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Foundations and Trends

®

in Robotics

Vol. 6, No. 1-2 (2017) 1–139

© 2017 F. Dellaert and M. Kaess

DOI: 10.1561/2300000043

Factor Graphs for Robot Perception

Frank Dellaert

Georgia Institute of Technology

dellaert@cc.gatech.edu

Michael Kaess

Carnegie Mellon University

kaess@cmu.edu

iii

2.5.4 Dogleg Minimization . . . . . . . . . . . . . . . . 26

2.6 Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . 28

3 Exploiting Sparsity 30

3.1 On Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 Motivating Example . . . . . . . . . . . . . . . . . 30

3.1.2 The Sparse Jacobian and its Factor Graph . . . . . 31

3.1.3 The Sparse Information Matrix and its Graph . . . 32

3.2 The Elimination Algorithm . . . . . . . . . . . . . . . . . 34

3.3 Sparse Matrix Factorization as Variable Elimination . . . . 37

3.3.1 Sparse Gaussian Factors . . . . . . . . . . . . . . . 37

3.3.2 Forming the Product Factor . . . . . . . . . . . . 38

3.3.3 Eliminating a Variable using Partial QR . . . . . . 39

3.3.4 Multifrontal QR Factorization . . . . . . . . . . . . 39

3.4 The Sparse Cholesky Factor as a Bayes Net . . . . . . . . 42

3.4.1 Linear-Gaussian Conditionals . . . . . . . . . . . . 42

3.4.2 Solving a Bayes Net is Back-substitution . . . . . . 43

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . 44

4 Elimination Ordering 47

4.1 Complexity of Elimination . . . . . . . . . . . . . . . . . . 47

4.2 Variable Ordering Matters . . . . . . . . . . . . . . . . . . 49

4.3 The Concept of Fill-in . . . . . . . . . . . . . . . . . . . . 51

4.4 Ordering Heuristics . . . . . . . . . . . . . . . . . . . . . 52

4.4.1 Minimum Degree Orderings . . . . . . . . . . . . . 52

4.4.2 Nested Dissection Orderings . . . . . . . . . . . . 53

4.5 Ordering Heuristics in Robotics . . . . . . . . . . . . . . . 54

4.6 Nested Dissection and SLAM . . . . . . . . . . . . . . . . 58

4.7 Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . 60

5 Incremental Smoothing and Mapping 62

5.1 Incremental Inference . . . . . . . . . . . . . . . . . . . . 64

5.2 Updating a Matrix Factorization . . . . . . . . . . . . . . 64

5.3 Kalman Filtering and Smoothing . . . . . . . . . . . . . . 67

5.3.1 Marginalization . . . . . . . . . . . . . . . . . . . 68

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2023-10-10

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