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

Ancient Documents Denoising and Decomposition Using Aujol and Chambolle Algorithm

With the improvement of printing technology since the 15th century, there is a huge amount of printed documents published and distributed. These documents are degraded by the time and require to be preprocessed before being submitted to image indexing strategy, in order to enhance the quality of images. This paper proposes a new pre-processing that permits to denoise these documents, by using a Aujol and Chambolle algorithm. Aujol and Chambolle algorithm allows to extract meaningful components from image. In this case, we can extract shapes, textures and noise. Some examples of specific processings applied on each layer are illustrated in this paper

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

Minimizing Quotient Regularization Model

Quotient regularization models (QRMs) are a class of powerful regularization techniques that have gained considerable attention in recent years, due to their ability to handle complex and highly nonlinear data sets. However, the nonconvex nature of QRM poses a significant challenge in finding its optimal solution. We are interested in scenarios where both the numerator and the denominator of QRM are absolutely one-homogeneous functions, which is widely applicable in the fields of signal processing and image processing. In this paper, we utilize a gradient flow to minimize such QRM in combination with a quadratic data fidelity term. Our scheme involves solving a convex problem iteratively.The convergence analysis is conducted on a modified scheme in a continuous formulation, showing the convergence to a stationary point. Numerical experiments demonstrate the effectiveness of the proposed algorithm in terms of accuracy, outperforming the state-of-the-art QRM solvers.Comment: 20 page

Decreasing time consumption of microscopy image segmentation through parallel processing on the GPU

The computational performance of graphical processing units (GPUs) has improved significantly. Achieving speedup factors of more than 50x compared to single-threaded CPU execution are not uncommon due to parallel processing. This makes their use for high throughput microscopy image analysis very appealing. Unfortunately, GPU programming is not straightforward and requires a lot of programming skills and effort. Additionally, the attainable speedup factor is hard to predict, since it depends on the type of algorithm, input data and the way in which the algorithm is implemented. In this paper, we identify the characteristic algorithm and data-dependent properties that significantly relate to the achievable GPU speedup. We find that the overall GPU speedup depends on three major factors: (1) the coarse-grained parallelism of the algorithm, (2) the size of the data and (3) the computation/memory transfer ratio. This is illustrated on two types of well-known segmentation methods that are extensively used in microscopy image analysis: SLIC superpixels and high-level geometric active contours. In particular, we find that our used geometric active contour segmentation algorithm is very suitable for parallel processing, resulting in acceleration factors of 50x for 0.1 megapixel images and 100x for 10 megapixel images

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

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