CSE 554 Lecture 6: Fairing and Simplification

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CSE 554 Lecture 6: Fairing and Simplification Fall 2012 CSE554 Fairing and simplification Slide 1 Review Iso-contours in grayscale images and volumes Piece-wise linear representations Polylines (2D)

1 CSE 554 Lecture 6: Fairing and Simplification Fall 2012 CSE554 Fairing and simplification Slide 1 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 CSE554 Fairing and simplification Slide 2 1

and Vectors Same representation x, y or x, y, z 2 Y p 1, 2 p 2, 2 v 1, 2 v 1, 2 Different meaning: 1 2 x Point: a fixed location

2 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 CSE554 Fairing and simplification Slide 3 Points and Vectors Same representation x, y or x, y, z 2 Y p 1, 2 p 2, 2 v 1, 2 v 1, 2 Different meaning: 1 2 x Point: a fixed location (relative to {0,0} or {0,0,0}) Vector: a direction and magnitude No location (any location is possible) CSE554 Fairing and simplification Slide 4 2

3 Point Operations Subtraction Result is a vector p 2 p 1 v p 2 x p 1x,p 2y p 1y Addition with a vector v p 2 Result is a point p 1 v p 2 p 1 x v x,p 1 y v y p 1 Can points add? Not yet CSE554 Fairing and simplification Slide 5 Vector Operations Addition/Subtraction Result is a vector v 1 v 2 v 1 x v 2x,v 1y v 2y v 2 Scaling by a scalar Result is a vector s v s v x,s v y Magnitude Result is a scalar v v 2 2 x v y v 1 v v 1 s* v v 2 A unit vector: v 1 To make a unit vector (normalization): v v CSE554 Fairing and simplification Slide 6 3

4 Adding Points Affine combinations Weighted addition of points where all weights sum to 1 n w i p i p, n where w i 1 p 1 v 2 p 2 Result is another point Same as adding scaled vectors to a point v 3 p v4 n n w i p i n n w i p i p 1 w i v i p 1 w i p 1 p 3 p 4 CSE554 Fairing and simplification Slide 7 Adding Points Affine combinations: examples Mid-point of two points p p 1 p 2 2 Linear interpolation of two points p 1 p 1 p 2 p p 1 p 2 p 1 p 1 p 2 Centroid of multiple points n p p i n p 1 p 2 p p 3 p 4 CSE554 Fairing and simplification Slide 8 4

5 Geometry Processing Fairing (smoothing) Relocating vertices to achieve a smoother appearance Simplification Reducing vertex count CSE554 Fairing and simplification Slide 9 Fairing (2D) Reducing bumpiness by changing the vertex locations CSE554 Fairing and simplification Slide 10 5

6 Fairing (2D) What is a bump? A vertex far from the mid-point of its two neighbors A big bump A small bump CSE554 Fairing and simplification Slide 11 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 Iterative fairing Controls how far p moves away from p At each iteration, update all vertices using locations in the previous iteration A p' 1 p p 1 p 2 close to 1 will create oscillation Typically p 1 p' p 1 p 2 p 1 p 2 2 CSE554 Fairing and simplification Slide 12 6

7 Fairing (2D) Drawback The initial shape is shrunk! 100 iterations 200 iterations 400 iterations CSE554 Fairing and simplification Slide 13 Fairing (2D) Non-shrinking mid-point averaging [Taubin 1995] Alternate between two kinds of iterations with different Odd iterations: Odd 0.5 (positive) Shrinking the shape Even iterations: Even 1 1 : typically 0.1 Odd (negative) Expanding the shape CSE554 Fairing and simplification Slide 14 7

8 Fairing (2D) The initial shape is no longer shrunk The result converges with increasing iterations 100 iterations 200 iterations 400 iterations CSE554 Fairing and simplification Slide 15 Fairing (3D) Fairing by centroid averaging Moving each vertex towards the centroid of its edge-adjacent neighbors (called the 1-ring neighbors) Linear interpolation p 1 p 5 m p i p' 1 p m centroid p 2 p p 4 Iterative, non-shrinking fairing Alternate between shrinking and expanding p 3 Same choices of as in 2D Each iteration updates all vertices using locations in the previous iteration CSE554 Fairing and simplification Slide 16 8

9 Fairing (3D) Example: fairing iso-surface of a binary volume CSE554 Fairing and simplification Slide 17 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. CSE554 Fairing and simplification Slide 18 9

10 More Vector Operations Dot product (in both 2D and 3D) Result is a scalar v 1 v 2 v 1 v 2 Cos In coordinates (simple!) v 1 a 2D: 3D: v 1 v 2 v 1 x v 2x v 1y v 2y v 1 v 2 v 1 x v 2x v 1y v 2y v 1z v 2z v 2 Matrix product between a row and a column vector CSE554 Fairing and simplification Slide 19 More Vector Operations Uses of dot products Angle between vectors: ArcCos v 1 v 2 v 1 v 2 v 1 a Orthogonal: v 1 v 2 0 v 2 Projected length of v 1 onto v 2 : h v 1 v 2 v 2 v 1 h v 2 CSE554 Fairing and simplification Slide 20 10

11 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: v 1 v 2 v 1 v 2 Sin In coordinates: v 1 v 2 v 1 y v 2z v 1z v 2y,v 1z v 2x v 1x v 2z,v 1x v 2y v 1y v 2x v1 v2 a v 2 v 1 CSE554 Fairing and simplification Slide 21 More Vector Operations Uses of cross products Getting the normal vector of the plane E.g., the normal of a triangle formed by v 1 v 2 Computing area of the triangle formed by v 1 v 2 v 1 v 2 Area 2 v1 v2 v 2 Testing if vectors are parallel: v 1 v 2 0 v 1 CSE554 Fairing and simplification Slide 22 11

