1. Optical Flow Estimation
Slides by Yael Pritch, taken from the image processing course
Many slides from Alexei Efros ans Steve Seitz

2. Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?
Why is it useful
Depth (3D) reconstruction
Motion detection - tracking moving objects
Compression
Mosaics
And more…

3. Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?
Solve pixel correspondence problem
given a pixel in H, look for nearby pixels of the same color in I
Key assumptions
color constancy: a point in H looks the same in I
For grayscale images, this is brightness constancy
small motion: points do not move very far
This is called the optical flow problem

4. Motion estimation: Optical flow
Goal: estimate the motion of each pixel

5. Classes of Techniques Feature-based methods Direct-methods
Extract visual features (corners, textured areas) and track them over multiple frames
Sparse motion fields, but possibly robust tracking
Suitable especially when image motion is large (10-s of pixels)
Direct-methods
Directly recover image motion from spatio-temporal image brightness variations
Motion vectors directly recovered without an intermediate feature extraction step
Dense motion fields, but more sensitive to appearance variations
Suitable for video and when image motion is small (< 10 pixels)

6. Optical flow constraints (grayscale images)
Let’s take a closer look at these constraints:
brightness constancy: Q: what’s the equation?
A: H(x,y) = I(x+u, y+v)
small motion: (u and v are less than 1 pixel)
suppose we take the Taylor series expansion of I:

7. Optical flow equation Combining these two equations
In the limit, as u and v go to zero, this approximation is good

8. Optical flow equation Intuitively, what does this constraint mean?
Q: How many equations and how many unknowns do we have per pixel?
Intuitively, what does this constraint mean?
The component of the flow in the gradient direction is determined
The component of the flow parallel to an edge is unknown
A: u and v are unknown, 1 equation
This explains the Barber Pole illusion

9. Getting more Equations
How to get more equations for a pixel?
Basic idea: impose additional constraints
most common is to assume that the flow field is smooth locally
one method: assume the pixel’s neighbors have the same (u,v)
If we use a 5x5 window, this gives us 25 equations per pixel!

10. Lukas-Kanade flow We have over constrained equation set:
Solution: solve least squares problem
minimum least squares solution given by solution (in d) of:
The summations are over all pixels in the K x K window
This technique was first proposed by Lukas & Kanade (1981)

11. When can we solve this ? When is This Solvable?
Optimal (u, v) satisfies Lucas-Kanade equation
When is This Solvable?
ATA should be invertible
The eigenvalues of ATA should not be too small (noise)
ATA should be well-conditioned
l1/ l2 should not be too large (l1 = larger eigenvalue)
ATA is invertible when there is no aperture problem

12. The Aperture Problem

13. The Aperture Problem

14. The Aperture Problem

15. The Aperture Problem and Let
Algorithm: At each pixel compute by solving
M is singular if all gradient vectors point in the same direction
e.g., along an edge
of course, trivially singular if the summation is over a single pixel or if there is no texture
i.e., only normal flow is available (aperture problem)
Corners and textured areas are OK

16. Local Patch Analysis

17. Edge
large gradients, all the same
large l1, small l2

18. Low texture region
gradients have small magnitude
small l1, small l2

19. High textured region gradients are different, large magnitudes
large l1, large l2

20. Observation This is a two image problem BUT
Can measure sensitivity/uncertainty by just looking at one of the images!
This tells us which pixels are easy to track, which are hard
very useful later on when we do feature tracking...

21. Errors in Lukas-Kanade
What are the potential causes of errors in this procedure?
Suppose ATA is easily invertible
Suppose there is not much noise in the image
When our assumptions are violated
Brightness constancy is not satisfied
The motion is not small
A point does not move like its neighbors
window size is too large
what is the ideal window size?

22. Iterative Refinement Iterative Lukas-Kanade Algorithm
Estimate velocity at each pixel by solving Lucas-Kanade equations
Warp H towards I using the estimated flow field
- use image warping techniques (easier said than done)
Repeat until convergence

23. Optical Flow: Iterative Estimation
estimate update
Initial guess:
Estimate: x0 x
(using d for displacement here instead of u)

24. Optical Flow: Iterative Estimationx x0
estimate update
Initial guess:
Estimate:

25. Optical Flow: Iterative Estimationx x0
estimate update
Initial guess:
Estimate:

26. Optical Flow: Iterative Estimationx0 x

27. Optical Flow: Iterative Estimation
Some Implementation Issues:
Warp one image, take derivatives of the other so you don’t need to re-compute the gradient after each iteration.
Often useful to low-pass filter the images before motion estimation (for better derivative estimation, and linear approximations to image intensity)

28. Optical Flow: Aliasing
Temporal aliasing causes ambiguities in optical flow because images can have many pixels with the same intensity.
I.e., how do we know which ‘correspondence’ is correct?
actual shift
estimated shift
nearest match is correct (no aliasing)
nearest match is incorrect (aliasing)
To overcome aliasing: coarse-to-fine estimation.

