an efficient central path algorithm for virtual navigation

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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.

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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. - PowerPoint PPT Presentation

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Page 1: An Efficient Central Path Algorithm For Virtual Navigation

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 2004Crete, Greece.

18th June, 2004.

Page 2: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 22Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Motivation

Navigation in virtual environments is needed in many applications such asVirtual SurgeryAutomatic flight planningComputer games

Page 3: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 33Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

The Problem

Given a three dimensional closed object and two points in the interior, find a path connecting those two points thatLies completely inside the objectStays away from the boundaryHas short length

Page 4: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 44Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Background

Topological thinning Pavlidis 1980, Paik et. al. 1998, Ge et. al.

1999, Bouix et. al. 2003, Telea & Vilanova 2003

Potential field based methods Hong 1995, Deschamps & Cohen 2001

Distance field based methods Bitter et. al. 2001, Wan et. al. 2001

Page 5: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 55Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Distance From Boundary (DFB)

Distance of a point from the nearest boundary.

Different measures of distance – Euclidean, City-block, Champher.

Find a path such that sum of DFB field at all points on the path is maximized.

Page 6: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 66Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Our Approach

Compute DFB field for a hierarchical subdivision as opposed to computing DFB for the entire object at the finest resolution.

Page 7: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 77Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Hierarchical Subdivision

Enclose the object in a bounding box

Page 8: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 88Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Subdivide the box into four equal parts

Hierarchical Subdivision

Page 9: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 99Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Keep subdividing the smaller parts till they are intersecting with the boundary of the object.

Hierarchical Subdivision

Page 10: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1010Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

The smallest size boxes are the voxels with size 1.

Size of a block b (size(b)) is the number of voxels on its side.

Hierarchical Subdivision

Page 11: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1111Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

DFB Field Computation

1

2

4

8

Page 12: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1212Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

DFB Field Computation

We compute the DFB for the cells by running a shortest path algorithm from the boundary to all the cells.

Page 13: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1313Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Path Computation

We can find the path between any two points by running a shortest path algorithm on the graph formed by the cells.

An edge between blocks b1 and b2 in the graph is now given a weight w as

W(b1,b2)=1/dfb(b1)+1/dfb(b2)

Page 14: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1414Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Path Computation

The algorithm returns a path in terms of connected blocks.

Page 15: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1515Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Path Computation

A path is obtained by joining the centres of the blocks.

Page 16: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1616Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Path Smoothening

Corner Cutting Splines

Page 17: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1717Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Result - Flythrough

Page 18: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1818Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Result – Computational Complexity

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.

The running time is O((n+hk)log(n+hk))

Page 19: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 1919Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Result – Computational Complexity

Page 20: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 2020Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

The Progressive Algorithm

We usually do not need to compute the DFB for the entire object.

The extraneous volume for which the DFB is computed becomes a bottleneck at times.

Page 21: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 2121Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

The Progressive Algorithm

Subdivide the region into coarse grid.

Choose a Region Of Interest (ROI) which contains the source and destination.

Compute the path for this ROI.

Page 22: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 2222Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

The Progressive Algorithm

Grow the ROI and recompute the path.

Continue growing until the change in path length falls below a threshold.

Page 23: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 2323Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Results - Flythrough

Page 24: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 2424Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Results – Running Time

Page 25: An Efficient Central Path Algorithm For Virtual Navigation

Slide Slide 2525Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in

Conclusion

DFB/Path computation is fast.Paths of multiple resolutions.Scaling the input does not adversely

affect the computation time.The subdivision grid also aids in

View Culling while rendering.Progressive extension makes it

more efficient.