lecture12 - EECS 442 Computer vision Fitting methods...

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Unformatted text preview: EECS 442 Computer vision Fitting methods Reading: [HZ] Chapters: 4, 11 [FP] Chapters: 16 Some slides of this lectures are courtesy of profs. S. Lazebnik & K. Grauman- Problem formulation- Least square methods- RANSAC- Hough transforms - Multi-model fitting- Fitting helps matching! Fitting Goal: Choose a parametric model to fit a certain quantity from data- Lines - Curves- Homographic transformation- Fundamental matrix- Shape model Example: Computing vanishing points H Example: Estimating an homographic transformation Example: Estimating F A Example: fitting an 2D shape template Example: fitting a 3D object model Fitting Goal: Choose a parametric model to fit a certain quantity from data Critical issues:- noisydata- outliers- missing data Critical issues: noisy data A Critical issues: noisy data (intra-class variability) H Critical issues: outliers Critical issues: missing data (occlusions) Fitting Goal: Choose a parametric model to fit a certain quantity from data Techniques: Least square methods RANSAC Hough transform EM (Expectation Maximization) [forthcoming lecture] Least squares methods- fitting a line - Data: ( x 1 , y 1 ), , ( x n , y n ) Line equation: y i = mx i + b Find ( m , b ) to minimize = = n i i i b x m y E 1 2 ) ( ( x i , y i ) y=mx+b 2 2 = = Y X XB X dB dE T T [ ] 2 2 n 1 n 1 n 1 i 2 i i XB Y b m 1 x 1 x y y b m 1 x y E = = = = M M M Normal equation = = n i i i b x m y E 1 2 ) ( Y X XB X T T = Least squares methods- fitting a line - ( ) Y X X X B T 1 T = ) XB ( ) XB ( Y ) XB ( 2 Y Y ) XB Y ( ) XB Y ( T T T T + = = b Ax = More equations than unknowns Look for solution which minimizes ||Ax-b|| = (Ax-b) T (Ax-b) Solve LS solution ) ( ) ( = i T x b Ax b Ax b A A A x T T 1 ) ( = Least squares methods- fitting a line - t 1 t A ) A A ( A + = U V A 1 1 = with equal to for all nonzero singular values and zero otherwise 1 + = pseudo-inverse of A Solving b A A A x t t 1 ) ( = Least squares methods- fitting a line - t V U A = U V A + + = = SVD decomposition of A Least squares methods- fitting a line - = = n 1 i 2 i i ) b x m y ( E ( x i , y i ) y=mx+b ( ) Y X X X B T 1 T = = b m B Not rotation-invariant Fails completely for vertical lines Limitations Distance between point ( x n , y n ) and line ax+by=d Find ( a , b , d ) to minimize the sum of squared perpendicular distances ax+by=d = + = n i i i d y b x a E 1 2...
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lecture12 - EECS 442 Computer vision Fitting methods...

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