in ^2 and get a complete description of its closure. We also give a sufficient condition for the closure of the skeleton to be of zero Lebesgue's measure. We then make a complete study of the PDE: du/dt +sign(u_0(x))(|Du|-1)=0 , which is closely related to the signed distance function. The existing literature provides no mathematical results for such PDEs. Indeed, we face the difficulty of considering a discontinuous Hamiltonian operator with respect to the space variable. We state an existence and uniqueness theorem, giving in particular an explicit Hopf-Lax formula for the solution as well as its asymptotic behaviour. This generalizes classical results for continous Hamitonian. We then get interested in a more general class of PDEs: du/dt +sign(u_0(x))H(D- u)=0, with H convex Under some technical but reasonable assumptions, we obtain the same kind of results. As far as we know, they are new for discontinuous Hamiltonians
Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations
In this paper, we first review some properties of the signed distance function. In particular, we examine the skeleton of a curve in ^2 and get a complete description of its closure. We also give a sufficient condition for the closure of the skeleton to be of zero Lebesgue's measure. We then make a complete study of the PDE: du/dt +sign(u_0(x))(|Du|-1)=0 , which is closely related to the signed distance function. The existing literature provides no mathematical results for such PDEs. Indeed, we face the difficulty of considering a discontinuous Hamiltonian operator with respect to the space variable. We state an existence and uniqueness theorem, giving in particular an explicit Hopf-Lax formula for the solution as well as its asymptotic behaviour. This generalizes classical results for continous Hamitonian. We then get interested in a more general class of PDEs: du/dt +sign(u_0(x))H(D- u)=0, with H convex Under some technical but reasonable assumptions, we obtain the same kind of results. As far as we know, they are new for discontinuous Hamiltonians
Mathematical Modeling of Textures: Application to Color Image Decomposition with a Projected Gradient Algorithm
International audienceIn this paper, we are interested in color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling
Structure and texture compression
In this paper, we tackle the problem of image compression. During the last past years, many algorithms have been proposed to take advantage of the geometry of the image. We intend here to propose a new compression algorithm which would take into account the structures in the image, and which would be powerful even when the original image has some textured areas. To this end, we first split our image into two components, a first one containing the structures of the image, and a second one the oscillating patterns. We then perform the compression of each component separately. Our final compressed image is the sum of these two compressed components. This new compression algorithm outperforms the standard biorthogonal wavelets compession
Modeling very oscillating signals. Application to image processing
This article is a companion paper of a previous work \cite{Aujol[3]} where we have developed the numerical analysis of a variational model first introduced by L. Rudin, S. Osher and E. Fatemi \cite{Rudin[1]} and revisited by Y. Meyer \cite{Meyer[1]} for removing the noise and capturing textures in an image. The basic idea in this model is to decompose f into two components (u+v) and then to search for (u,v) as a minimizer of an energy functional. The first component u belongs to BV and contains geometrical informations while the second one v is sought in a space G which contains signals with large oscillations, i.e. noise and textures. In Y. Meyer carried out his study in the whole ^2 and his approach is rather built on harmonic analysis tools. We place ourselves in the case of a bounded set of ^2 which is the proper setting for image processing and our approach is based upon functional analysis arguments. We define in this context the space G, give some of its properties and then study in this continuous setting the energy functional which allows us to recover the components u and v. model signals with strong oscillations. For instance, in an image, this space models noises and textures. case of a bounded open set of ^2 which is the proper setting for image processing. We give a definition of G adapted to our case, and we show that it still has good properties to model signals with strong oscillations. In \cite{Meyer[1]}, the author had also paved the way to a new model to decompose an image into two components: one in BV (the space of bounded variations) which contains the geometrical information, and one in G which consists in the noises ad the textures. An algorithm to perform this decomposition has been proposed in \cite{Meyer[1]}. We show here its relevance in a continuous setting
Some proximal methods for Poisson intensity CBCT and PET
International audienceCone-Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) are two complementary medical imaging modalities providing respectively anatomic and metabolic information on a patient. In the context of public health, one must address the problem of dose reduction of the potentially harmful quantities related to each exam protocol : X-rays for CBCT and radiotracer for PET. Two demonstrators based on a technological breakthrough (acquisition devices work in photon-counting mode) have been developed. It turns out that in this low-dose context, i.e. for low intensity signals acquired by photon counting devices, noise should not be approximated anymore by a Gaussian distribution, but is following a Poisson distribution. We investigate in this paper the two related tomographic reconstruction problems. We formulate separately the CBCT and the PET problems in two general frameworks that encompass the physics of the acquisition devices and the specific discretization of the object to reconstruct. We propose various fast numerical schemes based on proximal methods to compute the solution of each problem. In particular, we show that primal-dual approaches are well suited in the PET case when considering non differentiable regularizations such as Total Variation. Experiments on numerical simulations and real data are in favor of the proposed algorithms when compared with well-established methods
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