1. Some Mathematical Serendipities of Exploration and Biomedical Imaging August Lau, Apache Corporation UH 2008 Imaging Symposium Sponsored by Apache Corporation

2. 4 papers presented at SEG 2007

3. Data = Simple part + Complex part

4. A short history of imaging : Observations Explanation Simplification Complex Learning TOPOLOGY SEMIGROUP

5. SIMPLIFICATION (TOPOLOGY) THEORETICAL COMPUTATIONAL

6. SIMPLIFICATION (COMPUTATIONAL TOPOLOGY) OIL MEDICAL

7. COMPLEX LEARNING (SEMIGROUP) Apache support of Yale research THEORETICAL COMPUTATIONAL

8. 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.

9. INTERPRETATION NEEDS QUALITATIVE MATHEMATICS A short history of imaging : Observations Explanation Simplification Complex Learning TOPOLOGY SEMIGROUP

10. 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. “ Data = Simple part + Complex part TOPOLOGY and SEMIGROUP hold promise