Presentation05 - Computer Vision Lecture #5 Hossam...

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Computer Vision Lecture #5 Hossam Abdelmunim 1 2 1 Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt 2 Electerical and Computer Engineering Department, University of Louisville, Louisville, KY, USA ECE619/645 – Spring 2011
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Linear Algebra for Computer Vision ( Necessary Background )
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Matrix basics • Square matrix: number of rows = number of columns • Symmetric matrix A ij =A ji . • Skew symmetric matrix Aij=-Aji . • Identity I ij = δ ij – Kronecker delta δ ij =0 if i≠j δ ij =1 if i=j • Lower triangular Upper triangular
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Linear systems of equations Solve via Gauss elimination Reduce system to product of lower or upper triangular matrix and x
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LU Decomposition (Factorization) Any matrix can be written as a product of a lower triangular ( L ) and upper triangular matrix ( U ). Most used algorithm in linear algebra. A=LU When A is symmetric positive definite for all x t Ax>0 “Cholesky decomposition” A=LL t
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Distances/Metrics and Norms We would like to measure distances and directions in the vector space the same way that we do it in Euclidean 3D. Distance function d( u,v ) makes a vector space a metric space if it satisfies – d( u,v )>0 for u,v different – d( u,u )=0, d( u,v )=d( v,u ) – d( u,w )≤ d( u,v )+d( v,w ) (triangle inequality) Norm (“length”). – || u
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This note was uploaded on 01/12/2012 for the course ECE 618 taught by Professor Amini during the Spring '08 term at University of Louisville.

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Presentation05 - Computer Vision Lecture #5 Hossam...

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