CSE 554 Lecture 6: Fairing

1. CSE 554Lecture 6: Fairing Fall 2013

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. Image Processing • Filtering • Changing pixel values to blur the contents

5. Image As Signals (slides materials courtesy of Andy van Dam)

6. Down sampling • Reconstructing the same signal using fewer samples • How many samples do we need for a “good” reconstruction? (slides materials courtesy of Andy van Dam)

7. Waveform Analysis • A signal is the sum of phase-shifted sine curves at various scales (frequencies) (slides materials courtesy of Andy van Dam)

8. Frequency Domain • The decomposition of a signal into sines at different frequencies is unique • Can be plotted as a frequency spectrum (Fourier transform) (slides materials courtesy of Andy van Dam)

9. The Nyquist Limit • The sampling frequency cannot be lower than twice the highest frequency in the signal Signal: Sampling: (above Nyquist limit) Reconstruction: (slides materials courtesy of Andy van Dam)

10. The Nyquist Limit • The sampling frequency cannot be lower than twice the highest frequency in the signal Signal: Sampling: (under Nyquist limit) Reconstruction: (slides materials courtesy of Andy van Dam)

11. The Nyquist Limit • The sampling frequency cannot be lower than twice the highest frequency in the signal Signal: Sampling 1: (at Nyquist limit) Sampling 2: (at Nyquist limit) (slides materials courtesy of Andy van Dam)

12. Down sampling • Reconstructing the same signal using fewer samples • How many samples do we need for a “good” reconstruction? • At least twice the highest frequency • What if we don’t have enough samples? (slides materials courtesy of Andy van Dam)

13. Image Filtering • “Filter” the image to remove high frequencies, then sample. (slides materials courtesy of Andy van Dam)

14. Image Filtering • Low-pass filtering: a “box” filter in the frequency domain (slides materials courtesy of Andy van Dam)

15. Convolution • Continuous convolution • Given two functions, f and w • w is symmetric: w[x] = w[-x] • (f*w)[y] is: • Shift w so that it’s centered at y • Multiply f with (shifted) w • Find the area under the multiplied curve • Discrete convolution • f and w are two lists:

16. Image Filtering • Low-pass filtering: convolution with Sinc[x] • Removes all frequencies greater than 1/2

17. Image Filtering • Gaussian filtering: convolution with Gaussian function • Kernel size δ (standard deviation) controls the amount of smoothing • Repeated filtering using a small kernel is equivalent to a single filtering with a large kernel

18. Image Filtering • Discrete Gaussian filtering with kernel size 1 • Averaging by weights {1/4,1/2,/14} • Intuitively: the signal at a point is moved to half-way between the signal and the average signals of the two neighboring points. i-1 i i+1

19. Curve Fairing • Reducing “bumpiness” by changing the vertex locations

20. Curve Fairing • Curve as a signal • The “signal” at each vertex is its location {x,y}

21. 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

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

23. 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):

24. 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

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

26. Curve Fairing • 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

27. Curve Fairing • Drawback • The initial shape is shrunk! 100 iterations 200 iterations 400 iterations

28. Curve Fairing • 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

29. Curve Fairing • The initial shape is no longer shrunk • The result converges with increasing iterations 100 iterations 200 iterations 400 iterations

30. Surface Fairing • 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

31. Surface Fairing • Example: fairing iso-surface of a binary volume

32. 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.