The camera calibration problem consists in estimating the intrinsic and the extrinsic parameters. This problem can be solved by computing the fundamental matrix. The fundamental matrix can be obtained from a set of corresponding points. However in practice, corresponding points may be inaccurately estimated, falsely matched or badly located, due to occlusion and ambiguity, among others. On the other hand, if the set of corresponding points does not include information on different depth planes, the estimated fundamental matrix may not be able to correctly recover the epipolar geometry. In this paper a method for estimating the fundamental matrix is introduced. The estimation problem is posed as finding a set of corresponding points. Fundamental matrices are estimated using subsets of corresponding points and an optimisation criterion is used to select the best estimated fundamental matrix. The experimental evaluation shows that the least range of residuals is a tolerant criterion to large baselines.

1. AN APPROACH FOR ESTIMATING
THE FUNDAMENTAL MATRIX
Research work submitted for the degree of Master of Engineering in Computer Science
Daniel Barragan Calderon, Eng
Universidad del Valle, Cali - Colombia
If I have seen farther than others, it
is because I was standing on the
shoulders of giants
Albert Einstein

2. Content
 Motivation
 Camera Model
 Epipolar Geometry
 Camera Model to Epipolar Geometry Derivation
 3D Reconstruction Process
 State-of-the-art
 Problem Statement
 Research Objectives
 Proposed Approach
 Results
 Remarks, Conclusions
Universidad del Valle – School of Computer and Systems Engineering Slide 2

3. Motivation – Suite 1
Figure 1. 3D Applications. Source: Google Images
Universidad del Valle – School of Computer and Systems Engineering Slide 3

4. Motivation – Suite 2
?
Figure 2. Direct Problem, Inverse / Ill-posed Problem
Universidad del Valle – School of Computer and Systems Engineering Slide 4

5. Motivation – Suite 3
Figure 3. Stereo Capture [50]
Video 1. 3D Reconstruction
Universidad del Valle – School of Computer and Systems Engineering Slide 5

6. Camera Model
Figure 4. Extrinsic and Intrinsic Camera Parameters
Universidad del Valle – School of Computer and Systems Engineering Slide 6

7. Epipolar Geometry
Figure 5. Corresponding Points Figure 6. Epipolar Geometry
Universidad del Valle – School of Computer and Systems Engineering Slide 7

8. Camera Model to Epipolar Geometry
Derivation
 Points 𝒎 and 𝒎′ (homogeneous coordinates) can be related
through 𝑷 and 𝑷′
𝒎′ = 𝑷′𝑷+ 𝒎
 Epipolar line equation can be derived as follows
𝒍′ = 𝒆′ × 𝒎′ → 𝒍′ = 𝒆′ 𝒙 𝒎′
𝒍′ = 𝒆′ 𝒙 (𝑷′ 𝑷+ )𝒎
𝓕 = 𝒆′ 𝒙 𝑷′ 𝑷+
𝒍′ = 𝓕𝒎
 Epipolar equation
𝒎′ 𝑻 𝒍′ = 0 → 𝒎′ 𝑻 𝓕𝒎 = 0
Universidad del Valle – School of Computer and Systems Engineering Slide 8

9. 3D Reconstruction Process
Diagram 1. Illustration of 3D Reconstruction Workflow
Universidad del Valle – School of Computer and Systems Engineering Slide 9

10. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 10

11. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 11

12. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 12

13. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 13

14. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 14

15. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 15

16. State-of-the-art
Calibration
Extrinsic and
Epipolar
Intrinsic
Geometry
Parameters
One Camera Two Camera Natured Inspired
Robust Methods
Calibration Calibration Techniques
Genetic Bucketing
Natured Inspired Two Step Calibrating Two Natured Inspired Algorithms Basic Algorithms Algorithms
Techniques Techniques Times [30] Techniques
[37, 38] [43, 45]
Genetic Genetic
Tsai M-Estimators
Algorithms Algorithms
[32] [42]
[12-14,25,26] [35, 36]
Particle Swarm
Heikkila LMedS
Optimizer
[33] [40]
[15-18]
Neural Networks Zhang RANSAC
[19,22,29] [31] [44]
Diagram 2. State-of-the-art
Universidad del Valle – School of Computer and Systems Engineering Slide 16

17. Problem Statement – Suite 1
Translation and Rotation
Figure 7. Geometria Epipolar
Universidad del Valle – School of Computer and Systems Engineering Slide 17

18. Problem Statement – Suite 2
 Let 𝑺 be a set of corresponding points 𝒎 and
𝒎′ subject to:
 The points 𝒎 and 𝒎′ have to be true projections of
𝑴
 The 𝑢, 𝑣 𝑇 and 𝑢′, 𝑣′ 𝑇 coordinates have to
correspond to the true localisation of 𝒎 and 𝒎′ ,
respectively
 The cardinality of 𝑺 have to be in relation to depth
planes in the 3D scene
 The addressed problem consists in finding a set 𝑺
which fulfils the above criteria
Universidad del Valle – School of Computer and Systems Engineering Slide 18

19. Research Objetives
General Objective
 Proposing a correspondence selection method for the
fundamental matrix estimation
Specific Objectives
 Implementing techniques for correspondence selection
 Implementing techniques for the Fundamental Matrix
estimation
 Measuring the impact of correspondence selection on
Fundamental Matrix estimation
 Establishing a evaluation criterion for selecting the
algorithm with the more accurate Fundamental Matrix
Universidad del Valle – School of Computer and Systems Engineering Slide 19

20. Proposed Approach
 An algorithm for fundamental matrix estimation were
proposed
Diagram 3. Proposed Approach
Universidad del Valle – School of Computer and Systems Engineering Slide 20

21. Proposed Approach
 Clustering of Correspondences
Diagram 3. Proposed Approach
Universidad del Valle – School of Computer and Systems Engineering Slide 21

22. Proposed Approach
 Clustering of Correspondences
Diagram 4. Disparity-Based Clustering of Correspondences
Diagram 2. Proposed Genetic Method
Universidad del Valle – School of Computer and Systems Engineering Slide 22

23. Proposed Approach
 Clustering of Correspondences
 Disparity Estimation
𝒔𝒆𝒕 = ( 𝒎 𝟏 , 𝒎′ 𝟏 , … , 𝒎 𝒊 , 𝒎′ 𝒊 , … , (𝒎 𝒏 𝒄 , 𝒎′ 𝒏 𝒄 ))
𝒔𝒆𝒕 = (𝑢1 𝑣1 , 𝑢′1 𝑣′1 , … , 𝑢 𝑖 𝑣 𝑖 , 𝑢′ 𝑖 𝑣 ′ 𝑖 , … , (𝑢 𝑛 𝑐 𝑣 𝑛 𝑐 , 𝑢′ 𝑛 𝑐 𝑣′ 𝑛 𝑐 ))
𝒹 𝑢 𝑖 = 𝑢 𝑖 − 𝑢′𝑖
𝒹 𝑣 𝑖 = 𝑣 𝑖 − 𝑣 ′𝑖
𝓭 𝒊 = 𝒹 𝑢 𝑖, 𝒹 𝑣 𝑖
 Subtractive Clustering
𝑛 2
− 𝒅𝒊 − 𝒅𝒋
𝑃𝑜𝑡 𝑖 = exp( )
𝑟𝑎 2
𝑗=1
2
2
− 𝒅𝒊 − 𝒄 𝟏
𝑃𝑜𝑡 𝑖 = 𝑃𝑜𝑡 𝑖 − 𝑃𝑜𝑡𝑉𝑎𝑙(𝒄 𝟏 )exp(
𝑟𝑏 2
2
Universidad del Valle – School of Computer and Systems Engineering Slide 23

24. Proposed Approach
 Clustering of Correspondences
 Kmeans Clustering
(𝑡+1) 1
𝒄𝒋 = (𝑡)
𝒹𝑖
𝒮𝑗 (𝑡)
𝓭 𝒊 ∈𝒮 𝑗
(𝑡) (𝑡) (𝑡)
𝒮𝑗 = 𝓭 𝒊: 𝓭𝒊 − 𝒄𝒋 ≤ 𝓭 𝒊 − 𝒄 𝒋∗ 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ∗ = 1, … , 𝑘
𝒔𝒖𝒃𝒔𝒆𝒕 = ((𝑢1 𝑣1 , 𝑢′1 𝑣 ′1 ) 𝑟𝑎𝑛𝑑 , … , (𝑢λ𝑘 𝑣 𝜆𝑘 , 𝑢′ 𝜆𝑘 𝑣′ 𝜆𝑘 ) 𝑟𝑎𝑛𝑑 )
 Number of Subsets
℘ = 1 − [1 − (1 −∈) 𝑛 𝑐 ] 𝑛 𝑠
log 1 − ℘
𝑛𝑠 = 𝑛𝑐
log 1 − 1 −∈
Universidad del Valle – School of Computer and Systems Engineering Slide 24

25. Proposed Approach
 Correspondences Selection
Diagram 3. Proposed Approach
Universidad del Valle – School of Computer and Systems Engineering Slide 25

26. Proposed Approach
 Fundamental matrix estimation
Diagram 5. Correspondence Selection by GA
Diagram 3. Proposed Approach
Universidad del Valle – School of Computer and Systems Engineering Slide 26

27. Proposed Approach
 Correspondences Selection
 Population
𝜃 = (𝑥1 , … , 𝑥 𝑗 , … , 𝑥 𝑝 )
𝑥 𝑗 = (𝒎 𝑗 , 𝒎′ 𝑗 )
𝜃 = ((𝒎1 , 𝒎′1 ), … , (𝒎 𝑗 , 𝒎′ 𝑗 ), … , (𝒎 𝑝 , 𝒎′ 𝑝 ))
𝑥 𝑗 = (𝑢 𝑗 𝑣 𝑗 , 𝑢′ 𝑗 𝑣 ′ 𝑗 )
𝜃 = (𝑢1 𝑣1 , 𝑢′1 𝑣 ′1 , … (𝑢 𝑗 𝑣 𝑗 , 𝑢′ 𝑗 𝑣 ′ 𝑗 ) … , (𝑢 𝑝 𝑣 𝑝 𝑢′ 𝑝 𝑣 ′ 𝑝 ))
 Fitness
𝑛
𝑓 𝜃 = 𝑑 𝒎 𝒊 , 𝓕𝒎 𝒊 ′ + 𝑑 ′ (𝒎 𝒊 ′, 𝓕𝒎 𝒊 )
𝑖=1
𝜃0 = arg min 𝑓(𝜃)
 Selection (Roulette)
Universidad del Valle – School of Computer and Systems Engineering Slide 27

28. Proposed Approach
 Correspondences Selection
 Crossover
′
𝜃1 = 𝑠𝑢𝑏 𝜃1 , ℎ |𝑠𝑢𝑏 𝜃2 , 𝑝 − ℎ , ℎ = 𝒫𝑝
′
𝜃2 = 𝑠𝑢𝑏 𝜃2 , 𝑝 − ℎ |𝑠𝑢𝑏 𝜃1 , ℎ , ℎ = 𝒫𝑝
0.15 ≤ 𝒫 ≤ 0.85
 Mutation
𝑥′ = 𝑥𝑗 + 𝜉
𝑗
𝜉 : Mutation offset
0 1 2
7 𝑥𝑗 3
6 5 4
Universidad del Valle – School of Computer and Systems Engineering Slide 28

29. Proposed Approach
 Fundamental matrix
Diagram 3. Proposed Approach
Universidad del Valle – School of Computer and Systems Engineering Slide 29

30. Results – Suite 1
This section contains the results for the following tests:
 Results for different correspondences selection
methods and different fundamental matrix estimation
algorithms.
 Repeatability analysis for proposed GA-based
algorithm
 Performance evaluation using multiple datasets for
proposed GA-based algorithm
Universidad del Valle – School of Computer and Systems Engineering Slide 30

31. Results – Suite 2
 Results were evaluated using the following error
measure
residual
𝒎 𝒎′
Figure 8. Error Measure
Universidad del Valle – School of Computer and Systems Engineering Slide 31

32. Results – Suite 3
 Results were evaluated using the epipolar lines
Camera Camera
Left Right
Figure 9. Epipolar Lines
Universidad del Valle – School of Computer and Systems Engineering Slide 32

33. Results – Suite 4
Residual Estimation
Correspondence selection technique
Fundamental matrix
Random Buckets Proposed DBC*
estimation algorithm
Normalized 7 Points Algorithm 1,4482E-04 1,7010E-04 1,8253E-04
Normalized 8 Points Algorithm 1,1341E-07 2,0495E-09 1,2947E-06
Table 1. Residual Estimation
(a) (b)
Figure 10. (a) Norm. 7 Points + DBC , (b) Norm. 8 Points + DBC
*DBC: Disparity Based Clustering
Universidad del Valle – School of Computer and Systems Engineering Slide 33

34. Results – Suite 5
Residual Estimation
Correspondence selection technique
Fundamental matrix
Random Buckets Proposed DBC
estimation algorithm
LMedS 7,6743E-05 7,1070E-05 8,0746E-05
Proposed GA-based 8,9615E-06 1,5240E-05 2,4937E-05
Table 2. Residual Estimation (Robust Methods)
(a) (b)
Figure 11. (a) LMedS + DBC , (b) GA-Based+ DBC
Universidad del Valle – School of Computer and Systems Engineering Slide 34

35. Results – Suite 6
Residual Estimation
Correspondence selection technique
Fundamental matrix estimation
Random Buckets Proposed DBC
algorithm
Normalized 7 Points Algorithm 1,4482E-04 1,7010E-04 1,8253E-04
Normalized 8 Points Algorithm 1,1341E-07 2,0495E-09 1,2947E-06
LMedS 7,6743E-05 7,1070E-05 8,0746E-05
Proposed GA-based 8,9615E-06 1,5240E-05 2,4937E-05
Table 3. Residual Estimation
2,0000E-04
1,5000E-04
1,0000E-04
5,0000E-05
Random
0,0000E+00
Buckets
Proposed DBC
Chart 1. Residual Estimation
Universidad del Valle – School of Computer and Systems Engineering Slide 35

36. Results – Suite 7
Computing Time (Sec.)
Correspondence selection technique
Fundamental matrix estimation
Random Buckets Proposed DBC
algorithm
Normalized 7 Points Algorithm 2,654 2,794 3,547
Normalized 8 Points Algorithm 2,742 2,790 3,209
LMedS 3,563 3,620 4,002
Proposed GA-based 10,390 11,983 18,697
Table 4. Computing Time (Sec.) AMD 1,7GHz, 3Gb RAM
20,0000
Seconds 15,0000
10,0000
5,0000
0,0000 Random
Buckets
Proposed DBC
Chart 2. Computing Time (Sec.) AMD 1,7GHz, 3Gb RAM
Universidad del Valle – School of Computer and Systems Engineering Slide 36

37. Results – Suite 8
 Filtering the initial estimated corresponding points using
RANSAC and Guide Sampling [48] results were improved
Fundamental matrix estimation Residual Estimation Computing Time (Sec.)
algorithm
LMedS + Bucketing 7,1070E-05 3,620
Proposed GA-based 7,9477E-09 25,065
Table 5. Proposed GA-based + RANSAC + Guide Sampling
(a) (b)
Figure 12. (a) Bad Located and False Matches Filtering, (b) Epipolar Line for the
Proposed GA-based + RANSAC + Guide Sampling
Universidad del Valle – School of Computer and Systems Engineering Slide 37

38. Results – Suite 9
Dataset Residual Computing Time (Sec.)
1,5752E-09 25,967
2,4951E-09 37,333
Lab 9,6977E-10 67,101
1,0642E-09 32,820
1,4664E-10 20,284
Table 6. Repeatability Analysis for Proposed GA-based + RANSAC + Guide Sampling
Figure 13. Epipolar lines for the Proposed GA-based + RANSAC + Guide Sampling
Universidad del Valle – School of Computer and Systems Engineering Slide 38

39. Results – Suite 10
FM Estimation Computing
Dataset Residual
Algorithm Time (Sec.)
Bucketing + LMedS 1,8745E-04 3,0833
Lab
Proposed GA-based 2,1072E-06 20,4822
Bucketing + LMedS 1,5994E-05 3,3743
Corridor [49]
Proposed GA-based 1,3204E-09 7,3512
Bucketing + LMedS 1,2072E-04 45,4828
Raglan [49]
Proposed GA-based 1,6294E-10 59,2093
Bucketing + LMedS 1,8288E-04 2,9213
Kapel [49]
Proposed GA-based 6,0952E-09 37,5971
Table 7. Performance evaluation using multiple datasets
Universidad del Valle – School of Computer and Systems Engineering Slide 39

