We developed a fast algorithm to compute the elastic shape dissimilarity
metric between 2d closed planar curves. The algorithm performs the underlying
optimization to find the matching starting point, rotation and diffeomorphism
between two given curves in empirically subquadratic times with respect to
the number of nodes. The figure below shows a 100x100 shape distance matrix
computed for a subset of the MPEG7 shape data set.

Using a curve evolution approach to minimizing the Mumford-Shah
functional allows us to segment noisy images, which can be approximated
by piecewise smooth functions. This approach is very powerful, but also
computationally very expensive, because it requires solving elliptic
PDEs at each iteration of the curve evolution. We developed a fast
Newton-type algorithm for image segmentation with the Mumford-Shah
functional, using an adaptive Lagrangian shape optimization approach.
Below is an example of the evolution of the segmenting curves and the
evolution of the piecewise smooth approximation.
See paper for more information.

These pictures show the evolution of an initial curve to capture the
boundaries of cells in the given picture. The curves evolve in such a
way that they stop at a minimum of the piecewise smooth Mumford-Shah
energy.

These pictures show the evolution of the smooth approximations to the
original image, computed from the corresponding elliptic PDEs in the
domains defined by the curves.

We developed a fast Lagrangian method to segment images using the
Geodesic Active Contour model. The use of a Lagrangian representation
and second order energy sensitivity information results in a fast
segmentation algorithm. The curve representation adapts to the image and
the geometry by adjusting node density. It can also handle topological
changes in a very robust manner. Note that we do not use the level set
method or any other Eulerian representation to handle the topological
changes.

We also developed an adaptive Lagrangian algorithm for 3d image
segmentation using a minimal surface energy model. The method uses
second order energy sensitivies and judicious adaptation of the surface
triangulations to produce high quality segmentations fast. The low
computational cost is in stark contrast with a voxel-based approach,
such as the level set method.

In this example, we start with a coarse initial mesh. As the method
senses that it is close to an object, it adapts the triangulation to
capture the object accurately. It keeps the triangulation fine close to
sharp features, such as object edges and corners, but coarse in other
places, such as the smooth faces of the object, where small triangles
are not needed to resolve and represent the object.

This is another example showing the use of adaptive triangulated
surfaces, which do a good job in segmenting two touching balls with a
cusp in between.

In this work, we developed a multilevel inversion algorithm for
reconstruction of the source of an elliptic PDE in optimal time. The
algorithm is especially effective in the case of high fidelity
reconstructions. High fidelity reconstructions are obtained with low
regularization, which makes the inversion computationally expensive. Our
iterative multilevel algorithm resulted in substantial savings in computations
(see paper).

The following are examples of results from some of my work in the recent years.