An Efficient Central Path Algorithm For Virtual Navigation

Motivation 4 Navigation in virtual environments is needed in many applications such as Virtual

The Problem 4 Given a three dimensional closed object and two points in the

Background 4 Topological thinning - Pavlidis 1980, Paik et. al. 1998, Ge et. al.

Distance From Boundary (DFB) 4 Distance of a point from the nearest boundary. 4

Our Approach 4 Compute DFB field for a hierarchical subdivision as opposed to computing

Hierarchical Subdivision 4 Enclose the object in a bounding box Vision and Graphics Group,

Hierarchical Subdivision 4 Subdivide the box into four equal parts Vision and Graphics Group,

Hierarchical Subdivision 4 Keep subdividing the smaller parts till they are intersecting with the

Hierarchical Subdivision 4 The smallest size boxes are the voxels with size 1. 4

DFB Field Computation 1 2 4 8 Vision and Graphics Group, Department of Computer

DFB Field Computation 4 We compute the DFB for the cells by running a

Path Computation 4 We can find the path between any two points by running

Path Computation 4 The algorithm returns a path in terms of connected blocks. Vision

Path Computation 4 A path is obtained by joining the centres of the blocks.

Path Smoothening Corner Cutting Vision and Graphics Group, Department of Computer Science and Engineering,

Result - Flythrough Vision and Graphics Group, Department of Computer Science and Engineering, IIT

Result – Computational Complexity 4 It is proved that the number of voxels formed

Result – Computational Complexity Vision and Graphics Group, Department of Computer Science and Engineering,

The Progressive Algorithm 4 We usually do not need to compute the DFB for

The Progressive Algorithm 4 Subdivide the region into coarse grid. 4 Choose a Region

The Progressive Algorithm 4 Grow the ROI and recompute the path. 4 Continue growing

Results - Flythrough Vision and Graphics Group, Department of Computer Science and Engineering, IIT

Results – Running Time Vision and Graphics Group, Department of Computer Science and Engineering,

Conclusion 4 DFB/Path computation is fast. 4 Paths of multiple resolutions. 4 Scaling the

Slides: 25

Download presentation

An Efficient Central Path Algorithm For Virtual Navigation Parag Chaudhuri, Rohit Khandekar, Deepak Sethi, Prem Kalra Vision and Graphics Group, Department of Computer Science and Engineering, Indian Institute of Technology Delhi. Computer Graphics International 2004 Crete, Greece. 18 th June, 2004.

Motivation 4 Navigation in virtual environments is needed in many applications such as Virtual Surgery l Automatic flight planning l Computer games l Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 2

The Problem 4 Given a three dimensional closed object and two points in the interior, find a path connecting those two points that l Lies completely inside the object l Stays away from the boundary l Has short length Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 3

Background 4 Topological thinning - Pavlidis 1980, Paik et. al. 1998, Ge et. al. 1999, Bouix et. al. 2003, Telea & Vilanova 2003 4 Potential field based methods - Hong 1995, Deschamps & Cohen 2001 4 Distance field based methods - Bitter et. al. 2001, Wan et. al. 2001 Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 4

Distance From Boundary (DFB) 4 Distance of a point from the nearest boundary. 4 Different measures of distance – Euclidean, City-block, Champher. 4 Find a path such that sum of DFB field at all points on the path is maximized. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 5

Our Approach 4 Compute DFB field for a hierarchical subdivision as opposed to computing DFB for the entire object at the finest resolution. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 6

Hierarchical Subdivision 4 Enclose the object in a bounding box Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 7

Hierarchical Subdivision 4 Subdivide the box into four equal parts Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 8

Hierarchical Subdivision 4 Keep subdividing the smaller parts till they are intersecting with the boundary of the object. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 9

Hierarchical Subdivision 4 The smallest size boxes are the voxels with size 1. 4 Size of a block b (size(b)) is the number of voxels on its side. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 10

DFB Field Computation 1 2 4 8 Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 11

DFB Field Computation 4 We compute the DFB for the cells by running a shortest path algorithm from the boundary to all the cells. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 12

Path Computation 4 We can find the path between any two points by running a shortest path algorithm on the graph formed by the cells. 4 An edge between blocks b 1 and b 2 in the graph is now given a weight w as W(b 1, b 2)=1/dfb(b 1)+1/dfb(b 2) Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 13

Path Computation 4 The algorithm returns a path in terms of connected blocks. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 14

Path Computation 4 A path is obtained by joining the centres of the blocks. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 15

Path Smoothening Corner Cutting Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi Splines http: //vglab. cse. iitd. ac. in Slide 16

Result - Flythrough Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 17

Result – Computational Complexity 4 It is proved that the number of voxels formed in the final subdivision are O(n+hk) - n : number of voxels on the boundary. - h : number of holes in the object. - k : number of levels of subdivision. 4 The running time is O((n+hk)log(n+hk)) Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 18

Result – Computational Complexity Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 19

The Progressive Algorithm 4 We usually do not need to compute the DFB for the entire object. 4 The extraneous volume for which the DFB is computed becomes a bottleneck at times. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 20

The Progressive Algorithm 4 Subdivide the region into coarse grid. 4 Choose a Region Of Interest (ROI) which contains the source and destination. 4 Compute the path for this ROI. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 21

The Progressive Algorithm 4 Grow the ROI and recompute the path. 4 Continue growing until the change in path length falls below a threshold. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 22

Results - Flythrough Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 23

Results – Running Time Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 24

Conclusion 4 DFB/Path computation is fast. 4 Paths of multiple resolutions. 4 Scaling the input does not adversely affect the computation time. 4 The subdivision grid also aids in View Culling while rendering. 4 Progressive extension makes it more efficient. Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http: //vglab. cse. iitd. ac. in Slide 25