40. Results – Suite 11
Figure 14. Epipolar lines for multiple datasets [49]
Universidad del Valle – School of Computer and Systems Engineering Slide 40

41. Remarks – Suite 1
 The GA-based algorithm can be used in
applications that do not require successive fast
calibration of a stereo rig, for example: content
generation where calibration is done usually one
time at the beginning of the capture
 Parallel computing reduce estimation time for
robust algorithms when the computing time
dedicated to algorithm iterations is long compared
with the computing time dedicated to split tasks.
Test were made but they are not include in the
research work
Universidad del Valle – School of Computer and Systems Engineering Slide 41

42. Remarks – Suite 2
 Algorithms’ speed can be improved when
operations over vector of correspondences are
done through indexes
 Security systems that use multiple cameras are
based nowadays just on plain information from
images but not on their coordinate systems.
Unifying coordinate systems of cameras would
avoid many drawbacks of actual security systems
Universidad del Valle – School of Computer and Systems Engineering Slide 42

43. Conclusions – Suite 1
 Residual value does not provide reliable results as a
benchmarking for fundamental matrix estimation
when presence of outliers is high. It is necessary to
perform a previous filtering step in order to obtain
reliable residual values
 The GA (genetic algorithm) by itself is not able to
discard correspondence outliers, it is necessary to
include a previous filtering step when noise levels are
high in order to obtain satisfactory results for
fundamental matrix estimation
Universidad del Valle – School of Computer and Systems Engineering Slide 43

44. Conclusions – Suite 2
 Mathematically having 7 or 8 corresponding points is
enough to solve the equation system for fundamental
matrix estimation, but having 7 or 8 pairs free of false
matches and bad matches is a difficult task in real
problems. It is better to have a bigger number of
correspondences to include the variability inherent to
reality from different depth planes
Universidad del Valle – School of Computer and Systems Engineering Slide 44

45. Contributions
 Poster
Acerca del Algoritmo 8 Puntos
LatinAmerican Conference On Networked and Electronic Media 2009
Daniel Barragan, Maria Trujillo
 Paper Submitted and Oral Presentation
An Approach for Estimating the Fundamental Matrix
6th Colombian Computation Congress 2011
Daniel Barragan, Maria Trujillo
 Paper Submitted
A GA-based Method for Estimating the Fundamental Matrix
IEEE Congress on Evolutionary Computation 2011
Daniel Barragan, Ivan Cabezas, Maria Trujillo
 Paper Submitted
A GA-based Method for Estimating the Fundamental Matrix
22nd British Machine Vision Conference 2011
Daniel Barragan, Ivan Cabezas, Maria Trujillo
Universidad del Valle – School of Computer and Systems Engineering Slide 45

