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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
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
Shape recognition
Classifying shapes
When are two shapes close?
Shape retrieval
(invariance to size/rotation/translation + not too sensitive to noise + reasonable deformations...)
Applications
Handwriting recognition
Face recognition
Target recognition
DNA + molecule matching
Automatic filing/sorting/retrieving of pictures
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).
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.