## Convenient calibration method for unsynchronized camera networks using an inaccurate small reference object |

Optics Express, Vol. 20, Issue 23, pp. 25292-25310 (2012)

http://dx.doi.org/10.1364/OE.20.025292

Acrobat PDF (2132 KB)

### Abstract

In this paper, a new and convenient calibration algorithm is proposed for unsynchronized camera networks with a large capture volume. The proposed method provides a simple and accurate means of calibration using a small 3D reference object. Moreover, since the inaccuracy of the object is also compensated simultaneously, the manufacturing cost can be decreased. The extrinsic and intrinsic parameters are recovered simultaneously by capturing an object placed arbitrarily in different locations in the capture volume. The proposed method first resolves the problem linearly by factorizing projection matrices into the camera and the object pose parameters. Due to the multi-view constraints imposed on factorization, consistency of the rigid transformations among cameras and objects can be imposed. These consistent estimation results can be further refined using a non-linear optimization process. The proposed algorithm is evaluated via simulated and real experiments in order to verify that it is more efficient than previous methods.

© 2012 OSA

## 1. Introduction

3. T. Ueshiba and F. Tomita, “Plane-based calibration algorithm for multi-camera systems via factorization of homography matrices,” in “*Proc. IEEE International Conference on Computer Vision*,” (Nice, France, 2003), pp. 966–973. [CrossRef]

*et al.*suggested a more convenient method in which a freely moving bright spot is used as the calibration object [4

4. T. Svoboda, D. Martinec, and T. Pajdla, “A convenient multi-camera selfcalibration for virtual environments,” Presence: Teleop. Virt. Environ. **14**, 407–422 (2005). [CrossRef]

*Genlock*, this requirement cannot always be guaranteed. If synchronization is not ensured for the method, then a large number of still images should be captured, as in

*stop-motion*animation, which is a very difficult task. Consequently, such a method is unsuitable for unsynchronized systems, so in this case we would have no choice but to use large 2D or 3D reference objects, as in previous methods. Moreover, methods based on the self-calibration technique require a great deal of overlap among the many cameras to ensure stable calibration results. This implies the need for densely distributed cameras. The self-calibration technique also suffers from critical configurations of cameras and reference points, for which self-calibration is not possible [1].

*et al.*and Medeiros

*et al.*each proposed a method based on pairwise calibration [5

5. G. Kurillo, Z. Li, and R. Bajcsy, “Wide-area external multi-camera calibration using vision graphs and virtual calibration object,” in “Proc. ACM/IEEE International Conference on Distributed Smart Cameras,” (2008), pp. 1–9. [CrossRef]

6. H. Medeiros, H. Iwaki, and J. Park, “Online distributed calibration of a large network of wireless cameras using dynamic clustering,” in “Proc. ACM/IEEE International Conference on Distributed Smart Cameras,” (2008), pp. 1–10. [CrossRef]

*et al.*used an iterative approach to refine the calibration results from unsynchronized cameras [7]. However, all these methods assume that the intrinsic parameters are given

*a priori*, which contributes to stable calibration results for sparsely distributed cameras.

8. S. N. Sinha and M. Pollefeys, “Camera network calibration and synchronization from silhouettes in archived video,” Int. J. Comput. Vision **87**, 266–283 (2010). [CrossRef]

*frontier points*. The latter can be used to determine the epipolar geometry between any two views. This elegant approach has the advantage of not using any calibration object. However, many frontier points should be spread over the images (See Fig.8 in [8

8. S. N. Sinha and M. Pollefeys, “Camera network calibration and synchronization from silhouettes in archived video,” Int. J. Comput. Vision **87**, 266–283 (2010). [CrossRef]

*stop-motion*animation. Although this work also suggested a scheme for solving the problem of unsynchronized video streams, an approximate interpolation strategy was used to treat sub-frame discrepancy, and it can only be applied when the subject’s inter-frame motion is small.

