Lec06.Factorization-Duality-3DTransforms

Wikipediaorgwikisingularvaluedecomposition singular

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Unformatted text preview: omposition •  Let M be a m-by-n matrix whose entries are real numbers. Then M may be decomposed as M = U S VT where: –  U is an m-by-m orthonormal matrix –  S is an m-by-n matrix with non-negative numbers on the main diagonal and zeros elsewhere –  V is an n-by-n orthonormal matrix •  Example http://en.wikipedia.org/wiki/Singular_value_decomposition Singular Value Decomposition •  Implication: We can take the multiplicative part of any transform and describe it as a sequence of a rotation, scaling and another rotation •  2D Example: Decomposing an Affine Transformation M = 0.95 0.23 0 0.49 0.89 0 >> U * S * V' 1 >> [U, S, V] = svd(M(1:2, 1:2)) U= -0.78156 -0.62384 -0.62384 0.78156 S= 1.2904 0 0 0.56789 ans = 0.95 0.23 0.46 0.02 0.49 0.89 V= -0.68658 -0.72705 -0.72705 0.68658 Interpretation in terms of angles? Singular Value Decomposition •  Implications: Even a simple shear can be written as a rotation→scaling→rotation •  Try visualizing it to understand how… [Exercise] Summary: 2D Transformations •  Image Registration •  2D Transformations –  –  –  –  Scaling Shear Rotation Translation •  Inverse Transformations •  Rotation about an arbitrary point •  Concatenation of transformations •  Order of transformations •  Factorization of Transformati...
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This note was uploaded on 03/02/2014 for the course CS 436 taught by Professor Sohaibkhan during the Winter '13 term at Lahore University of Management Sciences.

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