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Lecture 8

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

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  1. Lecture 8 Gradient Descent Method

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Illusions

  8. Gradient Descent minimize (m,b) that minimize E Find minimum energy using gradient descent

  9. Gradient Descent Algorithm 1. Start at random point 2. Move one step in gradient descent 3. Repeat 2 until no change in E

  10. 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

  11. Gradient Descent E(m,b) E(x,y, z)

  12. Gradient Descent E(m,b) Chain Rule

  13. 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

  14. 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

  15. Noise Removal E =

  16. Noise Removal with missing data Iij - Observed Image Data Fij - Actual Signal Aij - 0 = has data 1 = no data

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