  This review discusses methods for learning parameters for image
reconstruction problems using bilevel formulations. Image reconstruction
typically involves optimizing a cost function to recover a vector of unknown
variables that agrees with collected measurements and prior assumptions.
State-of-the-art image reconstruction methods learn these prior assumptions
from training data using various machine learning techniques, such as bilevel
methods. One can view the bilevel problem as formalizing hyperparameter
optimization, as bridging machine learning and cost function based optimization
methods, or as a method to learn variables best suited to a specific task. More
formally, bilevel problems attempt to minimize an upper-level loss function,
where variables in the upper-level loss function are themselves minimizers of a
lower-level cost function.
  This review contains a running example problem of learning tuning parameters
and the coefficients for sparsifying filters used in a regularizer. Such
filters generalize the popular total variation regularization method, and
learned filters are closely related to convolutional neural networks approaches
that are rapidly gaining in popularity. Here, the lower-level problem is to
reconstruct an image using a regularizer with learned sparsifying filters; the
corresponding upper-level optimization problem involves a measure of
reconstructed image quality based on training data. This review discusses
multiple perspectives to motivate the use of bilevel methods and to make them
more easily accessible to different audiences. We then turn to ways to optimize
the bilevel problem, providing pros and cons of the variety of proposed
approaches. Finally we overview bilevel applications in image reconstruction.
