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STUDENT PROJECTS

Proposals

On-going Projects

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Conditions : The Bachelor Semester Projects and Master Semester Projects are only reserved for regular EPFL students or for students of enrolled in am official mobility program.

Project Proposals

All Project

Number of projects:16

Benchmarking of Proximal Algorithms for Solving Regularized Inverse Problems

Bachelor Semester Project or Master Semester Project: Available

Inverse problems with l1 regularization are popular method for signal reconstruction. This is due to the fact that they promote sparse solutions, i.e., with few nonzero coefficients, and the observation that many real-world signals are sparse in a certain basis. However, due to the non-differentiability of the l1 norm, such problems do not admit a close-form solution; they are thus typically solved using iterative algorithms based on the proximal operator of the l1 norm. In this project, we propose to benchmark two of these proximal algorithms, the standard and very popular alternating direction method of multipliers (ADMM) [1], and a primal-dual splitting algorithm introduced by Condat [2]. The goal will be to compare the performance of these algorithms in various settings. The student should have a strong interest in optimization.

References:

[1] Boyd, Stephen, et al. "Distributed optimization and statistical learning via the alternating direction method of multipliers." Foundations and Trends® in Machine learning 3.1 (2011): 1-122.

[2] Condat, Laurent. "A primal–dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms." Journal of Optimization Theory and Applications 158.2 (2013): 460-479.

Supervision:

Benchmarking of the Simplex Algorithm for Solving Regularized Inverse Problems

Bachelor Semester Project: Available

The simplex algorithm is the oldest but still one of the most popular optimization algorithms for solving linear programs. In short, it iterates through vertices of the feasible region of the linear program until it reaches a minimum of the cost function. Recently, it was been shown that it can be used to reach an extreme point of the solution set of inverse problems with l1 regularization [1]. This is relevant because these extreme point solutions are guaranteed to be sparse, i.e., they can be expressed with few nonzero coefficients [2]. Different methods can be applied to solve inverse problems with l1 regularization using the simplex, with different tradeoffs in terms of problem dimension, speed and perhaps numerical stability. The goal of this project is to benchmark these different methods in order to quantify the aforementioned tradeoffs. The student should have a strong interest in optimization.

References:

[1] Gupta, Harshit, Julien Fageot, and Michael Unser. "Continuous-Domain Solutions of Linear Inverse Problems with Tikhonov versus Generalized TV Regularization." IEEE Transactions on Signal Processing 66.17 (2018): 4670-4684.

[2] Unser, Michael, Julien Fageot, and Harshit Gupta. “Representer Theorems for Sparsity-Promoting l1 Regularization”, IEEE Transactions on Information Theory 62.9 (2016): 5167-5180.

Supervision:

Image analysis of organoids with deep learning

Master Semester Project: Reserved

Three-dimensional stem-cell cultures called organoids have emerged as an ideal ex vivo model in regenera- tive medicine, disease modeling, or studies of biological tissue. Large amount of phase images of organoids were acquired with a digital holography microscopy. This modality requires several image processing tasks to extract meaningful information. For instance, the pipeline will detect the organoids on the image, as well as segment them.

In this project, the student will develop a deep learning framework to analyze the data in a high-throughput manner. The student will need to master deep learning and pytorch. For further details, please contact us.

Supervision:

High speed imaging with any camera

: Available

High speed imaging can be achieved with any camera, as has recently been demonstrated with the Virtual Frame Technique. Using this method, any monotonic phenomenon that is imaged instantaneously with perfect contrast (such that the image is binary) can be recorded at high rates by increasing the exposure time of the camera. Thus, complex temporal dynamics can be recorded at high rates and high resolution, averting the traditional trade-off between size of the region of interest and imaging rate. In this project we explore the limits of this method, using a dual-approach of experimental demonstration and theoretical analysis. Project in collaboration with Engineering Mechanics of Soft Interfaces (EMSI) laboratory (https://www.epfl.ch/labs/emsi/).

Supervision:

Image reconstruction for optical diffraction tomography

Master Semester Project: Reserved

Optical diffraction tomography (ODT) allows one to quantitatively measure the distribution of the refractive index (RI) of the sample [1]. It proceeds by measuring the complex fields that are produced when the sample is illuminated with plane waves from different angles. This allows for the deployment of numerical methods to recover the RI. In this work, we are interested in the limited-angle regime where only some angles are available, which makes the problem ill-posed. To overcome it, it is common to add prior knowledge (i.e., regularization) to the sample during the reconstruction such as non-negativity constraint. The project aims at implementing and evaluating a new regularization for ODT. The code will be done on Matlab within the GlobalBioIm library [2]. Good skill in Matlab is required. Please contact us for further details.

Reference
[1] Soubies, E., Pham, T. A., & Unser, M. (2017). Efficient inversion of multiple-scattering model for optical diffraction tomography. Optics express, 25(18), 21786-21800.
[2] Soubies, E., Soulez, F., Mccann, M. T., Pham, T. A., Donati, L., Debarre, T., ... & Unser, M. (2019). Pocket guide to solve inverse problems with globalbioim. Inverse Problems, 35(10), 104006.

Supervision:

Learning Robust Neural Networks via Controlling their Lipschitz Regularity

Master Semester Project: Reserved

For adversarial robustness, it has been shown that augmenting adversarial perturbations during training, or adversarial training, makes the model more robust. However, it is computationally challenging to employ it on large-scale datasets. Adversarially trained models overfit to the specific attack type used for training, and the performance on unperturbed images drops. In addition, methods which improve robustness to non-adversarial corruptions are relatively less studied. Recently, we developed a framework for learning activations of deep neural networks with the motivation of controlling the global Lipschitz constant of the input-output relation. The goal of this project is to investigate the effect of our framework in the global robustness of the network within various setups. The student should have solid programming skills, in particular being familiar with PyTorch and a general understanding of the main concepts of deep learning.

Supervision:

Slice-based Dictionary Learning for Computed Tomography

Master Semester Project: Reserved

In computed tomography (CT), the goal is to reconstruct a 3D object from a set of its 2D projections. Typically, this reconstruction task is formulated as an optimization problem where one also exploits certain properties of the signal of interest such as sparsity. Dictionary learning is a technique which uses training data to find a basis in which our signal can be represented in a sparse manner. Standard dictionary learning approaches are patch-based and thus computationally inefficient for 3D data. The goal of this project is to explore an alternate framework where the dictionary is based on 2D slices extracted from the volume. The project will be implemented in MATLAB.

Supervision:

(Theoretical project) Continuous-Domain Compressed Sensing with Splines

Master Diploma Project: Reserved

The project is to study theoretically and numerically a sparse inverse problem over a continuous domain (i.e., without performing any discretization). The optimization problem considered is an extension of the well-known LASSO over a non-reflexive Banach space. The minimization problem is designed using the prior knowledge according to which one aims to recover sparse signals in a dictionary defined by a differential operator L (for example the derivative operator), consisting of periodic L-splines. This is possible through the use, in the objective function, of the total-variation norm defined over the space of Radon measures and extending to measures the l1-norm, known to promote sparsity. The problem is challenging and has recently called for a lot of attention. In particular, it was proven in [1] that L-splines, defined as weighted sums of a shifted Green function of L, are always admissible solutions. The goal of the project is to characterize precisely the solution set in the periodic setup and to derive simple conditions that can be checked in applications and lead to uniqueness. [1] Unser, M., Fageot, J., and Ward, J.P., 2017. Splines are universal solutions of linear inverse problems with generalized TV regularization. SIAM Review, 59(4), pp. 769-793.

Supervision:

(Theoretical project) Learning Piecewise-Constant Functions to Classify and Cluster

Master Diploma Project: Available

Classification and clustering are some of the most important objectives in supervised and unsupervised learning, respectively. Interestingly, in both scenarios, the learning scheme eventually produces a piecewise-constant function. This remarkable property allows one to analyze them jointly. The goal of this project is to develop a variational framework to estimate piecewise-constant functions and to derive an efficient learning algorithm, built as a module. One can then also use this module in deep neural networks and compare the performance with classical setups for various applications of classification and clustering.

Supervision:

(Theoretical project) Theory of Multi-Splines

Master Diploma Project: Available

Splines are piecewise-polynomial functions that satisfy a certain degree of smoothness at their junctions. Along with numerous theoretical properties, splines are also very relevant in practice as they provide a numerically feasible parametrization for continuous functions with a discrete sequence of coefficients. While the theory of splines has been extensively developed in the past 70 years, most published works focus on the case of a single spline space. However, in many applications (\textit{e.g.,} snakes, derivative sampling), it is desirable to consider a sum of spline spaces to increase the flexibility of the model. The goal of this project is to develop the theory of multi-splines, where one assumes that the target (continuously defined) function lies in a sum of multiple spline spaces. The design of optimal generators for these spaces, the development of efficient numerical algorithms, and the study of the fundamental properties (\textit{e.g.,} Riesz basis, approximation power) of these spaces are possible directions in this project.

Supervision:

(Theoretical project) Quantification of the Approximation Error of Sparse Stochastic-Process Generators

Master Diploma Project: Available

The theory of sparse stochastic processes has been developed to model the sparsity of continuous-domain signals. It is well-known that most natural signals and images are sparse in some transform domain. Hence, sparse stochastic processes are ideal candidates to model these signals. Recently, we have developed an algorithm to generate the trajectories of sparse processes. It provides a way to benchmark novel algorithms that are designed within this stochastic framework. While our method for the generation of continuous-domain sparse processes is based on a theoretical foundation and is numerically very efficient, very little is known about its speed of convergence. The goal of this project is to analyze our scheme and to search for upper and lower bounds for its convergence rate and, finally, to provide a theoretical comparison with the classical approaches that generate stochastic processes.

