Non-Iterative, Feature-Preserving Mesh Smoothing

Transcription

1 Non-Iterative, Feature-Preserving Mesh Smoothing Thouis R. Jones (MIT), Frédo Durand (MIT), Mathieu Desbrun (USC)

2 3D scanners are noisy... Why Smooth?

3 3D scanners are noisy... and have dropouts... Why Smooth?

4 Why Smooth? 3D scanners are noisy... and have dropouts... and usually require multiple scans.

5 Goals Fast smoothing of meshes Robust Geometrically: preserve features Topologically: no connectivity information Simple to implement

6 Goals Fast smoothing of meshes polygon soups Robust Geometrically: preserve features Topologically: no connectivity information Simple to implement

7 Fast Mesh Smoothing Previous Work on Smoothing Taubin 1995; Desbrun et al Feature Preserving Clarenz et al. 2000; Desbrun et al. 2000; Meyer et al. 2002; Zhang and Fiume 2002; Bajaj and Xu 2003 Diffusion on Normal Field Taubin 2001; Belyaev and Ohtake 2001; Ohtake et al. 2002; Tasdizen et al Wiener Filtering of Meshes Peng et al. 2001; Alexa 2002; Pauly and Gross 2001 (points)

8 Approach We cast feature-preserving filtering as a robust estimation problem on vertex positions. Extend Bilateral Filter to 3D. Smith and Brady 1997; 1998 Tomasi and Manduchi Use first-order predictors based on facets of model. Single pass.

9 Least Squares Error Norm Non-Robust Estimation 4 3 y x Outliers have unlimited influence on estimate.

10 Robust Error Norm 0.3 Robust Estimation y x Outliers have bounded influence on estimate.

11 Gaussian Filter (Non-robust) I s = p image I(p) spatial f(s p) I f I

12 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) I f g fg I

13 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) I f g fg I

14 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) I f g fg I

15 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) I f g fg I

16 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) I f g fg I

17 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) I f g fg I

18 Bilateral Filter (Robust) I s = 1 k s p image I(p) spatial f(s p) influence g(i s I p ) k s = p f(s p) g(i s I p )

19 Bilateral Filter Left: Jones and Jones 2003 Right: Bilaterally filtered.

20 Extending the Bilateral Filter to Meshes How to separate location and signal in a 3D model? Forming local frames requires a connected mesh. Instead, use first-order predictors based on facets: Π q( p) p No connectivity required between facets. q

21 Bilateral Filter for Meshes Estimate p, the new position for a vertex p p = 1 k(p) q S prediction Π q (p) spatial f( c q p ) influence g( Π q (p) p ) area a q

22 Bilateral Filter for Meshes Estimate p, the new position for a vertex p p = 1 k(p) q S prediction Π q (p) spatial f( c q p ) influence g( Π q (p) p ) area a q

23 Bilateral Filter for Meshes Estimate p, the new position for a vertex p p = 1 k(p) q S prediction Π q (p) spatial f( c q p ) influence g( Π q (p) p ) area a q

24 Bilateral Filter for Meshes Estimate p, the new position for a vertex p p = 1 k(p) q S prediction Π q (p) spatial f( c q p ) influence g( Π q (p) p ) area a q

25 Why we expect it to work Predictions across corners are ``outliers''.

26 Dealing with Noise Noise has a nonlinear effect on predictions. We must mollify (pre-smooth) normals.

27 Dealing with Noise Noise has a nonlinear effect on predictions. We must mollify (pre-smooth) normals.

28 Dealing with Noise Noise has a nonlinear effect on predictions. We must mollify (pre-smooth) normals.

29 Dealing with Noise Noise has a nonlinear effect on predictions. We must mollify (pre-smooth) normals.

30 Dealing with Noise Noise has a nonlinear effect on predictions. We must mollify (pre-smooth) normals.

31 Implementation 3K vertices / second (typical), 1.4 GHz Athlon. Gaussians for f and g. Optimizations Cutoff at twice spatial filter radius. Binning for spatially coherent computation. Data and non-optimized code available online.

32 Results - Smoothing Original Desbrun 1999 Our result

33 Results - Effect of g Original Without g Our result

34 Results - Effect of Mollification Original Without mollification Our result

35 Results - Connectivity 50% Original Smoothed All predictors

36 Results - Varying width of f and g Original Narrow spatial and influence

37 Results - Varying width of f and g Original Narrow spatial and wide influence

38 Results - Varying width of f and g Original Wide spatial and influence

39 Normalization factor k as ``Confidence'' Normalization term k(p) is sum of weights, and is a measure of confidence in the estimation at p. p = 1 k(p) q S Π q (p) f( c q p ) g( Π q (p) p ) a q k(p) = q S f( c q p ) g( Π q (p) p ) a q

40 Results - k as Confidence Low High

41 Results - k as Confidence Low High

42 Results - vs Wiener Filtering Original

43 Results - vs Wiener Filtering (Low Noise) Peng et al Our result

44 Results - vs Wiener Filtering (High Noise) Peng et al Our result

45 Results - vs Anisotropic Diffusion Original

46 Results - vs Anisotropic Diffusion Clarenz et al Our result

47 Similar Methods Bilateral Mesh Denoising, Fleishman et al (next talk) Iterative Local frame No mollification Different predictor Trilateral Filter, Cloudhury and Tumblin 2003 (EGSR) Images and Meshes Mollify normals, then vertices Different predictor

48 Future Work Extend to other types of data (point models, volume data). Using k to steer further processing. Iterative application.

49 Conclusions Fast, feature preserving filter. Simple to implement. Applicable to polygon soups. Take-home message: Robust estimation for smoothing. Points across features are outliers. First-order predictors remove connectivity requirements.

50 Acknowledgements SIGGRAPH reviewers, Caltech SigDraft and MIT pre-reviewers. Udo Diewald, Martin Rumpf, Jianbo Peng, Denis Zorin, and Jean-Yves Bouguet, and Stanford 3D Scanning Repository for models. Peter Shirley and Michael Cohen for comments on this presentation. We would like to thank the NSF (CCR , DMS , DMS , EEC , EIA ).