Contact: Adrian Jarret (LCAV)
Synopsis: Comparison of analysis and synthesis formulations, solution of analysis formulation using cycle spinning with different wavelet basis, solution of synthesis formulation with adjoint of penalty operator and proximal gradient method, extension to weighted penalty.
Level: Master (5-6 months)
Sections: IC
Description: Wavelet bases are known to produce sparse representations of images in a very accurate manner, due to their ability to encode components of different scales within the same basis. For more generalizability and applicability, it is common to look for representations of images over many wavelet bases at the same time (i.e. a redundant dictionary of wavelets). In this project, we want to reconstruct complex images of the sky using radio astronomy datasets, which involve different types of celestial sources, using wavelet decomposition to accurately represent objects with different scales.
To do so, we solve the imaging inverse problem (reconstruct the images from radio-interferometric complex-valued measurements) by means of the LASSO minimization problem, a sparse optimization technique (see [1] for instance). The problem can be expressed using either an analysis or a synthesis formulation, i.e. optimizing either in the image domain or in the wavelet coefficients domain respectively. The synthesis method is easier to solve in practice but has been reported to produce poorer representations ([3, 4]). On the other hand, the analysis counterpart requires slower and more complex algorithms.
In this project, we want to propose a new method to tackle the analysis formulation of the optimization problem, with a technology inspired from the cycle spinning technique for shift-invariant wavelet-based representation of images (see [2]). The student will have to implement this approach in Python and to compare it with the solution obtained with the synthesis formulation. We will work on realistic simulated data using the radio astronomy Python package RASCIL. The student will also rely on the Pycsou package, an optimization package developed by members of EPFL (Imaging Center and LCAV) providing efficient implementations for numerous classical elements of the convex optimization pipeline (and still growing).
Depending on the progress of the student, we will consider a more complex optimization problem introducing penalization parameters adapted to the wavelet bases in the reconstruction (a.k.a. weighted L1 regularization). Many extensions in this research direction will be possible (learning the penalty parameters).
Deliverables:
- Python code that implements the reconstruction pipeline, from running the simulations to displaying the solutions
- Quantitative and documented study of the regularization parameters of the model
- Project report that presents the theoretical context and summarizes the experiments and the findings
Prerequisites:
- Good knowledge in linear algebra and functional analysis
- Experience in numerical computing and object-oriented programming with Python (Numpy, Matplotlib)
- Basic kowledge in (convex) optimization
Type of Work :
- ~80% implementation: Python code using Pycsou, RASCIL
- ~20% theory: inverse problems, optimization, convergence of algorithms
(This repartition might be adapted according to the interests of the student.)
References: