Final Review Course web page: vision.cis.udel.edu/~cv May 21, 2003  Lecture 37

Announcements HW 6 due tonight by midnight Final: Thursday, May 29, 1-3 pm in this room

Outline Review of course since midterm Course evaluations (including TA)

Lecture Topics Probability Cameras Camera calibration Single view geometry Stereo Tracking Robust estimation Structure from motion Optical flow Segmentation Classification

Probability Random variables –Discrete –Continuous (probability density functions) Histograms as PDF representations Joint, conditional probability Probabilistic inference: Bayes’ rule –MAP, ML inference

Cameras Lenses –Advantages (vs. pinhole camera), disadvantages Discretization effects of image capture

Camera Calibration Estimating the camera matrix –Least-squares via Direct Linear Transform (DLT) Extracting the calibration matrix –Nonlinear least-squares Estimating radial distortion I won’t ask about steps of DLT in detail (for this and other estimation problems), but you should know: –(1) When a DLT-like method is applicable –(2) The basic approach (stacking equations given by constraints on points) –(3) Number of points required –(4) Degenerate configurations

Single View Metrology Homogeneous representation of 2-D lines, 3-D planes Vanishing points and lines Single view metrology –Cross ratio Distances between planes –Homology (homography) Lengths & areas on planes Rectification –Affine vs. using homography

Stereo Epipolar geometry –Baseline, epipolar lines, epipoles, epipolar pencil Point-to-line mapping: Fundamental matrix F –Estimating F DLT with manually chosen correspondences Nonlinear minimization –Essential matrix Texture mapping –Bilinear interpolation

Tracking Tracking as probabilistic inference –Measurement likelihood, prior probability Examples –Feature tracking –Snakes Filtering methods –Kalman filter –Particle filters Steps –Sampling –Predicting –Measuring Estimating state from particle set

Robust Estimation RANSAC –Purpose –Methods –Application to automatic fundamental matrix estimation

Structure from Motion Triangulation –Covariance of structure estimates based on camera motion Stratified reconstruction –Necessary information for “upgrades” Affine factorization

Optical Flow Motion field vs. optical flow Brightness constancy constraint –Aperture problem Computing optical flow –Smoothness constraint –Least-squares solution for small set of motion parameters Time to collision

Segmentation Definition of segmentation Gestalt grouping strategies –Bottom-up, top-down Segmentation applications –Detecting shot boundaries –Background subtraction Pixel covariance & Mahalanobis distance Clustering –k -means clustering –Graph-theoretic clustering Eigenvector methods for segmentation –Normalized cut Hough transform

Classification Classification terminology Methods for classifier construction –Known probability densities Decision boundaries for normal distributions –Unknown densities Nonparametric approximation: Kernel methods, k -nearest neighbors Performance measurement –Cross-validation Dimensionality reduction with PCA Face recognition –Nearest neighbor –Eigenfaces

Classification Linear discriminants –Two-class –Multicategory Criterion functions J for computing discriminants –Learning as minimization of J Generalized linear discriminants Neural networks –Application: Face finding