Given a forward mapping Φ : R^N → R^M and a point x* ∈ R^N , the region {x ∈ R^N , ||Φ(x) − Φ(x*)|| ≤ ε}, where ε ≥ 0 is a perturbation amplitude, represents the set of all possible inputs x that could have produced the measurement Φ(x*) within an acceptable error margin. This set is related to uncertainty analysis, a key challenge in inverse problems. In this work, we develop a numerical algorithm called Jackpot (Jacobian Kernel Projection Optimization) which approximates this set with a low-dimensional adversarial manifold. The proposed algorithm leverages automatic differentation, allowing it to handle complex, high dimensional mappings such as those found when dealing with dynamical systems or neural networks. We demonstrate the effectiveness of our algorithm on various challenging large-scale, non-linear problems including parameter identification in dynamical systems and blind image deblurring.

Abstract. Neural networks allow solving many ill-posed inverse problems with unprecedented performance. Physics informed approaches already progressively replace carefully hand-crafted reconstruction algorithms in real applications. However, these networks suffer from a major defect: when trained on a given forward operator, they do not generalize well to a different one. The aim of this paper is twofold. First, we show through various applications that training the network with a family of forward operators allows solving the adaptivity problem without compromising the reconstruction quality significantly. Second, we illustrate that this training procedure allows tackling challenging blind inverse problems. Our experiments include partial Fourier sampling problems arising in magnetic resonance imaging with sensitivity estimation and off-resonance effects, computerized tomography with a tilted geometry, and image deblurring with Fresnel diffraction kernels.

In this article, we propose new algorithms for estimating microvibrations and correcting their effects in satellite imaging with linear pushbroom camera. We first design an accurate model of the acquisition process with a linear pushbroom camera, which incorporates the satellite attitude as parameters. Then, we propose a two-stage reconstruction method based on an identification neural network to identify the attitude, followed by a deep unrolled network to correct the microvibrations. We then evaluate the proposed framework on synthetic and real data, showing promising results for this challenging problem. Our results highlight the critical role of the focal plane’s geometry, to improve the microvibrations identifiability and, therefore, the reconstruction quality.

We propose two efficient optimization approaches to correct jitter effects appearing in a specific type of line scanning microscopy. In this modality, even lines suffer from a non uniform and non integer distortion with respect to odd lines, creating significant visual artifacts. The huge image size make this problem highly challenging. To handle it, we propose two techniques. One is based on dynamic programming and has a complexity linear w.r.t. the number of pixels. The second is based on a convex relaxation and can be particularly efficient for parallel architectures. Both algorithms provide globally optimal solutions. The empirical reconstruction results are of high quality.

The Spreading Projection Algorithm for Rapid K-space sampLING, or SPARKLING, is an optimization-driven method that has been recently introduced for accelerated 2D MRI using compressed sensing. It has then been extended to address 3D imaging using either stacks of 2D sampling patterns or a local 3D strategy that optimizes a single sampling trajectory at a time. 2D SPARKLING actually performs variable density sampling (VDS) along a prescribed target density while maximizing sampling efficiency and meeting the gradient-based hardware constraints. However, 3D SPARKLING has remained limited in terms of acceleration factors along the third dimension if one wants to preserve a peaky point spread function (PSF) and thus good image quality. In this paper, in order to achieve higher acceleration factors in 3D imaging while preserving image quality, we propose a new efficient algorithm that performs optimization on full 3D SPARKLING. The proposed implementation based on fast multipole methods (FMM) allows us to design sampling patterns with up to${10}^{{7}}$k-space samples, thus opening the door to 3D VDS. We compare multi-CPU and GPU implementations and demonstrate that the latter is optimal for 3D imaging in the high-resolution acquisition regime ($600\mu $m isotropic). Finally, we show that this novel optimization for full 3D SPARKLING outperforms stacking strategies or 3D twisted projection imaging through retrospective and prospective studies on NIST phantom and in vivo brain scans at 3 Tesla taking the particular case of${T}_{{2}}$*-w imaging. Overall the proposed method allows for 2.5-3.75x shorter scan times compared to GRAPPA-4 parallel imaging acquisition at 3 Tesla without compromising image quality.

BACKGROUND: The segmentation of a 3D image is a task that can hardly be automatized in certain situations, notably when the contrast is low and/or the distance between elements is small. The existing supervised methods require a high amount of user input, e.g. delineating the domain in all planar sections. RESULTS: We present FitEllipsoid, a supervised segmentation code that allows fitting ellipsoids to 3D images with a minimal amount of interactions: the user clicks on a few points on the boundary of the object on 3 orthogonal views. The quantitative geometric results of the segmentation of ellipsoids can be exported as a csv file or as a binary image. The core of the code is based on an original computational approach to fit ellipsoids to point clouds in an affine invariant manner. The plugin is validated by segmenting a large number of 3D nuclei in tumor spheroids, allowing to analyze the distribution of their shapes. User experiments show that large collections of nuclei can be segmented with a high accuracy much faster than with more traditional 2D slice by slice delineation approaches. CONCLUSIONS: We designed a user-friendly software FitEllipsoid allowing to segment hundreds of ellipsoidal shapes in a supervised manner. It may be used directly to analyze biological samples, or to generate segmentation databases necessary to train learning algorithms. The algorithm is distributed as an open-source plugin to be used within the image analysis software Icy. We also provide a Matlab toolbox available with GitHub.

We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves, for instance. The proposed principle and algorithm are a natural generalization of previous results revolving around the generation of blue-noise point distributions, such as Lloyd's algorithm or more advanced techniques based on power diagrams. We analyze the convergence properties and propose new approaches to accelerate the generation of point distributions. We also design new algorithms to project curves onto spaces of curves with bounded length and curvature or speed and acceleration. We illustrate the algorithm's interest through applications in advanced sampling theory, nonphotorealistic rendering, and path planning.

Magnetic resonance imaging (MRI) is probably one of the most successful application fields of compressed sensing. Despite recent advances, there is still a large discrepancy between theories and most actual implementations. Overall, many important questions related to sampling theory remain open. In this paper, we attack one of them: given a set of sampling constraints (e.g., measuring Fourier coefficients along physically plausible trajectories), how to optimally design a sampling pattern? We first outline three aspects that should be carefully designed by inspecting the literature, namely admissibility, limit of the empirical measure, and coverage speed. To address them jointly, we then propose an original approach which consists of projecting a probability distribution onto a set of admissible measures. The proposed algorithm permits handling arbitrary constraints and automatically generates efficient sampling patterns for MRI as shown on realistic simulations. We achieve a 20-fold undersampling factor at very high 2D resolution (100 $\mu$m isotropic) on physically plausible sampling trajectories with a gain in SNR of 2--3 dB on reconstructed MR images as compared to more standard sampling patterns (e.g., radial, spiral).

Compressed sensing is a theory which guarantees the exact recovery of sparse signals from a small number of linear projections. The sampling schemes suggested by current compressed sensing theories are often of little practical relevance, since they cannot be implemented on real acquisition systems. In this paper, we study a new random sampling approach that consists of projecting the signal over blocks of sensing vectors. A typical example is the case of blocks made of horizontal lines in the 2-D Fourier plane. We provide the theoretical results on the number of blocks that are sufficient for exact sparse signal reconstruction. This number depends on two properties named intra- and inter-support block coherence. We then show that our bounds coincide with the best so far results in a series of examples, including Gaussian measurements or isolated measurements. We also show that the result is sharp when used with specific blocks in time-frequency bases, in the sense that the minimum required amount of blocks to reconstruct sparse signals cannot be improved up to a multiplicative logarithmic factor. The proposed results provide a good insight on the possibilities and limits of block compressed sensing in imaging devices, such as magnetic resonance imaging, radio-interferometry, or ultra-sound imaging.

Collecting the maximal amount of information in a given scanning time is a major concern in magnetic resonance imaging (MRI) to speed up image acquisition. The hardware constraints (gradient magnitude, slew rate, etc.), physical distortions (e.g., off-resonance effects) and sampling theorems (Shannon, compressed sensing) must be taken into account simultaneously, which makes this problem extremely challenging. To date, the main approach to design gradient waveform has consisted of selecting an initial shape (e.g., spiral, radial lines, etc.) and then traversing it as fast as possible using optimal control. In this paper, we propose an alternative solution which first consists of defining a desired parameterization of the trajectory and then of optimizing for minimal deviation of the sampling points within gradient constraints. This method has various advantages. First, it better preserves the density of the input curve which is critical in sampling theory. Second, it allows to smooth high curvature areas making the acquisition time shorter in some cases. Third, it can be used both in the Shannon and CS sampling theories. Last, the optimized trajectory is computed as the solution of an efficient iterative algorithm based on convex programming. For piecewise linear trajectories, as compared to optimal control reparameterization, our approach generates a gain in scanning time of 10% in echo planar imaging while improving image quality in terms of signal-to-noise ratio (SNR) by more than 6 dB. We also investigate original trajectories relying on traveling salesman problem solutions. In this context, the sampling patterns obtained using the proposed projection algorithm are shown to provide significantly better reconstructions (more than 6 dB) while lasting the same scanning time.

