Some Mathematical Serendipities of

Exploration and Biomedical Imaging

August Lau, Apache Corporation

UH 2008 Imaging Symposium

Sponsored by Apache Corporation

4 papers presented at SEG 2007

Data = Simple part + Complex part

A short history of imaging :

Observations Explanation Simplification Complex Learning

TOPOLOGY SEMIGROUP

SIMPLIFICATION(TOPOLOGY)

THEORETICAL

COMPUTATIONAL

OIL

MEDICAL

SIMPLIFICATION(COMPUTATIONAL TOPOLOGY)

COMPLEX LEARNING(SEMIGROUP)

Apache support of Yale research

THEORETICAL

COMPUTATIONAL

COMPLEX LEARNING(DIFFUSION SEMIGROUP)

INPUT OUTPUT

Apache support of Yale research

The images on the top left are shuffled out of order, By building an affinity matrix between the images , we obtain a first nontrivial eigenvector, this is the “Google rank” which we use to order the images by their rank relative to the top left image in the corner

This is a simple case where one parameter (angle of rotation) suffices ,and a single rank does the job.

A short history of imaging :

Observations Explanation Simplification Complex Learning

TOPOLOGY SEMIGROUP

INTERPRETATION NEEDS QUALITATIVE MATHEMATICS

My words:“I think that we will have a new approach in science which will be very different from reductionism and Newtonian derivative of reductionism,” … “The impetus could come from seismic imaging and biomedical imaging.  Seismic imaging shows that despite our powerful computers, we are still at a loss as to how to deal with multiple scattering.  By the same token, biomedical imaging could image the most detail units but the interaction of these units which give rise to macroscopic behavior is still to be found. “

INTERPRETATION NEEDS QUALITATIVE MATHEMATICS

TOPOLOGY and SEMIGROUP hold promise

Data = Simple part + Complex part