1. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 1 Geographic Information Processing Shortest Path Analysis

2. Path Costs Path costs are accumulated by traversal of a path Path costs may be distances, or attributes that are functions of position along the path A cost unit is accumulated when moving from one point (pixel) to another Changes in path taken between two endpoints can produce differences in accumulated cost A “shortest” path minimizes accumulated cost between the endpoints 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 2

3. Path Length (Cost) 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 3

4. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 4 Shortest Path Characteristics Path in space from start to goal Path choices can be made Cost metric accumulates along path Objective is minimum cost accumulation Suboptimal alternative paths are of interest

5. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 5 Routing Examples Pipelines Communications circuits Flight paths Hazard avoidance

6. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 6 Cost Metrics Cost of construction Distance over terrain Time of flight Exposure to hazard

7. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 7 Cost Models as Metrics Can include multiple factors –Terrain –Resources –Time –Environmental constraints (high cost) –Right-of-way purchases & negotiation –Risk

8. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 8 Terrain Multipliers Hills Soil conditions Vegetation Water conditions Parallel occupancy with highways or rails Urban congestion

9. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 9 Pipeline Cost Model Direct (resource and construction) costs Multipliers for access difficulties Right-of-Way purchases Shipping costs

10. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 10 Problem Preparation Terrain space is overlaid with a grid –Fineness of grid must be estimated –Path nodes and segments are labeled Costs are computed for each grid segment –Using cost model –Data inputs from terrain or other data sources

11. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 11 A Shortest Path Algorithm Method –Dynamic Programming (Richard Belman) –Ref: Description –Jefferis & Fegley paper –Ref:

12. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 12 References –R. E. Bellman, Dynamic Programming, Princeton Univ. Press., 1957. –R. P. Jefferis and K. A. Fegley, “Application of Dynamic Programming to Routing Problems,” IEEE Trans. On System Science and Cybernetics, Vol. SSC-1, No. 1 (November 1965)

13. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 13 Method Formulate recursion relation: –U i is optimal cost assigned to a node –C ij is cost of reaching node i from node ij Apply to cost grid of nodes indexed by ij

14. 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 14 Discussion Paper, containing example, will be distributed and discussed at length in class.

15. Tools Compute surface distance between successive points[meters] Compute earth radius using WGS 84 model [meters] Compute surface point given one point, angle, and distance [lat,lon] Trigonometric Arc-Functions over 360° 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 15

16. More Tools Generation of Gaussian filter coefficients for convolution array Laplacian filter coefficients for convolution array (accentuates curvature) Compute number of pixels to given point Result: {latPoints,lonPoints} 3/13/2016© 2009 Raymond P. Jefferis III Lect 11 - 16