*frontier points*, it requires good initialization of the camera parameters.

4. T. Svoboda, D. Martinec, and T. Pajdla, “A convenient multi-camera selfcalibration for virtual environments,” Presence: Teleop. Virt. Environ. **14**, 407–422 (2005). [CrossRef]

## 2. Related works

*et al.*proposed a method of placing a 3D calibration object in only one position in each pairwise view overlap [10

10. J. Kassebaum, N. Bulusu, and W.-C. Feng, “3-d target-based distributed smart camera network localization,” IEEE Trans. Image Process. **19**, 2530–2539 (2010). [CrossRef] [PubMed]

4. T. Svoboda, D. Martinec, and T. Pajdla, “A convenient multi-camera selfcalibration for virtual environments,” Presence: Teleop. Virt. Environ. **14**, 407–422 (2005). [CrossRef]

11. C. Tomasi and T. Kanade, “Shape and motion from image streams under orthography: a factorization method,” Int. J. Comput. Vision **9**, 137–154 (1992). [CrossRef]

3. T. Ueshiba and F. Tomita, “Plane-based calibration algorithm for multi-camera systems via factorization of homography matrices,” in “*Proc. IEEE International Conference on Computer Vision*,” (Nice, France, 2003), pp. 966–973. [CrossRef]

16. M. Wilczkowiak, P. Sturm, and E. Boyer, “Using geometric constraints through parallelepipeds for calibration and 3D modelling,” IEEE Trans. Pattern Anal. Mach. Intell. **27**, 194–207 (2005). [CrossRef] [PubMed]

**14**, 407–422 (2005). [CrossRef]

*et al.*[16

16. M. Wilczkowiak, P. Sturm, and E. Boyer, “Using geometric constraints through parallelepipeds for calibration and 3D modelling,” IEEE Trans. Pattern Anal. Mach. Intell. **27**, 194–207 (2005). [CrossRef] [PubMed]

## 3. Preliminaries

### 3.1. Camera parameterization

**X**in the 3D world coordinate system to a point

**x**in the 2D image of the

*i*th camera is expressed as follows: where

**x**and

**X**are homogeneous coordinates of the points and ’≅’ indicates equality up to scale.

**K**

*is the camera’s intrinsic matrix, given by [1]. The rotation matrix*

_{i}**R**

*and the vector*

_{i}**t**

*represent the camera’s orientation and position, respectively. The 3 × 4 matrix*

_{i}**M**

*encapsulates the camera’s intrinsic and extrinsic parameters.*

_{i}### 3.2. Object pose parameterization

**X**

_{ref}be the homogeneous coordinates of a reference point with respect to the coordinate system attached to the 3D object. When the object is located at the

*j*th position on the world coordinate system, the point is represented as follows: where the rotation matrix

**S**

*represents the object’s orientation and the vector*

^{j}**v**

*its position.*

^{j}## 4. Image measurements

### 4.1. Projection matrix

*projection matrix*. It represents a perspective projection that maps the reference points on the object located at the

*j*th position onto the corresponding imaged points of the

*i*th camera. This is illustrated in Fig 1. The rotation matrix

**R**

_{i}**S**

*and the vector*

^{j}**R**

_{i}**v**

*+*

^{j}**t**

*represent the*

_{i}*i*th camera’s rotation and position with respect to the coordinate system attached to the

*j*th object.

*j*th location and can be viewed from the

*i*th camera. Let

*k*th point on the reference object with respect to the coordinate system attached to the object. Let

*l*th row of

*n*point correspondences, a 2

*n*× 12 matrix Λ can be obtained by stacking up the Eq. (4). The projection matrix is computed up to scale by solving the set of equations

*i*th camera when the object is placed in the

*j*th location, and

### 4.2. Measurement matrix

*n*different locations and seen by

*m*cameras. Let

*i*th camera and the

*j*th object pose and

*m*cameras and

*n*object poses into the following single matrix: The matrix

**W̃**will be called the

*measurement matrix*. When the scale factors are recovered, the measurement matrix can be factorized as follows: However, since the projection matrices are obtained up to scale, the measurement matrix cannot be factorized at its current form.

