Those are the slides for the book: Visual Computing: Geometry, Graphics, and Vision. by Frank Nielsen (2005) http://www.sonycsl.co.jp/person/nielsen/visualcomputing/ http://www.amazon.com/Visual-Computing-Geometry-Graphics-Charles/dp/1584504277

1. © 2011Frank Nielsen
INF555
Fundamentals of 3D
Lecture 4:
Debriefing: ICP (kD-trees)
Homography
Graphics pipeline
Frank Nielsen
5 octobre 2011
nielsen@lix.polytechnique.fr

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

3. © 2011Frank Nielsen
ICP is a generic method
Example for curve registrations:
Curves before registration
Curves after registration

4. © 2011Frank 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:

5. © 2011Frank Nielsen
Align the center of mass of sets:

6. © 2011Frank Nielsen
Finding the rotation matrix:
Compute the singular value decomposition
Optimal transformation:
Recover the translation
once optimal rotation is found
Cross-covariance matrix:

7. © 2011Frank Nielsen
ICP: Monotonicity and convergence
The average squared Euclidean distance
decreases monotonously (→ convergence)
In fact (stronger result):
Each correspondence pair distance decreases
Drawback:
When does the local minimum is global?
Difficult to handle symmetry
(use texture, etc. to disambiguate)

8. © 2011Frank Nielsen
Best 3D transformation (with quaternions)
With respect to least squares...
SVD take into account reflections...
Minimize:

9. © 2011Frank Nielsen
Best 3D transformation (with quaternions)
Cross-covariance matrix:
Anti-symmetric matrix

10. © 2011Frank Nielsen
Best 3D transformation (with quaternions)
Take the unit eigenvector corresponding to the maximal eigenvalue:
Get the remaining translation (easy) as:

11. © 2011Frank Nielsen
Time complexity of ICP
Linear (fixed dimension) to find least square transformation
At each iteration, perform n nearest neighbor queries
Naive implementation: O(l*n*n) => slow for large n
http://meshlab.sourceforge.net/

12. © 2011Frank Nielsen
kD-trees for fast NN queries
Nearest neighbor (NN) queries in small dimensions...
http://en.wikipedia.org/wiki/Kd-tree

13. © 2011Frank Nielsen
Build time: O(dn log n) with O(dn) memory

14. © 2011Frank Nielsen

15. © 2011Frank Nielsen
Query complexity: From O(dlog n) to O(dn)

16. © 2011Frank Nielsen
kD-trees for fast NN queries
Kd-Trees are extremely useful data-structures (many applications)
But also: Approximate nearest neighbors
http://www.cs.umd.edu/~mount/ANN/

17. © 2011Frank Nielsen
http://donar.umiacs.umd.edu/quadtree/points/kdtree.html

18. © 2011Frank Nielsen
Homography (also called collineation)

19. © 2011Frank Nielsen
Homography (Collineation)
Assuming h33 is not zero, set it to 1 and get:

20. © 2011Frank Nielsen
Homography (Collineation)

21. © 2011Frank Nielsen
Homography (Collineation)
From 4 pairs of point correspondences:

22. © 2011Frank Nielsen
Homography (Collineation)
From n pairs of point correspondences:
Matrix pseudo-inverse

23. © 2011Frank Nielsen

24. © 2011Frank Nielsen
Homography (Collineation)
Matching planar surfaces...

25. © 2011Frank Nielsen
Homography (Collineation)
Matching perspective pictures acquired
from the same nodal point

26. © 2011Frank Nielsen
Homography (Collineation): Projective geometry
Play with duality point/line in projective space
It is more stable to select four pairs of lines instead of four points

27. © 2011Frank Nielsen
Graphics pipeline

28. © 2011Frank Nielsen
Graphics pipeline: Scene graph
Scaled rigid transformation:
High level API is Java3D:
https://java3d.dev.java.net/
OpenGL in Processing

29. © 2011Frank Nielsen
Graphics pipeline: Projection
Projections are irreversible transformations
Orthographic projection

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Perspective projection: Pinhole camera

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Graphics pipeline: Perspective projection

32. © 2011Frank Nielsen
Graphics pipeline: Perspective projection

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Graphics pipeline:
Perspective truncated pyramid
Perspective frustrum

34. © 2011Frank Nielsen
Graphics pipeline: Several viewports

35. © 2011Frank Nielsen
Graphics pipeline (matrix concatenation):
Mv: positionning transformation
V: viewing transformation
C: clipping transformation
Mv: viewing transformation
Mv: positionning transformation
V: viewing transformation
C: clipping transformation
Mv: viewing transformation

36. © 2011Frank Nielsen
OpenGL in Java: Processing

37. © 2011Frank Nielsen
public void display(GLAutoDrawable gLDrawable) {
final GL gl = gLDrawable.getGL();
gl.glClear(GL.GL_COLOR_BUFFER_BIT | GL.GL_DEPTH_BUFFER_BIT);
gl.glLoadIdentity();
gl.glTranslatef(-1.5f, 0.0f, -6.0f);
gl.glBegin(GL.GL_TRIANGLES); // Drawing Using Triangles
gl.glVertex3f(0.0f, 1.0f, 0.0f); // Top
gl.glVertex3f(-1.0f, -1.0f, 0.0f); // Bottom Left
gl.glVertex3f(1.0f, -1.0f, 0.0f); // Bottom Right
gl.glEnd(); // Finished Drawing The Triangle
gl.glTranslatef(3.0f, 0.0f, 0.0f);
gl.glBegin(GL.GL_QUADS); // Draw A Quad
gl.glVertex3f(-1.0f, 1.0f, 0.0f); // Top Left
gl.glVertex3f(1.0f, 1.0f, 0.0f); // Top Right
gl.glVertex3f(1.0f, -1.0f, 0.0f); // Bottom Right
gl.glVertex3f(-1.0f, -1.0f, 0.0f); // Bottom Left
gl.glEnd(); // Done Drawing The Quad
gl.glFlush();
}

38. © 2011Frank Nielsen
Processing: 3D color shapes
GLU (Utility)
GLUT (Utility Toolkit, including user interfaces.)
http://www.cs.umd.edu/~meesh/kmconroy/JOGLTutorial/