Supervision:

(Theoretical project) Sparse-Dictionary Learning in the Continuum

Master Diploma Project: Available

In recent years, a popular trend in inverse problems is sparse-dictionary learning, or sparse coding. The idea is to recover a signal that is sparse in a certain dictionary basis, which is unknown \textit{a priori}. The dictionary is inferred from some training data. This is conceptually a very natural approach since a tailored dictionary can only be better suited than a predefined one - putting computational difficulties aside. This project will consist in the implementation of a sparse-dictionary-learning algorithm for continuous-domain signals, where the learned dictionary atoms are Green’s functions of differential operators. The dictionary-learning problem is formulated as an optimization problem in a function space, which aims at the selection of atoms that can best represent the training data in a sparse way. The project will be implemented in Matlab.

Supervision:

(Theoretical project) Supervised Learning with a Family of Continuously Indexed Kernels

Master Diploma Project: Available

Kernel methods played a vital role in classical machine learning. Although blessed with strong mathematical foundations, these schemes have been outperformed by deep neural networks in recent years. This is mainly due to the low capacity of kernel-based models, which reduces the generalization power of the learning scheme. Recently, we have developed a variational framework to achieve learning with multiple kernels and overcome this issue by an increase in the generalization power of the learning method. The key element of our theory is a sparsity-promoting regularization that ensures the well-posedness of the problem, both theoretically and practically. The goal of this project is to extend this framework to a more general setup, where one considers a family of kernel functions with a continuous parametrization and develop a novel learning scheme that represents the target function by selecting few (sparse) kernels.

Supervision:

An off-the-grid algorithm in ImageJ for 3D single-molecule localization microscopy

Master Semester Project: Reserved

A new class of algorithms has recently emerged in the literature for the recovery of point source signals from altered and noisy measurements. These methods are able to perform the reconstruction, without requiring any discretization of the domain, by solving an infinite dimensional optimization problem. They interleave convex optimization updates (corresponding to adding a new point source) with non-convex optimization steps (corresponding to changing the intensities and positions of the point sources). Single-molecule localization microscopy is an imaging technique in fluorescence microscopy that is able to bypass the diffraction limit and reach nanoscale resolution for the imaging of sub-cellular structures in cells (e.g., microtubules). High performance numerical solvers are needed to locate precisely the positions of the fluorescent molecules. The previously mentioned class of algorithms are currently the one that obtain the state-of-the-art results in this application. The goal of this project is to implement one of these methods, called the Sliding Frank-Wolf algorithm, in Java as an ImageJ/Fiji plugin so that it can be usable by biologists. It will include a user interface and permit automatic processing of large datasets and output super-resolved images. This project is a continuation of the semester project of Amandine Evard. The code and material are available. The idea is to add new features to the algorithm and the plugin (spline interpolation of the point-spread function, tracking of molecules in time, new modalities...).

Supervision:

Homotopy Methods for Regularized Inverse Problems

Master Semester Project: Available

In an inverse problem, the objective is to reconstruct a signal from a set of measurements. This is typically achieved by solving an optimization problem with a data-fidelity term that enforces the consistency between the reconstructed signal with the measured data. When the problem is ill-posed, a common technique is to add a regularization term that is based on our prior knowledge on the form of the signal. A regularization parameter then balances the weight between the data fidelity and the regularization terms. The choice of this parameter is crucial, and it is typically hard to tune. Hence, homotopy methods aim to solve the optimization problem for all possible values of the regularization parameter, so that the user can choose a suitable one. The goal of this project is to investigate such methods for some specific discrete inverse problems, starting with a literature review, and to implement an algorithm in practice. As the project is somewhat exploratory, the student should be able to take initiative and to work autonomously. He or she should also have strong mathematical interest, particularly in the field of optimization.

Supervision:

Image-based quantification of cell blebbing

Master Semester Project: Available

Blebbing is a very dynamic phenomenon that plays an important role during apoptosis, cell migration, or cell division. Using time-lapsed microscopy techniques, phase contrast and fluorescence, biologists can observe blebs which are spherical protrusions which appear and disappear on the membrane of the cell. The goal of the project is to design and to implement image-analysis algorithms based on active contour and curve optimization take into account the blebbing. It requires a automatic segmentation of the cell over the multichannel sequence of images and a local extraction of the bulges to quantify blebbing. The project will be implemented in Java as an ImageJ plugin with an user interface allowing a manual edition of the outlines of the blebs.

Supervision:

2020 EPFL • webmaster.big@epfl.ch23.05.2020