## 5. Parameter estimation

### 5.1. Rescaling the measurement matrix (Method 1)

*reduced measurement*matrix can be factorized as follows:

*a*} have the values such that for all the

_{i}*i*’s, then From this observation, we can see that if all the

### 5.2. Rescaling the measurement matrix (Method 2)

**K**

*is an upper triangular matrix and (*

_{i}**R**

_{i}**S**

*) is also rotation matrix, we can think that*

^{j}**R**

_{i}**S**

*. Moreover, since (*

^{j}**K**

*)*

_{i}_{33}is 1, (Σ)

_{33}is

### 5.3. Analysis of the two rescaling methods

*a*,

*b*, and

*c*. Then, the scale factor from

*Method 2*is 1/

*c*. It is worthwhile to note that {(

**K**

*)*

_{i}_{11}, (

**K**

*)*

_{i}_{22}}(= {

*f*,

_{u}*f*}) = {

_{v}*a*/

*c*,

*b*/

*c*}. The determinant of

*abc*.

*Method 1*is (

*abc*)

^{−1/3}and

*Method 2*is (

*ccc*)

^{−1/3}. However, it is worthwhile to note that these scale factors are not ideal values because they are computed from

*a*,

*b*, and

*c*are also deviated from their true values. Since these variables are independent of each other, the product of

*a*,

*b*, and

*c*has noise reduction effect compared to

*ccc*. On the other hand, the product of three

*c*’s amplifies its noise.

*abc*and

*ccc*. For example, the relative error of

*abc*is ‖

*abc*−

*ãb̃c̃*‖/

*abc*, where {

*a*,

*b*,

*c*} are obtained from

*ã*,

*b̃*,

*c̃*} from contaminated

### 5.4. Factorization

**Ỹ**. Let the SVD of

**Ỹ**be given as:

**D**contains the singular values of

**Ỹ**:

*σ*

_{1}≥

*σ*

_{2}≥. . . ≥

*σ*

_{3}

*. In the absence of noise,*

_{n}**Ỹ**satisfying Eq. (10) has rank 3 and consequently

*σ*

_{4}=

*σ*

_{5}= . . . =

*σ*

_{3}

*= 0. If noise were present, this would not be the case. If we want to find the rank 3 matrix which is closest to*

_{n}**Ỹ**in the Frobenius norm, such a matrix can be obtained by setting all the singular values, besides the three largest ones, to zero. Let

**Ū**and

**V̄**be the matrices consisting of the first three columns of

**U**and

**V**, respectively. Then, the rank-3 factorization result can be given as: where

**T**is an arbitrary non-singular 3 × 3 matrix. To obtain the camera and object pose parameters, we have to resolve this affine ambiguity. This issue is considered in the next section. It is worthwhile noting that this ambiguity always exists even if the method described in section 5.2 is used.

### 5.5. Resolving affine ambiguity

**Û**

_{3m}

_{×3}and

**V̂**

_{2}

_{n}_{×3}in Eq. (16) be decomposed in the 3 × 3 and 2 × 3 submatrices, respectively:

*j*’s, it can be seen that

**TT**

*≅*

^{T}**D**because

**V̂**

^{T}**V̂**=

**D**and, consequently, that

**R**

*is an arbitrary rotation matrix representing the fact that the global Euclidean reference frame can be chosen arbitrarily.*

_{w}### 5.6. Reconstructing the camera and object pose parameters

**K**

*and*

_{i}**R**

*from RQ-decomposition of*

_{i}**U**

_{i}**T**and

**S**

*from the rotation matrix closest to*

_{j}18. K. S. Arun, T. S. Huang, and S. D. Blosten, “Least-squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. **PAMI-9**, 698–700 (1987). [CrossRef]

**t**

*and*

_{i}**v**

*.*

^{j}### 5.7. Filling missing entries

**Ỹ**are missing and the factorization cannot proceed. To fill the missing entries, the following simple deduction process can be used. where

*k*th camera and the

*l*th object position. The candidates for the intermediate

*k*th camera and the

*l*th object position may be multiple. In this case, all the candidates can be used simultaneously to compensate the image noise.