29. Revisiting the small motion assumption
Is this motion small enough?
Probably not—it’s much larger than one pixel (2nd order terms dominate)
How might we solve this problem?

30. Reduce the resolution!

31. Coarse-to-fine optical flow estimation
Gaussian pyramid of image H
Gaussian pyramid of image I
image I
image H
u=10 pixels
u=5 pixels
u=2.5 pixels
u=1.25 pixels
image H
image I

32. Coarse-to-fine optical flow estimation
Gaussian pyramid of image H
Gaussian pyramid of image I
image I
image H
run iterative L-K
warp & upsample
run iterative L-K .
image J
image I

33. Multi-resolution Lucas Kanade Algorithm
Compute Iterative LK at highest level
For Each Level i
Take flow u(i-1), v(i-1) from level i-1
Upsample the flow to create u*(i), v*(i) matrices of twice resolution for level i.
Multiply u*(i), v*(i) by 2
Compute It from a block displaced by u*(i), v*(i)
Apply LK to get u’(i), v’(i) (the correction in flow)
Add corrections u’(i), v’(i) to obtain the flow u(i), v(i) at the ith level, i.e., u(i)=u*(i)+u’(i), v(i)=v*(i)+v’(i)

34. Optical Flow: Iterative Estimation
Some Implementation Issues:
Warping is not easy (ensure that errors in warping are smaller than the estimate refinement)
Warp one image, take derivatives of the other so you don’t need to re-compute the gradient after each iteration.
Often useful to low-pass filter the images before motion estimation (for better derivative estimation, and linear approximations to image intensity)

35. Optical Flow Results

36. Optical Flow Results

37. Optical flow Results

38. More slides

39. Image Warping

40. Image Warping image filtering: change range of image g(x) = h(f(x))
image warping: change domain of image
g(x) = f(h(x)) f x f x h f x f x h

41. Image Warping image filtering: change range of image g(x) = h(f(x))
image warping: change domain of image
g(x) = f(h(x)) f g h f g h

42. Parametric (global) warping
Examples of parametric warps:
aspect
translation
rotation
perspective
affine
cylindrical

43. 2D coordinate transformations
translation: x’ = x + t x = (x,y)
rotation: x’ = R x + t
similarity: x’ = s R x + t
affine: x’ = A x + t
perspective: x’  H x x = (x,y,1) (x is a homogeneous coordinate)
These all form a nested group (closed w/ inv.)

44. Image Warping
Given a coordinate transform x’ = h(x) and a source image f(x), how do we compute a transformed image g(x’) = f(h(x))?
h(x) x x’
f(x)
g(x’)

45. Forward Warping What if pixel lands “between” two pixels?
Send each pixel f(x) to its corresponding location x’ = h(x) in g(x’)
What if pixel lands “between” two pixels?
h(x) x x’
f(x)
g(x’)

46. Forward Warping What if pixel lands “between” two pixels?
Send each pixel f(x) to its corresponding location x’ = h(x) in g(x’)
What if pixel lands “between” two pixels?
Answer: add “contribution” to several pixels, normalize later (splatting)
h(x) x x’
f(x)
g(x’)

47. Inverse Warping What if pixel comes from “between” two pixels?
Get each pixel g(x’) from its corresponding location x = h-1(x’) in f(x)
What if pixel comes from “between” two pixels?
h-1(x’) x x’
f(x)
g(x’)

48. Inverse Warping What if pixel comes from “between” two pixels?
Get each pixel g(x’) from its corresponding location x = h-1(x’) in f(x)
What if pixel comes from “between” two pixels?
Answer: resample color value from interpolated (prefiltered) source image x x’
f(x)
g(x’)

49. Interpolation Possible interpolation filters:
nearest neighbor
bilinear
bicubic (interpolating)
sinc / FIR
Needed to prevent “jaggies” and “texture crawl” (see demo)