Extracting geometrical information from large 2D or 3D biomedical images is important to better understand fundamental phenomena such as morphogenesis. We address the problem of automatically analyzing spatial organization of cells or nuclei in 2D or 3D images of tissues. This problem is challenging due to the usually low quality of microscopy images as well as their typically large sizes. The structure tensor is a simple and robust descriptor that was developed to analyze textures orientation. Contrarily to segmentation methods which rely on an object based modeling of images, the structure tensor considers the sample at a macroscopic scale, like a continuous medium. We show that this tool allows quantifying two important features of nuclei in tissues: their privileged orientation as well as the ratio between the length of their main axes. A quantitative evaluation of the method is provided for synthetic and real 2D and 3D images. As an application, we analyze the nuclei orientation and anisotropy on multicellular tumor spheroids cryosections. This analysis reveals that cells are elongated in a privileged direction that is parallel to the spheroid boundary. A MATLAB toolbox and an Icy plugin are available to use the proposed method.

Restoring images degraded by spatially varying blur is a problem encountered in many disciplines such as astrophysics, computer vision, and biomedical imaging. One of the main challenges in performing this task is to design efficient numerical algorithms to approximate integral operators. We introduce a new method based on a sparse approximation of the blurring operator in the wavelet domain. This method requires $\mathcal{O}(N \epsilon^{-d/M})$ operations to provide $\epsilon$-approximations, where $N$ is the number of pixels of a $d$-dimensional image and $M\geq 1$ is a scalar describing the regularity of the blur kernel. In addition, we propose original methods to define sparsity patterns when only the operator regularity is known. Numerical experiments reveal that our algorithm provides a significant improvement compared to standard methods based on windowed convolutions.

Reducing acquisition time is of fundamental importance in various imaging modalities. Variable density sampling (VDS) provides an appealing framework for addressing this issue. It was justified recently from a theoretical point of view in the compressed sensing (CS) literature. Unfortunately, the sampling schemes suggested by current CS theories may not be relevant since they do not take the acquisition constraints (for example, continuity of the acquisition trajectory in magnetic resonance imaging (MRI)) into account. In this paper, we propose a numerical method to perform variable density sampling with block constraints. Our main contribution is a new way to draw the blocks in order to mimic CS strategies based on isolated measurements. The basic idea is to minimize a tailored dissimilarity measure between a probability distribution defined on the set of isolated measurements and a probability distribution defined on a set of blocks of measurements. This problem turns out to be convex and solvable in high dimension. Our second contribution is to define an efficient minimization algorithm based on Nesterov's accelerated gradient descent in metric spaces. We carefully study the choice of the metrics and of the prox-function. We show that the optimal choice may depend on the type of blocks under consideration. Finally, we show that we can obtain better MRI reconstruction results using our sampling schemes than with standard strategies such as equiangularly distributed radial lines.

Reducing acquisition time is a crucial challenge for many imaging techniques. Compressed sensing (CS) theory offers an appealing framework to address this issue since it provides theoretical guarantees on the reconstruction of sparse signals by projection on a low-dimensional linear subspace. In this paper, we focus on a setting where the imaging device allows us to sense a fixed set of measurements. We first discuss the choice of an optimal sampling subspace allowing perfect reconstruction of sparse signals. Its design relies on the random drawing of independent measurements. We discuss how to select the drawing distribution and show that a mixed strategy involving partial deterministic sampling and independent drawings can help in breaking the so-called coherence barrier. Unfortunately, independent random sampling is irrelevant for many acquisition devices owing to acquisition constraints. To overcome this limitation, the notion of a variable density sampler (VDS) is introduced and defined as a stochastic process with a prescribed limit empirical measure. It encompasses samplers based on independent measurements or continuous curves. The latter are crucial to extend CS results to actual applications. We propose two original approaches to designing a continuous VDS, one based on random walks over the acquisition space and one based on the travelling salesman problem. Following theoretical considerations and retrospective CS simulations in magnetic resonance imaging, we intend to highlight the key properties of a VDS to ensure accurate sparse reconstructions, namely its limit empirical measure and its mixing time.

