1. Lecture 12 Modules Employing Gradient Descent Computing Optical Flow Shape from Shading

2. 2 Gibbs Sampler E/T is big is overflow

3. 3 Gibbs Sampler E max /T that will overflow = BIGGEST DOUBLE

4. 4 2 Modules that Employ Gradient Descent 1.Computing Optical Flow for Motion Using Gradient Based Approach 2.Shape from Shading

5. 5 Optical Flow Motion Field in Image Plane

6. 6 Optical Flow 2 Methods: 1.Featured Based - similar to stereo where you solve - correspondence (matching) problem between 2 consecutive frames 2.Gradient of Intensity Based - No matching needed - Works well when images have much texture - Dense map of (u,v) at each pixel

7. 7 Gradient of Intensity Based 1 23 I(x,y,t 1 ) I(x,y,t 2 ) I(x,y,t 3 ) I(x,y,t) 16/sec t ALIASING - Spatial Resolution (x,y) pixels per cm - Temporal Resolution frames per second

8. 8 Aliasing Problems noticeable when your sampling cannot truly estimate the underlying frequency Have to sample double the frequency

9. 9 Chain Rule : I(x,y,t) Assumption: “As an object moves, its intensity does not change”

10. 10 Specular Regions Specular regions are noise for Computer Vision 2 2

11. 11 Gradient of Intensity Based Ix uIy v It

12. 12 Gradient of Intensity Based

13. 13 Gradient of Intensity Based Unknowns : u at each (x,y) v at each (x,y)

14. 14 Gradient of Intensity Based Use Gradient Descent : E(u,v) Update Rule Highly Textured Knowns : I x, I y, I t at (x,y)

15. 15 Research Topics  Find (u,v) through gradient Method: Coarse-to-Fine  How to choose 1, 2 automatically  How to get the annealing schedule automatically T high Random Walk T low Greedy

16. 16 Shape from Shading Point Light at ∞ ping-pong viewer Image Observed: f (viewer position, camera model, shape of object, material of object, light color, light model, light position)

17. 17 Material of Object  Color  Shiny  Transparency  Texture  Bumpy

18. 18 Light Model  Ambient – light (constant) at each point  Spot  Omni – Neon – All Direction  Point Light - “Sun”

19. 19 Light  R,G,B I (x,y) = Ambient + Diffuse + Specular = I a k a + k d I d cos  + k s  s (cos  )  I a : Ambient Light I d : Diffuse Light – Main Light k a : Ambient Constant “glow in dark” k d : Main Color Diffuse Constant White is high, Black is low k s : Mirror Like, Specularity Constant k s = 0 for ping pong = 0.5 for apple = 1 for billioud

20. 20 Shininess Factor  = 20  = 1 Sharp Shiny Blurry Shiny

21. 21 Shininess Factor  : angle between V and R  : angle between L and N cosq = L.N = |L||N|cos  = cos 

22. 22 Shininess Factor Diffuse = k d I d cos  cos  decrease I brighter darker 0o0o 45 o 85 o

23. 23 Shape Shape = Normal at a surface (N x, N y, N z ) unit

24. 24 Normal Equation of Plane

25. 25 Normal Normal is different at every point

26. 26 Light Direction L is the same at every point Contour of Constant Intensity

27. 27 SFS: Data Constraint Data Constraint

28. 28 SFS: Energy Function  Known : Ia, kd, (a,b,1), I(x,y)  Unknown : p,q