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Landmark Selection for Vision-Based Navigation. Pablo L. Sala, U. of Toronto Robert Sim, U. of Toronto/U. of British Columbia Ali Shokoufandeh, Drexel U. Sven Dickinson, U. of Toronto. Intuitive Problem Formulation. Intuitive Problem Formulation. Intuitive Problem Formulation.
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Landmark Selection for Vision-Based Navigation Pablo L. Sala, U. of Toronto Robert Sim, U. of Toronto/U. of British Columbia Ali Shokoufandeh, Drexel U. Sven Dickinson, U. of Toronto
A Graph Theoretic Formulation Problem Definition: The -Minimum Overlapping Region Decomposition Problem (-MORDP) for a world instance <G=(V,E), F, {v}vV> consists of finding a minimum size -overlapping decomposition D = {R1, …, Rd} of V into regions such that:
A Graph Theoretic Formulation Problem Definition: The -Minimum Overlapping Region Decomposition Problem (-MORDP) for a world instance <G=(V,E), F, {v}vV> consists of finding a minimum size -overlapping decomposition D = {R1, …, Rd} of V into regions such that: Theorem 1: A -MORDP can be reduced to an equivalent 0-MOVRDP, and the solution to this latter problem can be extended to a solution for the original problem.
A Graph Theoretic Formulation Problem Definition: The -Minimum Overlapping Region Decomposition Problem (-MORDP) for a world instance <G=(V,E), F, {v}vV> consists of finding a minimum size -overlapping decomposition D = {R1, …, Rd} of V into regions such that: Theorem 1: A -MORDP can be reduced to an equivalent 0-MOVRDP, and the solution to this latter problem can be extended to a solution for the original problem. Theorem 2: The decision problem <0-MORDP, d> is NP-complete. (Proof by reduction from the Minimum Set Cover Problem.)
Heuristic Methods for 0-MORDP • 0-MORDP is intractable. • Can we efficiently find an effective approximation? • We developed and tested six greedy approximation algorithms.
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region:
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 25
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 25
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 19
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 19
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 19
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 19
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 17
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 17
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 14
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 14
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 11
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 11
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 9
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 8
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 8
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 6
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 4
Algorithm A.x: O(|V|2|F|) k = 4 Features commonly visible in region: 4
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region:
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 1
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 1
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 1
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 1
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 1
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 1
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 2
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 2
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 2
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 2
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 2
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 3
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 4
Algorithms B.x and C: O(k|V|2|F|) k = 5 Features commonly visible in region: 5
Results Simulated Data
Simulated Data (cont.) • Two types of Worlds: Irregular (Irreg) and Rectangular (Rect). • average diameter: 40m. • pose space sampled at 50 cm intervals. • average number of sides: 6. • average number of obstacles: 7. • Two types of Features: Short-Range and Long-Range. • visibility range N(0.65, 0.2) to N(12.5, 1) m, • and angular range N(25, 3) degrees. • Visibility range N(0.65, 0.2) to N(17.5, 2) m, • and angular range N(45, 4) degrees.