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. - PowerPoint PPT PresentationTRANSCRIPT
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.
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
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
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
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.
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.
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
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
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
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
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
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.
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)
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.
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.
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
Slide Slide 1717Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in
Result - Flythrough
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))
Slide Slide 1919Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in
Result – Computational Complexity
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.
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.
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.
Slide Slide 2323Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in
Results - Flythrough
Slide Slide 2424Vision and Graphics Group, Department of Computer Science and Engineering, IIT Delhi http://vglab.cse.iitd.ac.in
Results – Running Time
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.