Abstract

Abstract We present a complete classification of minimal problems for generic arrangements of points and lines in space observed partially by three calibrated perspective cameras when each line is incident to at most one point. This is a large class of interesting minimal problems that allows missing observations in images due to occlusions and missed detections. There is an infinite number of such minimal problems; however, we show that they can be reduced to 140616 equivalence classes by removing superfluous features and relabeling the cameras. We also introduce camera-minimal problems, which are practical for designing minimal solvers, and show how to pick a simplest camera-minimal problem for each minimal problem. This simplification results in 74575 equivalence classes. Only 76 of these were known; the rest are new. To identify problems having potential for practical solving of image matching and 3D reconstruction, we present several natural subfamilies of camera-minimal problems as well as compute solution counts for all camera-minimal problems which have less than 300 solutions for generic data.KeywordsMinimal problemsCalibrated cameras3D reconstruction

Highlights

  • Minimal problems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], which we study, are 3D reconstruction problems recovering camera poses and world coordinates from given images such that random input instances have a finite positive number of solutions

  • We provide a complete classification of minimal problems for generic arrangements of points and lines in space observed partially by three calibrated perspective cameras when each line is incident to at most one point

  • There is an infinite number of such minimal problems; we show that they can be reduced to 140616 equivalence classes of reduced minimal problems by removing superfluous features and relabeling the cameras

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Summary

Introduction

Minimal problems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], which we study, are 3D reconstruction problems recovering camera poses and world coordinates from given images such that random input instances have a finite positive number of solutions. We find all terminal camera-minimal problems with less than 300 solutions for generic data and present other interesting cases that might be important for practical solving of image matching and 3D reconstruction. We construct a long list of interesting problems under partial visibility, even with our restrictions, i.e. having exactly three cameras and having each line incident to at most one point. We construct a long list of interesting problems under partial visibility, even with our restrictions, i.e. having exactly three cameras and having each line incident to at most one point1 These restrictions make the task of enumerating problems tractable while making it still possible to account for very practical incidence cases where several existing feature detectors are applicable. SIFT [36] and LAF [37] provide quivers (points with one direction attached), which can be interpreted as lines through the points and used to compute relative camera poses [38]

Previous work
Problem Specification
Reduced PL1Ps
Balanced PL1Ps
Camera-Minimal PL1Ps
Checking minimality
Computing degrees
Result
Conclusion
10 Note on partial visibility in two views
11 Note on registration problems
12.1 Theorems 1 and 4
12.3 Theorem 6 and its corollaries
13 Computations
13.2 Checking minimality
13.3 Computing degrees
13.4 Finding subfamilies
Full Text
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