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One M@y He@r The Sh@pe of @ Drum

One M@y He@r The Sh@pe of @ Drum. Ben Haj Rhouma Mohamed Sultan Qaboos University-Oman Joint work with L. Hermi , Univ of Arizona & M.A. Khabou , Univ. of West Florida Queen Dido Conference May 2010. Can we ?. Mark Kac’s question. Can we hear the shape of a drum ?. No we can’t.

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One M@y He@r The Sh@pe of @ Drum

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  1. One M@y He@r The Sh@pe of @ Drum Ben HajRhouma Mohamed Sultan Qaboos University-Oman Joint work withL. Hermi, Univ of Arizona & M.A. Khabou, Univ. of West Florida Queen Dido Conference May 2010

  2. Can we ? Mark Kac’s question. Can we hear the shape of a drum?

  3. No we can’t

  4. Bilby and Hawk

  5. Eigenvalues of the Laplacian Operator D

  6. Some positive answers Ashbaugh and Benguria two-proofs of the PPW conjecture 1991 - 1992 . Steve Zelditch proved that the answer to Kac's question is positive if one imposes restrictions to certain convex planar regions with analytic boundary. It is not known whether two non-convex analytic domains can have the same eigenvalues. Invariance of ratios + Universal inequalities

  7. Shape recognition Classifying shapes When are two shapes close? Shape retrieval (invariance to size/rotation/translation + not too sensitive to noise + reasonable deformations...)

  8. Applications Handwriting recognition Face recognition Target recognition DNA + molecule matching Automatic filing/sorting/retrieving of pictures

  9. Eigenfaces and Eigenimages Eigenfaces are a set of eigenvectors used in Computer vision developed by Sirovich and Kirby (1987) and used by Turk and Pentland Eigenface generation : A large set of digitized images of human faces, taken under the same ( similar) lighting conditions, are normalized to line up the eyes and mouths. Each digitalized picture makes up a row in a large matrix from which Eigenfaces can be extracted out of the image data by means of principal component analysis (PCA).

  10. Can we guess the shape of a drum? Compute the ratio of eigenvalues of known shapes for Dirichlet, Neumann, Clamped plate and Buckling problem. Use part of the data (labeled) to train a neural network to classify shapes. Test the performance of the classifier on the unlabeled data.

  11. Computing the eigenvalues

  12. Testing simple shapes

  13. Real images

  14. Neumann & Stekloff

  15. Results on the Squid database

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