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

Bilby and Hawk

Eigenvalues of the Laplacian Operator

D

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.

Computing the eigenvalues

Testing simple shapes

Real images

Neumann & Stekloff

Results on the Squid database