for Vision-Based Navigation 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

Intuitive Problem Formulation

Intuitive Problem Formulation

Intuitive Problem Formulation

Intuitive Problem Formulation

Intuitive Problem Formulation

Intuitive Problem Formulation

A Graph Theoretic Formulation Problem Definition: The -Minimum Overlapping Region Decomposition Problem (-MORDP) for a world instance <G=(V,E), F, {v} vV> 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} vV> 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} vV> 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|) Features commonly visible in region:

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 1

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 1

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 1

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 1

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 1

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 1

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 2

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 2

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 2

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 2

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 2

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 3

Algorithms B.x and C: O(k|V|2|F|) Features commonly visible in region: 4

Algorithms B.x and C: O(k|V|2|F|) 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.

Simulated Data (cont.)

Real Data We applied the best-performing algorithm (B.2) to real feature visibility data. 0 90 180 270

Real Data (cont.) Data collected in 6m  3m area. Sampled at 25 cm intervals. Total of 897 visible features. Camera at 0, 90, 180, and 270 degree orientations. SIFT features.

Typical Feature Visibility Regions

Real Data Decompositions k =4,  =0

Real Data Decompositions (cont.) k =4,  =1

Real Data Decompositions (cont.) k =10,  =0

Real Data Decompositions (cont.) k =10,  =1

Conclusions We have introduced a novel graph theoretic formulation of the landmark acquisition problem, and have established its intractability. We have explored a number of greedy approximation algorithms, systematically testing them on synthetic worlds and demonstrating them on two real worlds. The resulting decompositions find large regions in the world in which a small number of features can be tracked to support efficient on-line localization. The formulation and solution are general, and can accommodate other classes of image features.