### 5.8. Refining the parameters

*k*th point on the reference object and

*i*th camera when the object is placed in the

*j*th location.

### 5.9. Compensating the inaccuracy of a reference object

**R**

*and the translation vectors*

_{f}**t**

*of the planes with respect to the coordinate attached to the object. These variables are described in Fig. 3 for one example plane. Since the planes at different positions can be viewed from more than two cameras concurrently, as in Fig. 3, and the variables are not changed while the object moves, they can be computed to satisfy overall consistency. The re-projection error function is extended from Eq. (20) as follows: where*

_{f}*f*(

*k*) is the index of the face including the

*k*th point.

**R**

*(*

_{f}*k*) and

**t**

*(*

_{f}*k*) are initialized according to the original drawing of the object. The other parameters are initialized with the estimation results obtained up to section 5.7.

### 5.10. Dealing with radial distortion

*x̆*,

*y̆*) are represented as follows: where (

*x*,

*y*) are distortion-free normalized image coordinates, and

*k*

_{1}and

*k*

_{2}are the coefficients of the radial distortion. An initial guess of

*k*

_{1}and

*k*

_{2}can be obtained through similar equations to Eq. (13) in [2], which can be modified to be proper for the proposed method or simply by setting them to 0.

## 6. Experimental results

### 6.1. Simulated experiments

*ENV1*and

*ENV2*, respectively.

*f*,

_{u}*f*,

_{v}*s*,

*u*

_{0},

*v*

_{0})=(1600, 1600, 0, 800, 600). Zero-mean Gaussian-distributed noise with standard deviation

*σ*was added to the image projections. It was assumed that there were no gross outliers owing to false correspondence. However, to consider outliers in feature point detection owing to perspective and photometric effect, 10% of the noise-vectors are replaced with those from another Gaussian distribution with 2

*σ*for all simulated experiments.

*f*,

_{v}*u*, and

_{o}*v*are not depicted here.

_{o}#### 6.1.1. Performance w.r.t noise level, object size, and number of object positions

*ENV2*, more object positions were needed than in

*ENV1*, because not all the cameras viewed the same region and the object was placed in several separate view overlaps. Fig. 5 shows the visibility map for the object positions in

*ENV2*when the number of object positions is 25. From this map, it can be seen that the results for

*ENV2*validate the process of filling the missing entries described in Section 5.7.

*et al.*[10

10. J. Kassebaum, N. Bulusu, and W.-C. Feng, “3-d target-based distributed smart camera network localization,” IEEE Trans. Image Process. **19**, 2530–2539 (2010). [CrossRef] [PubMed]

#### 6.1.2. Comparison of the two rescaling methods

*ENV2*. After carrying out the factorization step, the RPE results obtained from

*Method 1*were better than those obtained with the

*Method 2*. These results can be expected from the analysis of Section 5.3.

*σ*= 0.5) and a larger object (360mm), the final results after the refinement step are fortunately similar. However, the differences in the initial values affect the final results when the noise level increases or a smaller object (240mm) is used.

**Ỹ**rescaled by the two methods. Fig. 6 shows the magnitude of the singular values of rescaled

**Ỹ**for each environment of Table 2. To be close to rank 3, the ratio of the 3rd and 4th singular values should be large with respect to the ratio of the 1st and 3rd singular values. We can see that this ratio of

*Method 1*is larger than that of

*Method 2*. From these experimental results, it can be concluded that

*Method 1*is more desirable than

*Method 2*.

