  Bio-image analysis is challenging due to inhomogeneous intensity
distributions and high levels of noise in the images. Bayesian inference
provides a principled way for regularizing the problem using prior knowledge. A
fundamental choice is how one measures "distances" between shapes in an image.
It has been shown that the straightforward geometric L2 distance is degenerate
and leads to pathological situations. This is avoided when using Sobolev
gradients, rendering the segmentation problem less ill-posed. The high
computational cost and implementation overhead of Sobolev gradients, however,
have hampered practical applications. We show how particle methods as applied
to image segmentation allow for a simple and computationally efficient
implementation of Sobolev gradients. We show that the evaluation of Sobolev
gradients amounts to particle-particle interactions along the contour in an
image. We extend an existing particle-based segmentation algorithm to using
Sobolev gradients. Using synthetic and real-world images, we benchmark the
results for both 2D and 3D images using piecewise smooth and piecewise constant
region models. The present particle approximation of Sobolev gradients is 2.8
to 10 times faster than the previous reference implementation, but retains the
known favorable properties of Sobolev gradients. This speedup is achieved by
using local particle-particle interactions instead of solving a global Poisson
equation at each iteration. The computational time per iteration is higher for
Sobolev gradients than for L2 gradients. Since Sobolev gradients precondition
the optimization problem, however, a smaller number of overall iterations may
be necessary for the algorithm to converge, which can in some cases amortize
the higher per-iteration cost.
