Computer Visionand Robotics Introduction to Artificial Intelligence CS440/ECE448 Lecture 27 1-unit projects for grad students: Get in touch with me ! Last homework out today!! Next Tuesday: Review (everything since mid-term) Final: May 8, 1:30 to 3pm, here
Last lecture • Support vectors machines This lecture • Computer vision • Robotics Reading • Chapters 24 and 25
support hyperplanes Linearly Separable Classes support vectors What is the maximum-margin separating plane?
w.x+b = 1 w.x+b = -1 Support Vector Machines (Boser, Guyon & Vapnik, 1992; Vapnik 1995) w.x+b = 0 positive examples negative examples What is the maximum-margin separating plane?
Support vector machines ctd • Examples are of the form ( xi, yi ), where yi = ¨ 1. • They all verify yi ( w.xi + b ) ≥ 1. • The distance between the separating plane is 2 / | w |. • Thus finding the maximum margin plane amounts to • Minimizing: ½ | w |^2 • subject to: yi ( w.xi + b ) ≥ 1 for i = 1, … ,n. • A quadratic programming problem!
Support Vector Machines ctd The dual formulation of the problem is: • Maximize: i=1ni -1/2 i,j=1nyiyjij ( xi¢xj ) • subject to: i=1nyii =0, and i≥ 0 for i=1, … , n. Note 1: The weitghts i are nonzero only for support vectors. Note 2: The data points only appear in the optimization problem via their dot product K( xi , xj ) = xi¢xj . Note 3: This allows classes that are not linearly separable to be handled via nonlinear mappings and appropriate kernelsK.
non-linear mapping ( X1, X2, X3 ) Kernel Machines x12+x22=1 X1+X2=1 ( x1, x2)
Computer Vision Tasks • Stereo, structure from motion, shape from X: What is the 3D shape of the objects present in the image? • Segmentation: Separate objects from background. • Recognition: Identify the objects present in the image.
Pompei painting, 2000 years ago. Brunelleschi, 1415 Van Eyk, XIVth Century Massaccio’s Trinity, 1425
Pinhole Perspective Equation NOTE:z is always negative..
Affine projection models: Weak perspective projection is the magnification. When the scene relief is small compared its distance from the Camera, m can be taken constant: weak perspective projection.
Affine projection models: Orthographic projection When the camera is at a (roughly constant) distance from the scene, take m=1.
What is the image of a sphere? Planar pinhole perspective Orthographic projection Spherical pinhole perspective
Diffraction effects in pinhole cameras. Shrinking pinhole size Use a lens!
Lenses Snell’s law n1 sina1 = n2 sin a2 Descartes’ law
Photography (Niepce, “La Table Servie,” 1822) Milestones: • Daguerréotypes (1839) • Photographic Film (Eastman, 1889) • Cinema (Lumière Brothers, 1895) • Color Photography (Lumière Brothers, 1908) • Television (Baird, Farnsworth, Zworykin, 1920s) CCD and CMOS Devices (1970)
Image Formation: Radiometry The light source(s) The sensor characteristics The surface normal The surface properties The optics What determines the brightness of an image pixel?
2 Minimize |w-w’|. Correlation Methods (1970--) Slide the window along the epipolar line until w.w’ is maximized. Normalized Correlation: minimize q instead.
Human/Felix Bug Barbara Steele Face Joe Camel Problem: Recognizing instances Recognizing categories
Candidate parts Matching Response scores Part dictionary Validation images Learning validation images Classifier Valildation parts Test image Part detection Testing response vector Decision First steps toward category-level object recognition (Lazebnik et al., 2006) Training pairs … … … …
ILM Toyota What is it all for?
Copan Courtesy of S. Leigh Courtesy of G. Robinson & M.S. Sharma