  This tutorial covers biomedical image reconstruction, from the foundational
concepts of system modeling and direct reconstruction to modern sparsity and
learning-based approaches.
  Imaging is a critical tool in biological research and medicine, and most
imaging systems necessarily use an image-reconstruction algorithm to create an
image; the design of these algorithms has been a topic of research since at
least the 1960's. In the last few years, machine learning-based approaches have
shown impressive performance on image reconstruction problems, triggering a
wave of enthusiasm and creativity around the paradigm of learning. Our goal is
to unify this body of research, identifying common principles and reusable
building blocks across decades and among diverse imaging modalities.
  We first describe system modeling, emphasizing how a few building blocks can
be used to describe a broad range of imaging modalities. We then discuss
reconstruction algorithms, grouping them into three broad generations. The
first are the classical direct methods, including Tikhonov regularization; the
second are the variational methods based on sparsity and the theory of
compressive sensing; and the third are the learning-based (also called
data-driven) methods, especially those using deep convolutional neural
networks. There are strong links between these generations: classical
(first-generation) methods appear as modules inside the latter two, and the
former two are used to inspire new designs for learning-based
(third-generation) methods. As a result, a solid understanding of all of three
generations is necessary for the design of state-of-the-art algorithms.