46. References – Suite 1
 [1] J. Bazin, I. Kweon, C. Demonceaux, y P. Vasseur, “Automatic calibration of catadioptric cameras in urban
environment,” 2008, págs. 3108-3114.
 [2] P. Krsek, M. Spanel, M. Svub, V. Stancl, O. Siler, y R. Barton, “Network collaborative environment supporting
3D medicine,” Engineering in Medicine and Biology Society, 2009. EMBC 2009. Annual International Conference of
the IEEE, 2009, págs. 2164-2167.
 [3] Jiuxiang Hu, A. Razdan, y J. Zehnder, “An Algorithm to Calibrate Field Cameras for Stereo Clouds,” 2008,
págs. II-1048-II-1051.
 [4] L. Ray, “Monocular 3D vision for a robot assembly environment,” IEEE International Conference on Systems
Engineering, Pittsburgh, PA, USA: , págs. 430-434.
 [5] V. Maz'ya y T.O. Shaposhnikova, Jacques Hadamard, AMS Bookstore, 1999.
 [6] G. Xú y Z. Zhang, Epipolar geometry in stereo, motion, and object recognition, Springer, 1996.
 [7] J. Weng, P. Cohen, y M. Herniou, “Camera Calibration with Distortion Models and Accuracy Evaluation,”
IEEE Trans. Pattern Anal. Mach. Intell., vol. 14, 1992, págs. 965-980.
 [8] Richard Hartley y Andrew Zisserman, Multiple view geometry in computer vision, Cambridge University
Press, 2003.
 [9] H.C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature, vol.
293, 1981, págs. 133-135.
 [10] R. Hartley, “In defense of the eight-point algorithm,” Pattern Analysis and Machine Intelligence, IEEE
Transactions on, vol. 19, 1997, págs. 580-593.
 [11] Q. Luong y O.D. Faugeras, “The fundamental matrix: Theory, algorithms, and stability analysis,” International
Journal of Computer Vision, vol. 17, Ene. 1996, págs. 43-75.
 [12] A. Abellard, M. Bouchouicha, y Mohamed Moncef Ben Khelifa, “A genetic algorithm application to stereo
calibration,” Computational Intelligence in Robotics and Automation, 2005. CIRA 2005. Proceedings. 2005 IEEE
International Symposium on, 2005, págs. 285-290.
 [13] M. Bouchouicha, M. Khelifa, y W. Puech, “A non-linear camera calibration with genetic algorithms,” 2003,
págs. 189-192 vol.2.
Universidad del Valle – School of Computer and Systems Engineering Slide 46

47. References – Suite 2
 [14] Z. Yang, F. Chen, y J. Zhao, “A novel camera calibration method based on genetic algorithm,” 2008, págs.
2222-2227.
 [15] Dechao Wang, Yaqing Tu, y Tienan Zhang, “Research on the application of PSO algorithm in non-linear
camera calibration,” Intelligent Control and Automation, 2008. WCICA 2008. 7th World Congress on, 2008, págs.
4495-4500.
 [16] Xiaona Song, Bo Yang, Zhiquan Feng, Ting Xu, Deliang Zhu, y Yan Jiang, “Camera Calibration Based on
Particle Swarm Optimization,” Image and Signal Processing, 2009. CISP '09. 2nd International Congress on, 2009,
págs. 1-5.
 [17] J. Ze-Tao, W. Wenhuan, y W. Min, “Camera Autocalibration from Kruppa's Equations Using Particle Swarm
Optimization,” Proceedings of the 2008 International Conference on Computer Science and Software Engineering -
Volume 01, IEEE Computer Society, 2008, págs. 1032-1034.
 [18] H. Gao, B. Niu, Y. Yu, y L. Chen, “An Improved Two-Stage Camera Calibration Method Based on Particle
Swarm Optimization,” Emerging Intelligent Computing Technology and Applications. With Aspects of Artificial
Intelligence, 2009, págs. 804-813.
 [19] M. Ahmed, E. Hemayed, y A. Farag, “A neural approach for single- and multi-image camera calibration,”
1999, págs. 925-929 vol.3.
 [20] Qiang Ji y Yongmian Zhang, “Camera calibration with genetic algorithms,” Systems, Man and Cybernetics,
Part A: Systems and Humans, IEEE Transactions on, vol. 31, 2001, págs. 120-130.
 [21] Junghee Jun y Choongwon Kim, “Robust camera calibration using neural network,” TENCON 99.
Proceedings of the IEEE Region 10 Conference, 1999, págs. 694-697 vol.1.
 [22] M. Ahmed, E. Hemayed, y A. Farag, “Neurocalibration: a neural network that can tell camera calibration
parameters,” Computer Vision, 1999. The Proceedings of the Seventh IEEE International Conference on, 1999, págs.
463-468 vol.1.
 [23] K. Bilal y J. Qureshi, “Nature inspired optimization techniques for Camera calibration,” Emerging
Technologies, 2008. ICET 2008. 4th International Conference on, 2008, págs. 27-31.
 [24] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley
Professional, 1989.
Universidad del Valle – School of Computer and Systems Engineering Slide 47