Additive or multiplicative stationary noise recently became an important issue in applied fields such as microscopy or satellite imaging. Relatively few works address the design of dedicated denoising methods compared to the usual white noise setting. We recently proposed a variational algorithm to tackle this issue. In this paper, we analyze this problem from a statistical point of view and provide deterministic properties of the solutions of the associated variational problems. In the first part of this work, we demonstrate that in many practical problems, the noise can be assimilated to a colored Gaussian noise. We provide a quantitative measure of the distance between a stationary process and the corresponding Gaussian process. In the second part, we focus on the Gaussian setting and analyze denoising methods which consist in minimizing the sum of a total variation term and an $l^2$ data fidelity term. While the constrained formulation of this problem allows us to easily tune the parameters, the Lagrangian formulation can be solved more efficiently since the problem is strongly convex. Our second contribution consists in providing analytical values of the regularization parameter in order to approximately reach a given noise level.

Parallel transmission is a very promising candidate technology to mitigate the inevitable radio-frequency (RF) field inhomogeneity in magnetic resonance imaging at ultra-high field. For the first few years, pulse design utilizing this technique was expressed as a least squares problem with crude power regularizations aimed at controlling the specific absorption rate (SAR), hence the patient safety. This approach being suboptimal for many applications sensitive mostly to the magnitude of the spin excitation, and not its phase, the magnitude least squares (MLS) problem then was first formulated in 2007. Despite its importance and the availability of other powerful numerical optimization methods, the MLS problem yet has been faced almost exclusively by the pulse designer with the so-called variable exchange method. In this paper, we investigate various two-stage strategies consisting of different initializations and nonlinear programming approaches, and incorporate directly the strict SAR and hardware constraints. Several schemes such as sequential quadratic programming, interior point methods, semidefinite programming and magnitude squared least squares relaxations are studied both in the small and large tip angle regimes with RF and static field maps obtained in vivo on a human brain at 7T. Convergence and robustness of the different approaches are analyzed, and recommendations to tackle this specific problem are finally given. Small tip angle and inversion pulses are returned in a few seconds and in under a minute respectively while respecting the constraints, allowing the use of the proposed approach in routine.

A framework and an algorithm are presented in order to remove stationary noise from images. This algorithm is called variational stationary noise remover. It can be interpreted both as a restoration method in a Bayesian framework and as a cartoon+texture decomposition method. In numerous denoising applications, the white noise assumption fails. For example, structured patterns such as stripes appear in the images. The model described here addresses these cases. Applications are presented with images acquired using different modalities: scanning electron microscope, FIB-nanotomography, and an emerging fluorescence microscopy technique called selective plane illumination microscopy.

We analyze the illumination invariance of the level lines of an image. We show that if the scene surface has Lambertian reflectance and the light is directed, then a necessary and sufficient condition for the level lines to be illumination invariant is that the three-dimensional scene be developable and that its albedo satisfy some geometrical constraints. We then show that the level lines are “almost” invariant for piecewise developable surfaces. Such surfaces fit most of the urban structures. This allows us to devise a fast and simple algorithm that detects changes between pairs of remotely sensed images of urban areas, independently of the lighting conditions. We show the effectiveness of the algorithm both on synthetic OpenGL scenes and real QuickBird images. The synthetic results illustrate the theory developed in this paper. The two real QuickBird images show that the proposed change detection algorithm is discriminant. For easy scenes it achieves a rate of 85% detected changes for 10% false positives, while it reaches a rate of 75% detected changes for 25% false positives on demanding scenes.

Many problems in remote sensing can be modeled as the minimization of the sum of a data term and a prior term. We propose to use a new complex wavelet based prior and an efficient scheme to solve these problems. We show some results on a problem of image reconstruction with noise, irregular sampling and blur. We also show a comparison between two widely used priors in image processing: sparsity and regularity priors.

Many compression algorithms consist of quantizing the coefficients of an image in a linear basis. This introduces compression noise that often look like ringing. Recently some authors proposed variational methods to reduce those artifacts. They consists of minimizing a regularizing functional in the set of antecedents of the compressed image. In this paper we propose a fast algorithm to solve that problem. Our experiments lead us to the conclusion that these algorithms effectively reduce oscillations but also reduce contrasts locally. To handle that problem, we propose a fast contrast enhancement procedure. Experiments on a large dataset suggest that this procedure effectively improves the image quality at low bitrates.

We propose an algorithm that equalizes the contrast of grayscale image pairs to simplify the task of change detection. To ensure robustness of the detection under different illumination conditions, some authors recently proposed algorithms that compare the level lines of the images. We show - using ideas from the ldquoshape from shadingrdquo community - that under directed light, a necessary condition for the level lines to be illumination invariant is that the underlying surfaces be developable. The surfaces of cities can be modeled as piece-wise smooth developable surfaces, and it is therefore sensible to make use of the level lines for change detection. Our algorithm is robust and efficient both on synthetic OpenGL scenes and natural Quickbird images.