  The random walker (RW) algorithm is used for both image segmentation and
registration, and possesses several useful properties that make it popular in
medical imaging, such as being globally optimizable, allowing user interaction,
and providing uncertainty information. The RW algorithm defines a weighted
graph over an image and uses the graph's Laplacian matrix to regularize its
solutions. This regularization reduces to solving a large system of equations,
which may be excessively time consuming in some applications, such as when
interacting with a human user. Techniques have been developed that precompute
eigenvectors of a Laplacian offline, after image acquisition but before any
analysis, in order speed up the RW algorithm online, when segmentation or
registration is being performed. However, precomputation requires certain
algorithm parameters be fixed offline, limiting their flexibility. In this
paper, we develop techniques to update the precomputed data online when RW
parameters are altered. Specifically, we dynamically determine the number of
eigenvectors needed for a desired accuracy based on user input, and derive
update equations for the eigenvectors when the edge weights or topology of the
image graph are changed. We present results demonstrating that our techniques
make RW with precomputation much more robust to offline settings while only
sacrificing minimal accuracy.
