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Fairing and Simplification in Geometry Processing

This lecture covers the concepts of fairing (smoothing) and simplification in geometry processing, including techniques for reducing bumpiness, relocating vertices, and reducing vertex count. It also explores operations on points and vectors and their applications in fairing and simplification.

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Fairing and Simplification in Geometry Processing

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

  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. 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. Points and Vectors • 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) Y 2 x 1 2

  5. Point Operations • Subtraction • Result is a vector • Addition with a vector • Result is a point • Can points add? • Not yet…

  6. Vector Operations • 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):

  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. Adding Points • Affine combinations: examples • Mid-point of two points • Linear interpolation of two points • Centroid of multiple points

  9. Geometry Processing • Fairing (smoothing) • Relocating vertices to achieve a smoother appearance • Simplification • Reducing vertex count

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

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

  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. Fairing (2D) • Drawback • The initial shape is shrunk! 100 iterations 200 iterations 400 iterations

  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. Fairing (2D) • The initial shape is no longer shrunk • The result converges with increasing iterations 100 iterations 200 iterations 400 iterations

  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. Fairing (3D) • Example: fairing iso-surface of a binary volume

  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. More Vector Operations • 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

  20. h More Vector Operations • Uses of dot products • Angle between vectors: • Orthogonal: • Projected length of onto :

  21. More Vector Operations • 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:

  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. More Vector Operations (Sign change!)

  24. Geometry Processing • Fairing (smoothing) • Relocating vertices to achieve a smoother appearance • Simplification • Reducing vertex count

  25. Simplification (2D) • Representing the shape with fewer vertices (and edges) 200 vertices 50 vertices

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

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

  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. 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 convexquadratic function of p, the minimizing p is where the derivative of QEM is zero, which is a linear equation

  30. Row vector Matrix transpose [Eq. 1] Matrix (dot) product 2x2 matrix 1x2 column vector Scalar Simplification (2D) • Minimizing QEM • Writing QEM in matrix form

  31. Simplification (2D) • Minimizing QEM • Solving the zero-derivative equation: • A linear system with 2 equations and 2 unknowns (px,py) • Using Gaussian elimination, or matrix inversion: [Eq. 2]

  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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 3x3 matrix 1x3 column vector Scalar 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 (px,py,pz)

  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. Simplification (3D) Degenerate triangles after collapse • 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 Duplicate edges after collapse Collapse

  45. Simplification (3D) • Example: 5600 vertices 500 vertices

  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

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