12 More Vector Operations Dot Product Cross Product Distributive? v v 1 v 2 v v 1 v v 2 v v 1 v 2 v v 1 v v 2 Commutative? v 1 v 2 v 2 v 1 v 1 v 2 v 2 v 1 (Sign change!) Associative? v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 CSE554 Fairing and simplification Slide 23 Geometry Processing Fairing (smoothing) Relocating vertices to achieve a smoother appearance Simplification Reducing vertex count CSE554 Fairing and simplification Slide 24 12

13 Simplification (2D) Representing the shape with fewer vertices (and edges) 200 vertices 50 vertices CSE554 Fairing and simplification Slide 25 Simplification (2D) If I want to replace two vertices with one, where should it be? CSE554 Fairing and simplification Slide 26 13

14 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: CSE554 Fairing and simplification Slide 27 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: p q n Projection of vector (p-q) onto n This distance has a sign Above or under of the line n p We will use the distance squared Line q CSE554 Fairing and simplification Slide 28 14

15 Simplification (2D) Closed point to multiple lines Sum of squared distances from p to all lines (Quadratic Error Metric, QEM) Input lines: q 1, n 1,..., q m, n m m QEM p p q i n i 2 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 QEM p p 0 CSE554 Fairing and simplification Slide 29 Simplification (2D) Minimizing QEM Writing QEM in matrix form m QEM p p q i n i 2 p p x p y Row vector Matrix transpose QEM p p a p T 2p b c [Eq. 1] Matrix (dot) product a m n i x n ix m n i x n iy m n i x n iy n i y n iy m b m n i x n i q i m n i y n i q i m c n i q i 2 2x2 matrix 1x2 column vector Scalar CSE554 Fairing and simplification Slide 30 15

16 Simplification (2D) Minimizing QEM QEM p p a p T 2p b c Solving the zero-derivative equation: QEM p p 2 a p T 2 b 0 a p T b [Eq. 2] m n i x n ix n i x n iy m m n i x n iy n i y n iy m p x p y m n i x n i q i n i y n i q i m A linear system with 2 equations and 2 unknowns (p x,p y ) Using Gaussian elimination, or matrix inversion: p T a 1 b CSE554 Fairing and simplification Slide 31 Simplification (2D) What vertices to merge first? Pick the ones that lie on flat regions, or whose replacing vertex introduces least QEM error. CSE554 Fairing and simplification Slide 32 16

17 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. CSE554 Fairing and simplification Slide 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. 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. CSE554 Fairing and simplification Slide 34 17

18 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. CSE554 Fairing and simplification Slide 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. Step 3: Repeat step 2, until a desired number of vertices is left. CSE554 Fairing and simplification Slide 36 18

19 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. CSE554 Fairing and simplification Slide 37 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 p QEM coefficients (a, b, c) Used for fast update in Step 2 Stored at the edge: a, b, c, p, QEM p CSE554 Fairing and simplification Slide 38 19

20 Simplification (2D) Step 2: Collapsing an edge Remove the edge and its vertices a 1, b 1, c 1, p 1, QEM p 1 a, b, c, p, QEM p a 2, b 2, c 2, p 2,QEM p 2 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 a a 1, b b 1, c c 1, p 1,QEM p 1 p Collapse a a 2, b b 2, c c 2, p 2,QEM p 2 p 1,p 2 : new minimizer locations computed from the updated coefficients CSE554 Fairing and simplification Slide 39 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 CSE554 Fairing and simplification Slide 40 20

21 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: p q n Projection of vector (p-q) onto n This distance has a sign n above or below the plane p We use its square q CSE554 Fairing and simplification Slide 41 Simplification (3D) Closest point to multiple planes Input planes: q 1,n 1,..., q m, n m QEM (same as in 2D) m QEM p p q i n i 2 In matrix form: p p x p y p z QEM p p a p T 2p b c a b m c m n i x n ix n i y n ix m m n i z n ix m n i x n iy n i y n iy m m m n i x n i q i n i y n i q i m m n i z n i q i n i q i 2 n i z n iy 3x3 matrix 1x3 column vector Scalar m n i x n iz n i y n iz m m n i z n iz Find p that minimizes QEM: a p T b A linear system with 3 equations and 3 unknowns (p x,p y,p z ) CSE554 Fairing and simplification Slide 42 21

22 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. CSE554 Fairing and simplification Slide 43 Simplification (3D) Step 2: Collapsing an edge Degenerate triangles after collapse 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 CSE554 Fairing and simplification Slide 44 22

Simplification (3D) Example: 5600 vertices 500 vertices CSE554 Fairing and simplification Slide 45 Further Readings Fairing: A signal processing approach to fair surface design, by G.

23 Simplification (3D) Example: 5600 vertices 500 vertices CSE554 Fairing and simplification Slide 45 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 CSE554 Fairing and simplification Slide 46 23

Applications. Oversampled 3D scan data. ~150k triangles ~80k triangles

Applications. Oversampled 3D scan data. ~150k triangles ~80k triangles Mesh Simplification Applications Oversampled 3D scan data ~150k triangles ~80k triangles 2 Applications Overtessellation: E.g. iso-surface extraction 3 Applications Multi-resolution hierarchies for efficient