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## Computer Vision and Robotics

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**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