48. References – Suite 3
 [25] Qiang Ji y Yongmian Zhang, “Camera calibration with genetic algorithms,” Systems, Man and Cybernetics,
Part A: Systems and Humans, IEEE Transactions on, vol. 31, 2001, págs. 120-130.
 [26] M. Roberts y A. Naftel, “A genetic algorithm approach to camera calibration in 3D machine vision,” 1994,
págs. 12/1-12/5.
 [27] J. Kennedy y R. Eberhart, “Particle swarm optimization,” Neural Networks, 1995. Proceedings., IEEE
International Conference on, 1995, págs. 1942-1948 vol.4.
 [28] C.M. Bishop, Neural networks for pattern recognition, Oxford University Press, 1995.
 [29] Junghee Jun y Choongwon Kim, “Robust camera calibration using neural network,” 1999, págs. 694-697
vol.1.
 [30] Jean-Yves Bouguet, “Camera Calibration Toolbox for Matlab,” Toolbox, California Institute of Technology
CALTECH.
 [31] Z. Zhang, “A Flexible New Technique for Camera Calibration,” IEEE Transactions on Pattern Analysis and
Machine Intelligence, vol. 22, 2000, págs. 1330-1334.
 [32] R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-
the-shelf TV cameras and lenses,” Robotics and Automation, IEEE Journal of, vol. 3, 1987, págs. 323-344.
 [33] J. Heikkilä, “Geometric Camera Calibration Using Circular Control Points,” IEEE Trans. Pattern Anal. Mach.
Intell., vol. 22, 2000, págs. 1066-1077.
 [34] W. Sun y J. Cooperstock, “An empirical evaluation of factors influencing camera calibration accuracy using
three publicly available techniques,” Machine Vision and Applications, vol. 17, Abr. 2006, págs. 51-67.
 [35] S. Kumar, M. Thakur, B. Raman, y N. Sukavanam, “Stereo camera calibration using real coded genetic
algorithm,” TENCON 2008 - 2008 IEEE Region 10 Conference, 2008, págs. 1-5.
 [36] Yingjie Xing, Qiao Liu, Jing Sun, y Long Hu, “Camera Calibration Based on Improved Genetic Algorithm,”
Automation and Logistics, 2007 IEEE International Conference on, 2007, págs. 2596-2601.
 [37] J. Chai y S. Ma, “Robust epipolar geometry using genetic algorithm,” Computer Vision — ACCV'98, 1997,
págs. 272-279.
Universidad del Valle – School of Computer and Systems Engineering Slide 48

49. References – Suite 4
 [38] G.R. Whitehead, “Estimating Intrinsic Camera Parameters from the Fundamental Matrix Using an
Evolutionary Approach,” EURASIP Journal on Applied Signal Processing, vol. vol. 2004, 2004, págs. pp. 1113-1124.
 [39] P.H.S. Torr, “Bayesian Model Estimation and Selection for Epipolar Geometry and Generic Manifold Fitting,”
Int. J. Comput. Vision, vol. 50, 2002, págs. 35-61.
 [40] Z. Zhang, “Determining the Epipolar Geometry and its Uncertainty: A Review,” Int. J. Comput. Vision, vol. 27,
1998, págs. 161-195.
 [41] Z. Zhang, R. Deriche, O. Faugeras, y Q. Luong, “A robust technique for matching two uncalibrated images
through the recovery of the unknown epipolar geometry,” Artif. Intell., vol. 78, 1995, págs. 87-119.
 [42] R. Subbarao y P. Meer, “Beyond RANSAC: User Independent Robust Regression,” Computer Vision and
Pattern Recognition Workshop, 2006. CVPRW '06. Conference on, 2006, pág. 101.
 [43] Z. Zhang, “Determining the Epipolar Geometry and its Uncertainty: A Review,” Int. J. Comput. Vision, vol. 27,
1998, págs. 161-195.
 [44] P.H.S. Torr y D.W. Murray, “The Development and Comparison of Robust Methodsfor Estimating the
Fundamental Matrix,” Int. J. Comput. Vision, vol. 24, 1997, págs. 271-300.
 [45] Yi-Jun Huang y Wei-Jun Liu, “Robust estimation for the fundamental matrix based on LTS and bucketing,”
2009, págs. 486-491.
 [46] M. Trujillo and E. Izquierdo (UK), “Robust Estimation of the Fundamental Matrix by Exploiting Disparity
Redundancies,” ACTA Press, Sep. 2003.
 [47] M.A. Fischler y R.C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to
image analysis and automated cartography,” Communications of the ACM, vol. 24, Jun. 1981, págs. 381–395.
 [48] B. Tordoff y D. Murray, “Guided Sampling and Consensus for Motion Estimation,” ACM Digital Library, 2002.
 [49] Robotics Research Group Visual Geometry Group. Multi-view and oxford colleges building reconstruction.
<http://www.robots.ox.ac.uk/ vgg/data/data-mview.html>.
 [50] <www.mathworks.com/image-video-processing>
Universidad del Valle – School of Computer and Systems Engineering Slide 49

