(slides 4) Visual Computing: Geometry, Graphics, and Vision
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The document presents an overview of the Iterative Closest Point (ICP) algorithm, detailing its purpose in 3D shape registration and optimization for matching point correspondences. It discusses key aspects such as finding the optimal transformation, convergence properties, and the challenges of local minima. Additionally, the document covers related topics like KD-trees for efficient nearest neighbor queries and aspects of the graphics pipeline, including homography and perspective projections.
(slides 4) Visual Computing: Geometry, Graphics, and Vision
- 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
- 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/
- 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
- 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
- 30. © 2011Frank Nielsen Perspective projection: Pinhole camera
- 31. © 2011Frank Nielsen Graphics pipeline: Perspective projection
- 32. © 2011Frank Nielsen Graphics pipeline: Perspective projection
- 33. © 2011Frank Nielsen 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/