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(slides 3) Visual Computing: Geometry, Graphics, and Vision

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Frank Nielsen
Frank Nielsen

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

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© 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
© 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
© 2011 Frank Nielsen
© 2011 Frank Nielsen Harris-Stephens edge detector Aim at finding good feature
© 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
© 2011 Frank Nielsen
© 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
© 2011 Frank Nielsen Edge hysteresis
© 2011 Frank Nielsen Homogeneous coordinates and duality point/line homogenization Homogeneous vectorInhomogeneous vector dehomogenization (also known as Perspective division)
© 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
© 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
© 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
© 2011 Frank Nielsen Overview of duality in projective geometry The determinant of three points represent the volume of their parallepiped.
© 2011 Frank Nielsen 2D Transformations using homogeneous coordinates
© 2011 Frank Nielsen 2D Transformations using homogeneous coordinates
© 2011 Frank Nielsen Cartesian coordinate systems in 3D
© 2011 Frank Nielsen 3D Transformations using homogeneous coordinates Be careful: Gimbal lock
© 2011 Frank Nielsen Euler rotation
© 2011 Frank Nielsen Cross-product/outer product Consider the cross-product as a matrix multiplication: Outer-product
© 2011 Frank Nielsen Arbitrary matrix rotation: Rodrigues' formula Equivalent to:
© 2011 Frank Nielsen
© 2011 Frank Nielsen
© 2011 Frank Nielsen Rigid transformations Concatenation (non-commutative!)
© 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/
© 2011 Frank Nielsen Quaternions: 1D real+3D imaginary Real part (1D) Imaginary part (3D, i j k vectors) Multiplication: Norm (l2)
© 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
© 2011 Frank Nielsen Unit quaternions for rotations
© 2011 Frank Nielsen Conversion rotation matrix to quaternion
© 2011 Frank Nielsen Spherical linear interpolation (SLERP) LERP is non-sense for rotation matrices: SLERP is using quaternion algebra:
© 2011 Frank Nielsen Spherical linear interpolation (SLERP) Useful for computer graphics animation (bone, skinning at articulation)
© 2011 Frank Nielsen Bilateral filtering Edge-preserving smoothing Videos/demo
© 2011 Frank Nielsen Gaussian filtering: Blur everything Traditional spatial gaussian filtering
© 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...
© 2011 Frank Nielsen Domain filtering Range filtering
© 2011 Frank Nielsen Iterative Closest Point (ICP) Align point sets. For example, terrains (DEMs) Robotics
© 2011 Frank Nielsen Align point sets obtained from range scanners http://www-graphics.stanford.edu/projects/mich/
© 2011 Frank Nielsen Solve stone jigsaws... ICP for solving jigsaws
© 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
© 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:
© 2011 Frank Nielsen Align the center of mass of sets:
© 2011 Frank Nielsen Finding the rotation matrix: Compute the singular value decomposition Optimal transformation:
© 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.)
© 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
© 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
© 2011 Frank Nielsen Installez Processing.org → Rendu des TD1, 2 et 3 pour le Mardi 4 Octobre 17h (Upload work sur la page du cours)

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(slides 3) Visual Computing: Geometry, Graphics, and Vision

  • 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 Installez Processing.org → Rendu des TD1, 2 et 3 pour le Mardi 4 Octobre 17h (Upload work sur la page du cours)