Compressed sensing and related areas: bridging theory with practice

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Starting in the mid 1980s, the applied and computational harmonic analysis community constructed building blocks (e.g., wavelets and curvelets) that allow for efficient decomposition of many signals of interest (e.g., natural images and audio). The activity in this area led to fundamental changes in industry that have contributed to the digital revolution. Wavelet techniques have penetrated many related areas, including image compression, pattern recognition, radar, medical imaging, and geophysics. One of the main motivations when constructing these building blocks was to obtain sparse approximations for signals of practical interest. This research opened the door to a new technology: compressed sensing (CS).******The central observation of CS can be stated simply: signals that have (approximately) sparse representations can be recovered from few measurements via tractable algorithms. The challenge lies in the non-linear and combinatorial nature of sparse signals. This idea was soon generalized to allow for other non-linear structures including low-rank matrices and manifolds. Structured signals are ubiquitous, and the ideas of CS promised to revolutionize several different fields and applications. Indeed, the last decade has seen the construction of a comprehensive foundation for the theory of CS. Yet, the practice of CS has not penetrated industry to the same degree that wavelets have.******The main theme of my research programme is bridging the CS theory with real applications that have various rigid constraints. These often result in difficult, practically relevant mathematical problems. Below are some specific directions I intend to (continue to) explore:******(1) Analogue-to-digital conversion: For CS to be established as a viable signal acquisition scheme, it needs to be able to produce high accuracy estimates of the acquired signal. In a digital world, this must be achieved from quantized measurements. While there is some work in this area, there are several fundamental questions left to answer. We will build on our previous work to answer these and contribute to have compressed sensing fulfill one of its original goals: merging sensing with compression. ******(2) Model and data based recovery: CS typically does not assume any prior knowledge about the target signal other than sparsity. However, in many applications there is prior information that can incorporated to the measurement and/or the reconstruction schemes. For example, in a video frame sequence, consecutive frames are usually similar. We wish to explore various approaches for incorporating prior information into the reconstruction method, while keeping sensing non-adaptive. ******(3) Model and data based acquisition: We will focus on the joint design of model based acquisition and tailored recovery methods for CS. We will also address various "model mismatch" issues both by explicit modeling and by implicit modeling via deep learning.