12 Robust fitting (RANSAC and others).pdfVIP

Model Fitting Computer Vision CS 143, Brown James Hays 10/03/11 Slides from Silvio Savarese, Svetlana Lazebnik, and Derek Hoiem Fitting: find the parameters of a model that best fit the data Alignment: find the parameters of the transformation that best align matched points Example: Computing vanishing points Slide from Silvio Savarese H Example: Estimating an homographic transformation Slide from Silvio Savarese Example: Estimating “fundamental matrix” that corresponds two views Slide from Silvio Savarese A Example: fitting an 2D shape template Slide from Silvio Savarese Example: fitting a 3D object model Slide from Silvio Savarese Critical issues: noisy data Slide from Silvio Savarese A Critical issues: intra-class variability “All models are wrong, but some are useful.” Box and Draper 1979 Slide from Silvio Savarese H Critical issues: outliers Slide from Silvio Savarese Critical issues: missing data (occlusions) Slide from Silvio Savarese Fitting and Alignment ? Design challenges – Design a suitable goodness of fit measure ? Similarity should reflect application goals ? Encode robustness to outliers and noise – Design an optimization method ? Avoid local optima ? Find best parameters quickly Slide from Derek Hoiem Fitting and Alignment: Methods ? Global optimization / Search for parameters – Least squares fit – Robust least squares – Iterative closest point (ICP) ? Hypothesize and test – Generalized Hough transform – RANSAC Slide from Derek Hoiem Simple example: Fitting a line Slide from Derek Hoiem Least squares line fitting ?Data: (x1, y1), …, (xn, yn) ?Line equation: yi = m xi + b ?Find (m, b) to minimize 022 yAApA TT dB dE )()()(2 1 1 1 2 2 11 1 2 ApApyApyy yAp TTT nn n i ii y y b m x x y b m xE ??? n i ii bxmyE 1 2)( (xi, yi) y=mx+b yAAApyAApA TTTT 1 Matlab: p = A \ y; Modified from S. Lazebnik Least squares: Robustness to noise Least squares fit to the red points: