1. CSE554Fairing and simplificationSlide 1 CSE 554 Lecture 6: Fairing and Simplification Fall 2012

2. CSE554Fairing and simplificationSlide 2 Review Iso-contours in grayscale images and volumes – Piece-wise linear representations Polylines (2D) and meshes (3D) – Primal and dual methods Marching Squares (2D) and Cubes (3D) Dual Contouring (2D,3D) – Acceleration using trees Quadtree (2D), Octree (3D) Interval trees

3. CSE554Fairing and simplificationSlide 3 Geometry Processing Fairing (smoothing) – Relocating vertices to achieve a smoother appearance Simplification – Reducing vertex count Deformation – Relocating vertices guided by user interaction or to fit onto a target

4. CSE554Fairing and simplificationSlide 4 Same representation Different meaning: – Point: a fixed location (relative to {0,0} or {0,0,0}) – Vector: a direction and magnitude No location (any location is possible) Points and Vectors x Y 1 2 2

5. CSE554Fairing and simplificationSlide 5 Subtraction – Result is a vector Addition with a vector – Result is a point Can points add? – Not yet… Point Operations

6. CSE554Fairing and simplificationSlide 6 Addition/Subtraction – Result is a vector Scaling by a scalar – Result is a vector Magnitude – Result is a scalar A unit vector: To make a unit vector (normalization): Vector Operations

7. CSE554Fairing and simplificationSlide 7 Adding Points Affine combinations – Weighted addition of points where all weights sum to 1 – Result is another point Same as adding scaled vectors to a point

8. CSE554Fairing and simplificationSlide 8 Adding Points Affine combinations: examples – Mid-point of two points – Linear interpolation of two points – Centroid of multiple points

9. CSE554Fairing and simplificationSlide 9 Geometry Processing Fairing (smoothing) – Relocating vertices to achieve a smoother appearance Simplification – Reducing vertex count

10. CSE554Fairing and simplificationSlide 10 Fairing (2D) Reducing “bumpiness” by changing the vertex locations

11. CSE554Fairing and simplificationSlide 11 Fairing (2D) What is a bump? – A vertex far from the mid-point of its two neighbors A big bumpA small bump

12. CSE554Fairing and simplificationSlide 12 Fairing (2D) Fairing by mid-point averaging – Moving each vertex towards the mid-point of its two neighbors Using linear interpolation : some value between 0 and 1 – Controls how far p’ moves away from p – Iterative fairing At each iteration, update all vertices using locations in the previous iteration A close to 1 will create oscillation – Typically

13. CSE554Fairing and simplificationSlide 13 Fairing (2D) Drawback – The initial shape is shrunk! 100 iterations200 iterations400 iterations

14. CSE554Fairing and simplificationSlide 14 Fairing (2D) Non-shrinking mid-point averaging [Taubin 1995] – Alternate between two kinds of iterations with different Odd iterations: (positive) – Shrinking the shape Even iterations: (negative) – : typically 0.1 – Expanding the shape

15. CSE554Fairing and simplificationSlide 15 Fairing (2D) The initial shape is no longer shrunk – The result converges with increasing iterations 100 iterations200 iterations400 iterations

16. CSE554Fairing and simplificationSlide 16 Fairing (3D) Fairing by centroid averaging – Moving each vertex towards the centroid of its edge-adjacent neighbors (called the 1-ring neighbors) Linear interpolation – Iterative, non-shrinking fairing Alternate between shrinking and expanding – Same choices of as in 2D Each iteration updates all vertices using locations in the previous iteration centroid

17. CSE554Fairing and simplificationSlide 17 Fairing (3D) Example: fairing iso-surface of a binary volume

18. CSE554Fairing and simplificationSlide 18 Fairing Implementation Tips – At each iteration, keep two copies of locations of all vertices Store the smoothed location of each vertex in another list separate from the current locations – Building an adjacency table storing the neighbors of each vertex would be helpful, but not necessary Initialize the centroid as {0,0,0} at each vertex, and its neighbor count as 0. For each triangle, add the coordinates of each vertex to the centroids stored at the other two vertices and increment their neighbor count. – The neighbor count is twice the actual # of edge neighbors For each vertex, divide the centroid by its neighbor count.

19. CSE554Fairing and simplificationSlide 19 Dot product (in both 2D and 3D) – Result is a scalar – In coordinates (simple!) 2D: 3D: Matrix product between a row and a column vector More Vector Operations

20. CSE554Fairing and simplificationSlide 20 Uses of dot products – Angle between vectors: Orthogonal: – Projected length of onto : More Vector Operations h

21. CSE554Fairing and simplificationSlide 21 Cross product (only in 3D) – Result is another 3D vector Direction: Normal to the plane where both vectors lie (right-hand rule) Magnitude: – In coordinates: More Vector Operations

22. CSE554Fairing and simplificationSlide 22 More Vector Operations Uses of cross products – Getting the normal vector of the plane E.g., the normal of a triangle formed by – Computing area of the triangle formed by Testing if vectors are parallel:

23. CSE554Fairing and simplificationSlide 23 More Vector Operations Dot ProductCross Product Distributive? Commutative? Associative? (Sign change!)

24. CSE554Fairing and simplificationSlide 24 Geometry Processing Fairing (smoothing) – Relocating vertices to achieve a smoother appearance Simplification – Reducing vertex count

25. CSE554Fairing and simplificationSlide 25 Simplification (2D) Representing the shape with fewer vertices (and edges) 200 vertices50 vertices

26. CSE554Fairing and simplificationSlide 26 Simplification (2D) If I want to replace two vertices with one, where should it be?

27. CSE554Fairing and simplificationSlide 27 Simplification (2D) If I want to replace two vertices with one, where should it be? – Shortest distances to the supporting lines of involved edges After replacement:

28. CSE554Fairing and simplificationSlide 28 Simplification (2D) Distance to a line – Line represented as a point q on the line, and a perpendicular unit vector (the normal) n To get n: take a vector {x,y} along the line, n is {-y,x} followed by normalization – Distance from any point p to the line: Projection of vector (p-q) onto n – This distance has a sign “Above” or “under” of the line We will use the distance squared Line

29. CSE554Fairing and simplificationSlide 29 Simplification (2D) Closed point to multiple lines – Sum of squared distances from p to all lines (Quadratic Error Metric, QEM) Input lines: – We want to find the p with the minimum QEM Since QEM is a convex quadratic function of p, the minimizing p is where the derivative of QEM is zero, which is a linear equation

30. CSE554Fairing and simplificationSlide 30 Simplification (2D) Minimizing QEM – Writing QEM in matrix form 2x2 matrix1x2 column vectorScalar [Eq. 1] Matrix (dot) product Row vector Matrix transpose

31. CSE554Fairing and simplificationSlide 31 Simplification (2D) Minimizing QEM – Solving the zero-derivative equation: – A linear system with 2 equations and 2 unknowns (p x,p y ) Using Gaussian elimination, or matrix inversion: [Eq. 2]

32. CSE554Fairing and simplificationSlide 32 Simplification (2D) What vertices to merge first? – Pick the ones that lie on “flat” regions, or whose replacing vertex introduces least QEM error. 

33. CSE554Fairing and simplificationSlide 33 Simplification (2D) The algorithm – Step 1: For each edge, compute the best vertex location to replace that edge, and the QEM at that location. Store that location (called minimizer) and its QEM with the edge.

34. CSE554Fairing and simplificationSlide 34 Simplification (2D) The algorithm – Step 1: For each edge, compute the best vertex location to replace that edge, and the QEM at that location. Store that location (called minimizer) and its QEM with the edge. – Step 2: Pick the edge with the lowest QEM and collapse it to its minimizer. Update the minimizers and QEMs of the re-connected edges.

