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Publication# From Classical to Unsupervised-Deep-Learning Methods for Solving Inverse Problems in Imaging.

Abstract

In this thesis, we propose new algorithms to solve inverse problems in the context of biomedical images. Due to ill-posedness, solving these problems require some prior knowledge of the statistics of the underlying images. The traditional algorithms, in the field, assume prior knowledge related to smoothness or sparsity of these images. Recently, they have been outperformed by the second generation algorithms which harness the power of neural networks to learn required statistics from training data. Even more recently, last generation deep-learning-based methods have emerged which require neither training nor training data. This thesis devises algorithms which progress through these generations. It extends these generations to novel formulations and applications while bringing more robustness. In parallel, it also progresses in terms of complexity, from proposing algorithms for problems with 1D data and an exact known forward model to the ones with 4D data and an unknown parametric forward model. We introduce five main contributions. The last three of them propose deep-learning-based latest-generation algorithms that require no prior training. 1) We develop algorithms to solve the continuous-domain formulation of inverse problems with both classical Tikhonov and total-variation regularizations. We formalize the problems, characterize the solution set, and devise numerical approaches to find the solutions. 2) We propose an algorithm that improves upon end-to-end neural-network-based second generation algorithms. In our method, a neural network is first trained as a projector on a training set, and is then plugged in as a projector inside the projected gradient descent (PGD). Since the problem is nonconvex, we relax the PGD to ensure convergence to a local minimum under some constraints. This method outperforms all the previous generation algorithms for Computed Tomography (CT). 3) We develop a novel time-dependent deep-image-prior algorithm for modalities that involve a temporal sequence of images. We parameterize them as the output of an untrained neural network fed with a sequence of latent variables. To impose temporal directionality, the latent variables are assumed to lie on a 1D manifold. The network is then tuned to minimize the data fidelity. We obtain state-of-the-art results in dynamic magnetic resonance imaging (MRI) and even recover intra-frame images. 4) We propose a novel reconstruction paradigm for cryo-electron-microscopy (CryoEM) called CryoGAN. Motivated by generative adversarial networks (GANs), we reconstruct a biomolecule's 3D structure such that its CryoEM measurements resemble the acquired data in a distributional sense. The algorithm is pose-or-likelihood-estimation-free, needs no ab initio, and is proven to have a theoretical guarantee of recovery of the true structure. 5) We extend CryoGAN to reconstruct continuously varying conformations of a structure from heterogeneous data. We parameterize the conformations as the output of a neural network fed with latent variables on a low-dimensional manifold. The method is shown to recover continuous protein conformations and their energy landscape.

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Related concepts (41)

Inverse problem

An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source r

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Neural network

A neural network can refer to a neural circuit of biological neurons (sometimes also called a biological neural network), a network of artificial neurons or nodes in the case of an artificial neur

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Learning to embed data into a space where similar points are together and dissimilar points are far apart is a challenging machine learning problem. In this dissertation we study two learning scenarios that arise in the context of learning embeddings and one scenario in efficiently estimating an empirical expectation. We present novel algorithmic solutions and demonstrate their applications on a wide range of data-sets.
The first scenario deals with learning from small data with large number of classes. This setting is common in computer vision problems such as person re-identification and face verification. To address this problem we present a new algorithm called Weighted Approximate Rank Component Analysis (WARCA), which is scalable, robust, non-linear and is independent of the number of classes. We empirically demonstrate the performance of our algorithm on 9 standard person re-identification data-sets where we obtain state of the art performance in terms of accuracy as well as computational speed.
The second scenario we consider is learning embeddings from sequences. When it comes to learning from sequences, recurrent neural networks have proved to be an effective algorithm. However there are many problems with existing recurrent neural networks which makes them data hungry (high sample complexity) and difficult to train. We present a new recurrent neural network called Kronecker Recurrent Units (KRU), which addresses the issues of existing recurrent neural networks through Kronecker matrices. We show its performance on 7 applications, ranging from problems in computer vision, language modeling, music modeling and speech recognition.
Most of the machine learning algorithms are formulated as minimizing an empirical expectation over a finite collection of samples. In this thesis we also investigate the problem of efficiently estimating a weighted average over large data-sets. We present a new data-structure called Importance Sampling Tree (IST), which permits fast estimation of weighted average without looking at all the samples. We show successfully the evaluation of our data-structure in the training of neural networks in order to efficiently find informative samples.

Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.

2020Learning to embed data into a space where similar points are together and dissimilar points are far apart is a challenging machine learning problem. In this dissertation we study two learning scenarios that arise in the context of learning embeddings and one scenario in efficiently estimating an empirical expectation. We present novel algorithmic solutions and demonstrate their applications on a wide range of data-sets. The first scenario deals with learning from small data with large number of classes. This setting is common in computer vision problems such as person re-identification and face verification. To address this problem we present a new algorithm called Weighted Approximate Rank Component Analysis (WARCA), which is scalable, robust, non-linear and is independent of the number of classes. We empirically demonstrate the performance of our algorithm on 9 standard person re-identification data-sets where we obtain state of the art performance in terms of accuracy as well as computational speed. The second scenario we consider is learning embeddings from sequences. When it comes to learning from sequences, recurrent neural networks have proved to be an effective algorithm. However there are many problems with existing recurrent neural networks which makes them data hungry (high sample complexity) and difficult to train. We present a new recurrent neural network called Kronecker Recurrent Units (KRU), which addresses the issues of existing recurrent neural networks through Kronecker matrices. We show its performance on 7 applications, ranging from problems in computer vision, language modeling, music modeling and speech recognition. Most of the machine learning algorithms are formulated as minimizing an empirical expectation over a finite collection of samples. In this thesis we also investigate the problem of efficiently estimating a weighted average over large data-sets. We present a new data-structure called Importance Sampling Tree (IST), which permits fast estimation of weighted average without looking at all the samples. We show successfully the evaluation of our data-structure in the training of neural networks in order to efficiently find informative samples.