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Unformatted text preview: CSE454, PSU Robert Collins Lecture 21 RANSAC Estimation external readings/tutorials if you are interested: http://cmp.felk.cvut.cz/ransaccvpr2006/tutorial/tutorial.htm CSE454, PSU Robert Collins Summary: Transformations projective affine similarity Euclidean CSE454, PSU Robert Collins Parameter Estimation We will talk about estimating parameters of 1) Geometric models (e.g. lines, planes, surfaces) 2) Geometric transformations (any of the parametric transformations we have been talking about) Leastsquares is a general strategy to address both! CSE454, PSU Robert Collins Parameter Estimation: Fitting Geometric Models General Idea: • Want to fit a model to raw image features (data) (the features could be points, edges, even regions) • Parameterize model such that model instance is an element of R n i.e. model instance = (a 1 ,a 2 ,…,a n ) • Define an error function E(model i , data) that measures how well a given model instance describes the data • Solve for the model instance that minimizes E CSE454, PSU Robert Collins Example : Line Fitting General Idea: • Want to fit a model to raw image features (data) (the features could be points, edges, even regions) • Parameterize model such that model instance is an element of R n i.e. model instance = (a 1 ,a 2 ,…,a n ) • Define an error function E(model i , data) that measures how well a given model instance describes the data • Solve for the model instance that minimizes E CSE454, PSU Robert Collins Point Feature Data pts = [... 17 81; 23 72; 35 73; 37 58; 45 50; 57 56; 61 36; 70 32; 80 32; 84 19] Point features = {(x i ,y i )  i = 1,…,n} x y CSE454, PSU Robert Collins Example : Line Fitting General Idea: • Want to fit a model to raw image features (data) (the features could be points, edges, even regions) • Parameterize model such that model instance is an element of R n i.e. model instance = (a 1 ,a 2 ,…,a n ) • Define an error function E(model i , data) that measures how well a given model instance describes the data • Solve for the model instance that minimizes E CSE454, PSU Robert Collins Line Parameterization y = m * x + b m = slope b = yintercept (where b crosses the y axis) Model instance = (m,b) x y b dy dx m = dy/dx (The astute student will note a problem with representing vertical lines) CSE454, PSU Robert Collins Example : Line Fitting General Idea: • Want to fit a model to raw image features (data) (the features could be points, edges, even regions) • Parameterize model such that model instance is an element of R n i.e. model instance = (a 1 ,a 2 ,…,a n ) • Define an error function E(model i , data) that measures how well a given model instance describes the data • Solve for the model instance that minimizes E CSE454, PSU Robert Collins Least Squares • Given line (m,b) • distance of point (x i ,y i ) to line is vertical distance • E is sum of squared distances over all points d i x i y i m x i + b (m,b) (L.S. is just one type of error function) CSE454, PSU...
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 Least Squares, Regression Analysis, Robert Collins

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