Michaël Unser and Pol del Aguila Pla

Biomedical imaging plays a key role in medicine and biology. Its range of applications and its impact in research and medical practice have increased steadily during the past 4 decades. Part of the astonishing improvements in image quality and resolution is due to the use of increasingly sophisticated signal-processing techniques. This, in itself, would justify the tutorial. Nonetheless, the field is now transitioning towards the deep-learning era, where disruptive improvements and lack of theoretical background go hand-in-hand. To harness the power of these new techniques without suffering from their pitfalls, a structured understanding of the field is fundamental.

We start the tutorial by presenting the building blocks of an image-reconstruction problem, from the underlying image that lives in function spaces to its observed discrete measurements. Most importantly, we detail the small collection of forward and sampling operators that allow one to model most biomedical imaging problems, including magnetic resonance imaging, bright-field microscopy, structured-illumination microscopy, x-ray computed tomography, and optical diffraction tomography. This leads up to our exposition of 1st-generation methods (e.g., filtered back-projection, Tikhonov regularization), the regimes in which they are most attractive, and how to implement them efficiently.

We then transition to 2nd-generation methods (non-quadratic regularization, sparsity, and compressive sensing) and show how advanced signal processing allows image reconstruction with smaller acquisition times, less invasive procedures, and lower radioactive and irradiation dosage. We expose the foundations of these methods (results in compressed-sensing recovery, representer theorems, infinite-divisible distributions) and the most useful algorithms in imaging (proximal operators, projected gradient descent, alternate-direction method of multipliers), again exemplifying their efficient implementation.

Finally, we present the state of the art in 3rd-generation methods (deep-learning reconstruction of images), categorizing them using the building-block terminology introduced throughout the tutorial. In this manner, we emphasize the links to 1st- and 2nd-generation methods in order to provide intuition and guidelines to devise and understand novel 3rd-generation methods. Furthermore, we state the benefits of each proposal and give cautionary examples of the dangers of overreliance on training data.

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