The document discusses key concepts in 3D transformations and computer graphics, including edge detection using Harris-Stephens algorithm, quaternions for rotations, and the Iterative Closest Point (ICP) algorithm for aligning point sets. It highlights techniques like bilateral filtering and the challenges of nearest neighbor computations in high dimensions. Additionally, it covers the mathematical foundations behind projective geometry, including duality and intersection of lines.

1. © 2011 Frank Nielsen
INF555
Fundamentals of 3D
Lecture 3:
Debriefing: Lecture 2
Rigid transformations/Quaternions
Iterative Closest Point (+Kd-trees)
Frank Nielsen
nielsen@lix.polytechnique.fr
28 Septembre 2011

2. © 2011 Frank Nielsen
• Depth discontinuity
• Surface orientation
discontinuity
• Reflectance
discontinuity (i.e.,
change in surface
material properties)
• Illumination
discontinuity (e.g.,
shadow)
Harris-Stephens' combined corner/edge detector

3. © 2011 Frank Nielsen

4. © 2011 Frank Nielsen
Harris-Stephens edge detector
Aim at finding good feature

5. © 2011 Frank Nielsen
Harris-Stephens edge detector
Measure the corner response as
Algorithm:
– Find points with large corner response function R
(R > threshold)
– Take the points of local maxima of R

6. © 2011 Frank Nielsen

7. © 2011 Frank Nielsen
Edge thresholding hysterisis
● If a pixel value is above the high threshold, it is an edge.
● If a pixel value is below the low threshold, it is not an edge.
● If a pixel value is between the low and high thresholds,
it is an edge if it is connected to another edge pixel,
otherwise it is interpreted as noise.
Two thresholds: low and high
Single threshold value for edges -> Streaking

8. © 2011 Frank Nielsen
Edge hysteresis

9. © 2011 Frank Nielsen
Homogeneous coordinates and duality point/line
homogenization
Homogeneous vectorInhomogeneous vector
dehomogenization (also known as
Perspective division)

10. © 2011 Frank Nielsen
Projective plane
Equivalence class:
is equivalent to
Line coefficients stored in an inhomogeneous vector
Equation of the line:
Point and line have same homogeneous representation:
A point can be interpreted as the coefficients of the line
Point and line have same homogeneous representation:
A point can be interpreted as the coefficients of the line

11. © 2011 Frank Nielsen
Intersection of lines
Cross-product of two vectors:
Intersection point of two lines is obtained from their cross-product:
Line passing through two « points » l1* and l2*:
l=p*=l1* x l2*
p= l1 x l2
Duality

12. © 2011 Frank Nielsen
Application:
Detection of line segment intersection
l1=CrossProduct(p,q);
l2=CrossProduct(r,s);
// intersection point is the cross-product
//of the line coefficients (duality)
intersection=CrossProduct(l1,l2);
intersection.Normalize(); // to get back Euclidean point

13. © 2011 Frank Nielsen
Overview of duality in projective geometry
The determinant of three points represent the volume of their parallepiped.

14. © 2011 Frank Nielsen
2D Transformations using homogeneous coordinates

15. © 2011 Frank Nielsen
2D Transformations using homogeneous coordinates

16. © 2011 Frank Nielsen
Cartesian coordinate systems in 3D

17. © 2011 Frank Nielsen
3D Transformations using homogeneous coordinates
Be careful: Gimbal lock

18. © 2011 Frank Nielsen
Euler rotation

19. © 2011 Frank Nielsen
Cross-product/outer product
Consider the cross-product as a matrix multiplication:
Outer-product

20. © 2011 Frank Nielsen
Arbitrary matrix rotation:
Rodrigues' formula
Equivalent to:

21. © 2011 Frank Nielsen

22. © 2011 Frank Nielsen

23. © 2011 Frank Nielsen
Rigid transformations
Concatenation (non-commutative!)

