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

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

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

3/13/2016© 2009 Raymond P. Jefferis III Lect 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

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

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

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

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

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

3/13/2016© 2009 Raymond P. Jefferis III Lect 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

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

3/13/2016© 2009 Raymond P. Jefferis III Lect References –R. E. Bellman, Dynamic Programming, Princeton Univ. Press., –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)

3/13/2016© 2009 Raymond P. Jefferis III Lect 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

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

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

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