35. CSE554Fairing and simplificationSlide 35 Simplification (2D) The algorithm – Step 1: For each edge, compute the best vertex location to replace that edge, and the QEM at that location. Store that location (called minimizer) and its QEM with the edge. – Step 2: Pick the edge with the lowest QEM and collapse it to its minimizer. Update the minimizers and QEMs of the re-connected edges.

36. CSE554Fairing and simplificationSlide 36 Simplification (2D) The algorithm – Step 1: For each edge, compute the best vertex location to replace that edge, and the QEM at that location. Store that location (called minimizer) and its QEM with the edge. – Step 2: Pick the edge with the lowest QEM and collapse it to its minimizer. Update the minimizers and QEMs of the re-connected edges. – Step 3: Repeat step 2, until a desired number of vertices is left.

37. CSE554Fairing and simplificationSlide 37 Simplification (2D) The algorithm – Step 1: For each edge, compute the best vertex location to replace that edge, and the QEM at that location. Store that location (called minimizer) and its QEM with the edge. – Step 2: Pick the edge with the lowest QEM and collapse it to its minimizer. Update the minimizers and QEMs of the re-connected edges. – Step 3: Repeat step 2, until a desired number of vertices is left.

38. CSE554Fairing and simplificationSlide 38 Simplification (2D) Step 1: Computing minimizer and QEM on an edge – Consider supporting lines of this edge and adjacent edges – Compute and store at the edge: The minimizing location p (Eq. 2) QEM (substitute p into Eq. 1) – Used for edge selection in Step 2 QEM coefficients (a, b, c) – Used for fast update in Step 2 Stored at the edge:

39. CSE554Fairing and simplificationSlide 39 Simplification (2D) Step 2: Collapsing an edge – Remove the edge and its vertices – Re-connect two neighbor edges to the minimizer of the removed edge – For each re-connected edge: Increment its coefficients by that of the removed edge – The coefficients are additive! Re-compute its minimizer and QEM Collapse : new minimizer locations computed from the updated coefficients

40. CSE554Fairing and simplificationSlide 40 Simplification (3D) The algorithm is similar to 2D – Replace two edge-adjacent vertices by one vertex Placing new vertices closest to supporting planes of adjacent triangles – Prioritize collapses based on QEM

41. CSE554Fairing and simplificationSlide 41 Simplification (3D) Distance to a plane (similar to the line case) – Plane represented as a point q on the plane, and a unit normal vector n For a triangle: n is the cross-product of two edge vectors – Distance from any point p to the plane: Projection of vector (p-q) onto n – This distance has a sign “above” or “below” the plane We use its square

42. CSE554Fairing and simplificationSlide 42 Simplification (3D) Closest point to multiple planes – Input planes: – QEM (same as in 2D) In matrix form: – Find p that minimizes QEM: A linear system with 3 equations and 3 unknowns (p x,p y,p z ) 3x3 matrix 1x3 column vector Scalar

43. CSE554Fairing and simplificationSlide 43 Simplification (3D) Step 1: Computing minimizer and QEM on an edge – Consider supporting planes of all triangles adjacent to the edge – Compute and store at the edge: The minimizing location p QEM[p] QEM coefficients (a, b, c) The supporting planes for all shaded triangles should be considered when computing the minimizer of the middle edge.

44. CSE554Fairing and simplificationSlide 44 Simplification (3D) Step 2: Collapsing an edge – Remove the edge with least QEM – Re-connect neighbor triangles and edges to the minimizer of the removed edge Remove “degenerate” triangles Remove “duplicate” edges – For each re-connected edge: Increment its coefficients by that of the removed edge Re-compute its minimizer and QEM Collapse Degenerate triangles after collapse Duplicate edges after collapse

45. CSE554Fairing and simplificationSlide 45 Simplification (3D) Example: 5600 vertices500 vertices

46. CSE554Fairing and simplificationSlide 46 Further Readings Fairing: – “A signal processing approach to fair surface design”, by G. Taubin (1995) No-shrinking centroid-averaging Google citations > 1000 Simplification: – “Surface simplification using quadric error metrics”, by M. Garland and P. Heckbert (1997) Edge-collapse simplification Google citations > 2000