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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:9

(Theoretical project) Continuous-Domain Compressed Sensing with Splines

Master Diploma Project: Available

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.


(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.


(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.


(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.


(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 Greenís 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.


(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.


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...).


Dictionary Learning for Limited Angle Computed Tomography

Master Semester Project: Reserved

Computed tomography (CT) is a method for creating a 3D image of a sample. It proceeds by computationally combining many projection images (e.g., the X-ray image a doctor might take of a broken bone), taken from different angles. In some applications, a full 180į view of the object is not feasible. In the last decade, iterative procedures which exploit sparsity have been used for limited angle CT reconstructions. In this project, the student will apply a dictionary learning algorithm for limited angle CT to exploit redundancies in the measurements. Good Matlab skills are required for this project. For further details, donít hesitate to send a mail.


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.


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