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Lecture 8. Gradient Descent Method. Computer Vision. Scene Description as Output “Inverse of Computer Graphics”. Computer Vision. Computer Graphics. CV. Image. Scence Description -shape -color - identity objects - position - time to contact - tracking. CG. Image. Scence -shape
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Lecture 8 Gradient Descent Method
Computer Vision • Scene Description as Output • “Inverse of Computer Graphics” Computer Vision Computer Graphics CV Image Scence Description -shape -color - identity objects - position - time to contact - tracking CG Image Scence -shape -material – color, shiny, transparency, texture, etc. -light -camera
Optimization- Finding Best Solution • Regression -(Pseudo Inverse) - Linear • Hough Transform • Gradient Descent • Non-linear • Newton’s method • Simulated Annealing • Gibbs Sampler • Evolutioning/Genetic Algorithm • Most Computer Vision Problems => Optimization Problems
Computer Vision problems 1) Computing Optical Flow - Motion 2) Stereo Disparity - Shape 3) Shape from Shading - 4) Structure/Shape from Motion 5) Shape from regular Texture 6) Shape from Contours
Regression Best fit line (x1,y1), (x2,y2), (x3,y3), (x4,y4) ... Find m,b so that is minimum computed y observed y We have learned Pseudo Inverse
Overconstraint - more equation than unknowns Under constrained. Therefore, cannot be done. Assumption about the world are needed to solve such problems by adding constraints (equations) causes loss of 1-dimension of information 2-D 3-D COMPUTER GRAPHICS 3-D 2-D COMPUTER VISION
Gradient Descent minimize (m,b) that minimize E Find minimum energy using gradient descent
Gradient Descent Algorithm 1. Start at random point 2. Move one step in gradient descent 3. Repeat 2 until no change in E
Gradient Descent 1. m0 = random (-max, +max) 2. b0 = random (-max, +max) 3. Direction of Gradient a is step size - must be small enough 4. Repeat 2-3 until E has no change, or m,b no change
Gradient Descent E(m,b) E(x,y, z)
Gradient Descent E(m,b) Chain Rule
Iij - Observed Image Data Fij - Actual Signal Iij - Fij + G(m,s) Noise Removal Objective = Given Iij, find best estimate of Fij Smoothness Constraint Data Constraint l high for noisy data, l low for reliable data
Noise Removal 100 Algorithm: 1. Random Fij = [0..255] 2. Update Rule 3. Repeat step 2 until no change in E Fij 100 Unknowns : 100 x 100 = 10,000 Fij For each pixel i,j
Noise Removal E =
Noise Removal with missing data Iij - Observed Image Data Fij - Actual Signal Aij - 0 = has data 1 = no data