24. © 2011 Frank Nielsen
Quaternions for rotations
● Easy to invert scalars
● For 2D vectors, invert using complex numbers...
● For 3D vectors??? (-> 4D quaternions)
● For dD vectors??? (-> 8D octonions)
● Easy to invert scalars
● For 2D vectors, invert using complex numbers...
● For 3D vectors??? (-> 4D quaternions)
● For dD vectors??? (-> 8D octonions)
Provide rotation operator that is invertible
Sir William Rowan Hamilton
Lectures on Quaternions
http://digital.library.cornell.edu/

25. © 2011 Frank Nielsen
Quaternions: 1D real+3D imaginary
Real part (1D) Imaginary part (3D, i j k vectors)
Multiplication:
Norm (l2)

26. © 2011 Frank Nielsen
Unit quaternions
For a given 3D point p, we compute its rotation Rp as
Rotation theta around an axis u: Quaternion representation:
conjugate

27. © 2011 Frank Nielsen
Unit quaternions
for rotations

28. © 2011 Frank Nielsen
Conversion rotation matrix to quaternion

29. © 2011 Frank Nielsen
Spherical linear interpolation (SLERP)
LERP is non-sense for rotation matrices:
SLERP is using quaternion algebra:

30. © 2011 Frank Nielsen
Spherical linear interpolation (SLERP)
Useful for computer graphics animation
(bone, skinning at articulation)

31. © 2011 Frank Nielsen
Bilateral filtering
Edge-preserving smoothing
Videos/demo

32. © 2011 Frank Nielsen
Gaussian filtering: Blur everything
Traditional spatial gaussian filtering

33. © 2011 Frank Nielsen
Bilateral filtering
New! gaussian on the intensity difference filtering
Bilateral Filtering for Gray and Color Images, Tomasi and Manduchi 1998
.... SUSAN feature extractor...

34. © 2011 Frank Nielsen
Domain
filtering
Range filtering

35. © 2011 Frank Nielsen
Iterative Closest Point (ICP)
Align point sets. For example, terrains (DEMs)
Robotics

36. © 2011 Frank Nielsen
Align point sets obtained from range scanners
http://www-graphics.stanford.edu/projects/mich/

37. © 2011 Frank Nielsen
Solve stone jigsaws...
ICP for solving jigsaws

38. © 2011 Frank Nielsen
ICP: Algorithm at a glance
● Start from a not too far transformation
● Match the point of the target to the source
● Compute the best transformation from point correspondence
● Reiterate until the mismatch error goes below a threshold
In practice, this is a very fast registration method...
A Method for Registration of 3-D Shapes. by: Paul J Besl, Neil D Mckay.
IEEE Trans. Pattern Anal. Mach. Intell., Vol. 14, No. 2. (February 1992

39. © 2011 Frank Nielsen
ICP: Finding the best rigid transformation
Given point correspondences, find the best rigid transformation.
Source/ModelObservation/Target
Find (R,t) that minimizes the squared euclidean error:

40. © 2011 Frank Nielsen
Align the center of mass of sets:

41. © 2011 Frank Nielsen
Finding the rotation matrix:
Compute the singular value decomposition
Optimal transformation:

42. © 2011 Frank Nielsen
Registration of many point sets to a common atlas
Scoliotic Spine (Atlas of 307 patients)
Many variants of ICP method (truncated, robust, etc.)Many variants of ICP method (truncated, robust, etc.)

43. © 2011 Frank Nielsen
In theory,
ICP may provably run very slowly for well-constructedwell-constructed point sets...
What computational geometers say
David Arthur; Sergei Vassilvitskii
Worst-case and Smoothed Analysis of the ICP Algorithm, with an Application to the
k-means Method
FOCS 2006 => O(n/d)^d iterations (exponential)
... but smooth analysis of ICP is polynomial

44. © 2011 Frank Nielsen
Computing nearest neighbors in ICP...
● Naive linear-time algorithm
● Tree-like algorithm using kd-trees
● Tree-like algorithm using metric ball trees
●...
Challenging problem in very high-dimensions
(common to work up to dimension > 1000 nowadays)
Nearest neighbor of q

45. © 2011 Frank Nielsen
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