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Geometric Reconstruction from Partial Data
Geometric inverse problems refer to the general class of problems
concerned with the reconstruction of a domain from partial geometric
information. For instance, the area of the shadow that a 3D object
casts onto a plane is called the brightness function, from which
we may be interested to reconstruct the shape of the object.
In a related setting, the partial information may be distilled into moments
of the underlying shapes we seek to reconstruct. Over the past several
years, we have developed computational algorithms for solving such
problems and considered fundamental performance bounds encountered
in reconstruction from partial geometric data.
Related Journal Publications

R.J. Gardner, Markus Kiderlen, and Peyman Milanfar,
“Convergence of algorithms for reconstructing convex bodies and directional measures”,
The Annals of Statistics, vol. 34, no. 3, pp.
13311374, June 2006.

A. Poonawala, P. Milanfar, R. Gardner,
“Shape Estimation from Support and Diameter Functions”,
Journal of Mathematical Imaging and Vision 24: pp. 229244, March 2006.

A. Cuyt, G.H. Golub, P. Milanfar, B. Verdonk,
“Multidimensional Integral Inversion, with Applications in Shape Reconstruction”,
SIAM Journal on Scientific Computing, vol. 27, no. 3, pp.10581070, 2005.

M. Elad, P. Milanfar, G.H. Golub,
“Shape From Moments—An Estimation Theory Perspective”,
IEEE Transactions on Signal Processing, vol. 52, no. 7, pp. 18141829, July 2004.

R.J. Gardner, P. Milanfar,
“Reconstruction of Convex Bodies from Brightness Functions”,
Discrete and Computational Geometry, 29:279303, 2003.

B. Gustafsson, C. He, P. Milanfar, M. Putinar,
“Reconstructing Planar Domains From Their Moments”,
Inverse Problems, vol. 16, no. 4, pp. 10531070, August 2000.

G. Golub, P. Milanfar, J. Varah,
“A Stable Numerical Method for Inverting Shape from Moments”,
SIAM Journal on Scientific Computing, vol. 21, no. 4, pp. 12221243, December 1999.

P. Milanfar, G.C. Verghese, W.C. Karl, A.S. Willsky,
“Reconstructing Polygons from Moments with Connections to Array Processing”,
IEEE Transactions on Signal Processing, vol. 43, no. 2, pp. 432443, February 1995.
Related Conference Publications and Presentations

A. Cuyt, G. Golub, P. Milanfar, B. Verdonk,
“Inverting a Multidimensional Shape From Moments”,
Proceedings of the International Conference on Numerical Analysis and Applied Mathematics / Simos T.E.
[edit.], Weinheim, Wiley, p. 436439, 2004.

A. Poonawala, P. Milanfar, R. Gardner,
“On the Uncertainty Analysis of Shape Reconstruction from Areas of Silhouettes”,
to appear in Proceedings of Fifth International Conference on Advances In Pattern Recognition, Calcutta, India, December 2003.

A. Poonawala, P. Milanfar, R. Gardner,
“A Statistical Analysis of Shape Reconstruction From Areas of Shadows”,
Invited paper, Proceedings of the 36th Asilomar Conference on Signals, Systems and Computers, Pacific
Grove, CA, November 2002.

M. Elad, P. Milanfar, G.H. Golub,
“Shape From Moments as an Inverse Problem”,
Proceedings of the 36th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, November 2002.

P. Milanfar, R. J. Gardner,
“Shape Reconstruction from Brightness Functions”,
Invited paper, Proceedings of SPIE Vol. 4474, SPIE Conference on Advanced Signal Processing Algorithms,
Architectures, and Implementations X, August 2001, San Diego, CA.

P. Milanfar, M. Putinar, J. Varah, B. Gustafsson, G. Golub,
“Shape Reconstruction From Moments: Theory, Algorithms, and Applications”,
Invited paper, Proceedings of SPIE Vol. 4116, SPIE Conference on Advanced Signal Processing Algorithms,
Architectures, and Implementations X, August 2000, San Diego, CA.

P. Milanfar, W.C. Karl, A.S. Willsky, G.C. Verghese,
“Polygon Reconstruction from Moments Using Array Processing”,
Proceedings of the Sixth IEEE Digital Signal Processing Workshop, Yosemite National Park, CA,
October 1994.