50. Indoors by Iván Cabezas
THANKS
Universidad del Valle – School of Computer and Systems Engineering Slide 50

51. Derivation of Epipolar
Geometry from Proyective
Matrixes
Appendix A

52. Epipolar Geometry
 Relation between 𝒎 and
𝒎′ through 𝑷 and 𝑷′
𝒎 = 𝑷𝑴
𝑷+ 𝒎 = 𝑷+ 𝑷𝑴
𝑷+ 𝒎 = 𝑴
𝒎′ = 𝑷′𝑴
𝒎′ = 𝑷′𝑷+ 𝒎 Figure A1. Epipolar Geometry
Universidad del Valle – School of Computer and Systems Engineering Slide 52

53. Epipolar Geometry
 Relation between 𝒎 and
𝒎′ through 𝑷 and 𝑷′
𝒎′ = 𝑷′𝑷+ 𝒎
 Epipolar line equation
𝒍′ = 𝒆′ × 𝒎′
𝒍′ = 𝒆′ 𝒙 𝒎′
𝒍′ = 𝒆′ 𝒙 (𝑷′ 𝑷+ )𝒎 Figure A1. Epipolar Geometry
𝓕 = 𝒆′ 𝒙 𝑷′ 𝑷+
𝒍′ = 𝓕𝒎
Universidad del Valle – School of Computer and Systems Engineering Slide 53

54. Epipolar Geometry
 Epipolar line equation
𝒍′ = 𝓕𝒎
 Fundamental matrix
equation
𝒎′ 𝑻 𝒍′ = 0
𝒎′ 𝑻 𝓕𝒎 = 0 Figure A1. Epipolar Geometry
Universidad del Valle – School of Computer and Systems Engineering Slide 54

55. Epipolar Geometry 3D
Appendix B
Universidad del Valle – School of Computer and Systems Engineering

56. Epipolar Geometry 3D
Figure B1. Epipolar Geometry in 3D
Universidad del Valle – School of Computer and Systems Engineering Slide 56

57. Results Corridor Stereo Pair
Source: http://www.robots.ox.ac.uk/
Appendix C
Universidad del Valle – School of Computer and Systems Engineering

58. Results Corridor Stereo Pair
Figure C1. Disparity-Based Clustering of Correspondences
Universidad del Valle – School of Computer and Systems Engineering Slide 58

59. Results Corridor Stereo Pair
Figure C2. Elitistic Set of Correspondences
Universidad del Valle – School of Computer and Systems Engineering Slide 59

60. Results Corridor Stereo Pair
Figure C3. Epipolar Lines
Universidad del Valle – School of Computer and Systems Engineering Slide 60

61. Accuracy and Precision
Appendix D
Universidad del Valle – School of Computer and Systems Engineering

62. Accuracy and Precision
Figure D1. Accuracy and Precision
Universidad del Valle – School of Computer and Systems Engineering Slide 62

63. Stereo Capture
Appendix E
Universidad del Valle – School of Computer and Systems Engineering

64. Stereo Capture
Video E1. Stereo Rig and Corresponding Points
Universidad del Valle – School of Computer and Systems Engineering Slide 64

65. Correspondences Filtering
Appendix F
Universidad del Valle – School of Computer and Systems Engineering

66. Correspondences Filtering
Figure F1. Bad Located and False Matches Filtering
Figure F2. Epipolar Line for the Proposed GA-based + RANSAC + Guide Sampling
Universidad del Valle – School of Computer and Systems Engineering Slide 66