Subdivision Surface Fitting to A Dense Mesh Using Ridges and Umbilics

Transcription

1 Subdivision Surface Fitting to A Dense Mesh Using Ridges and Umbilics Xinhui Ma*, Simeon Keates, Yong Jiang, Jiří Kosinka *Department of Computer Science University of Hull, Hull HU6 7RX, UK. 1

2 Subdivision Surface Fitting Dense triangle mesh Coarse control mesh Subdivision surface Introduction 2

3 Outline Choosing features for control mesh Constructing connectivity Hausdorff distance including curvature vectors Computing the control mesh Examples and Conclusions Introduction 3

4 Existing Methods Mesh simplification Segmentation Global parameterisation Curvature lines Introduction 4

5 Our Method The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of control mesh for subdivision. Preserving and aligning with the salient features Curvature sensitive distance metric for automatic construction of connectivity Introduction 5

6 Features on Surfaces Ridges on implicit polynomial surfaces (a) Soap shape; (b) Rounded cube; (c) Rounded octahedron. Choosing features for control mesh 6

7 Feature Extraction , y x O y c xy c y x c y x c x c y b xy b y x b x b y k x k z -and -axes are principal directions x y Monge form : A local surface can be expressed as 7 Choosing features for control mesh

8 Ridges, Umbilics and Ridge-crossings Ridges are points of extrema of principal curvature along their curvature lines. b 0 = dk 1 /dx vanishes, i.e. b 0 = 0; or b 3 = dk 2 /dy vanishes, i.e. b 3 = 0. Umbilics are points where the two principal curvatures have the same value k 1 = k 2 The ridge-crossings are intersections of two ridges b 0 = b 3 = 0 Choosing features for control mesh 8

9 Classification of Umbilics by Ridge Configuration (a) Hyperbolic umbilic (b) Symmetric elliptic umbilic (c) Un-symmetric elliptic umbilic Choosing features for control mesh 9

10 Example of Extracted Features Extracted ridges, umbililcs and ridge-crossings from a Catmull-Clark subdivision surface of a car model. Choosing features for control mesh 10

11 Why Ridges and Umbilics? Ridges and umbilicus are geometrically and perceptually salient surface features, since ridges are points of extrema of principal curvatures along their curvature lines. Ridges and umbilicus are a decomposition of the curved surfaces as natural as the decomposition of the polyhedral surfaces into faces, edges and vertices (Thirion 1996). Their topological relationship provides a natural connectivity for control meshes. Choosing features for control mesh 11

12 Filtering of Features Choosing features for control mesh 12

13 Distance Transform and Voronoi Diagram Constructing connectivity 13

14 Algorithm of Connectivity Construction using Distance Transform 14

15 Distance Fields Constructing connectivity 15

16 Distance Metrics Mesh distance Sum of edge length Unwrap distance (~ mesh geodesic distance ) Hausdorff distance including curvature vectors (~ surface geodesic distance ) Hausdorff distance including curvature vectors 16

17 Arc-length Estimation s 1 cos θ 2 q + l2 4 k q p + l2 4 k p Hausdorff distance including curvature vectors 17

18 Hausdorff Distance including Principal Curvature Vectors s 1 cos θ 2 max max min i=1,2 j=1,2 max min j=1,2 i=1,2 q + l2 4 k qi p + l2 4 k pj, q + l2 4 k qi p + l2 4 k pj Hausdorff distance including curvature vectors 18

19 Ridge Alignment We can observe that the arc-length is determined by the term for a given chord length Hausdorff distance including curvature vectors 19

20 Ridge Alignment Consequently, two points on a ridge are more likely to be connected than two points on either side of the ridge if they have the same exact geodesic distance. The reason is that points on a ridge have lower variation of principal curvatures ( dk 1 dx = 0, dk 2 dy = 0), hence smaller estimated geodesic distance. Hausdorff distance including curvature vectors 20

21 Ridge Alignment (a) Unwrapped Euclidean distance (b) the proposed Hausdorff distance Connectivity of grassfire algorithms using different metrics Hausdorff distance including curvature vectors 21

22 Computing of Control Mesh Loop masks Linear system Computing of Control Mesh 22

23 Examples: Subdivision Surface Fitting Examples and Conclusions 23

24 Examples: Subdivision Surface Fitting Examples and Conclusions 24

25 Fitting Errors Examples and Conclusions 25

26 Ridge Preserving (a) Original (b) Proposed (c) Simplification (d) Kanai Examples and Conclusions 26

27 Ridge Preserving (a) (b) (c) (d) (a) Original (b) Proposed (c) Simplification (d) Kanai Examples and Conclusions 27

28 Computational Efficiency Examples and Conclusions 28

29 Discussions Accurate surface fitting with feature alignment Accuracy can be improved using more robust feature extraction algorithm and optimal feature filtering method. Although the proposed method sacrifices some efficiency to preserve the accuracy and features, it is still reasonably efficient with total computation times of few minutes. Examples and Conclusions 29

30 Conclusions Salient ridge features are well preserved The connectivity of control mesh follows the natural ridge-joined connectivity of umbilics and ridge-crossings. Curvature sensitive Hausdorff distance metric improves feature alignment. Quad mesh construction could be the future work Examples and Conclusions 30

31 Acknowledgments The authors would like to thank Malcolm Sabin, Kai Hormann and Neil Dodgson for valuable suggestions. The authors are pleased to acknowledge the financial support of EPSRC grant EP/H030115/1. 31

32 Reference Ma, X., Keates, S., Jiang, Y., Kosinka J., Subdivision surface fitting to a dense mesh using ridges and umbilics, Computer Aided Geometric Design, 32(1), Thank you! 32