#### 6.1.3. Performance w.r.t object inaccuracy and radial distortion

*ENV1*and 25 for

*ENV2*, respectively. The results are summarized in Table 3. It can be seen that the proposed method can be used even if the reference object is inaccurate. The radial distortion parameters were also well estimated. The proposed algorithm could not converge for the inaccuracy level more than (

*θ̂*, ‖

_{f}**t̂**

*‖)=(3°, 5mm) in*

_{f}*ENV1*and (1°, 3mm) in

*ENV2*(See Table 3 for an explanation of the parameters). Since there were many missing entries in

*ENV2*compared with

*ENV1*, error propagation during the filling process caused this difference. However, this performance means that we can allow a considerable degree of inaccuracy with the object and decrease the manufacturing cost.

#### 6.1.4. Comparison with the method of [44. T. Svoboda, D. Martinec, and T. Pajdla, “A convenient multi-camera selfcalibration for virtual environments,” Presence: Teleop. Virt. Environ. **14**, 407–422 (2005). [CrossRef]

]

**14**, 407–422 (2005). [CrossRef]

**14**, 407–422 (2005). [CrossRef]

**14**, 407–422 (2005). [CrossRef]

**14**, 407–422 (2005). [CrossRef]

*ENV2*look down at the capture volume, the points on the upper plane of the object can be viewed at three adjacent cameras simultaneously. This is the reason why the calibration results can be obtained in

*ENV2*compared to the results in

*ENV1*when the number of cameras is small (5 or 6). To identify the effects of the number of cameras, more cameras were located evenly in the environments while fixing the number of object positions. As mentioned above, more accurate calibration results were obtained. This implies that the method of [4

**14**, 407–422 (2005). [CrossRef]

### 6.2. Real image experiments

#### 6.2.1. Manufacturing a reference object

#### 6.2.2. Extraction of feature points

19. A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. **21**, 476–480 (1999). [CrossRef]

#### 6.2.3. Actual camera network environments

*ENV1*in Fig. 4(a) and

*ENV2*in Fig. 4(b). We refer to these environments as

*Network1*and

*Network2*, respectively. Fig. 8 shows the illustration of the camera networks. Table 5 shows the detailed specifications of the camera networks.

*Network1*is shown in the left-hand image of Fig. 8. Images captured from some cameras of

*Network1*are shown in Fig. 9. Fig. 10 shows the recovered cameras and objects. From these figure, it can be seen that the qualitative positions of cameras and objects were estimated correctly. Since the ground truth is not available in real image experiments, the re-projection results are good indices of the performance of the proposed method. The average re-projection error(RPE) of the reference points was 0.0835 pixels. When the inaccuracy of the object was not compensated by the technique described in Section 5.9, the average RPE was 0.284 pixels. This implies that the object had inaccuracy and the pose discrepancy of the object’s faces from an original drawing was well estimated. The average of the estimated discrepancy of the position was 2.32mm.

*Network2*. Fig. 11 shows the images of the reference object captured from some cameras. Fig. 12 shows the visibility map for the object positions in

*Network2*. The filling process described in Section 5.7 was conducted successfully in this experiment. The visualization of the calibration results are shown in Fig. 13. The average RPE was 0.0473 pixels. When the inaccuracy of the object was not compensated, the average RPE is 0.148 pixels.

*Network1*and

*Network2*. The calibration results are shown in Table 6 and 7. We can see that the differences between the results from the subgroups of the cameras and those from all the cameras are quite small. These results imply that the proposed algorithm provides consistent and stable estimation results.

**14**, 407–422 (2005). [CrossRef]

*Network1*or

*Network2*. In these environments, no features could be viewed from more than two cameras simultaneously due to the sparse distribution of the cameras.

## 7. Conclusions

**14**, 407–422 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | R. Hartley and A. Zisserman, |

2. | Z. Zhang, “A flexible new technique for camera calibration,” Tech. Rep. MSR-TR-98-71, Microsoft Corporation (1998). |

3. | T. Ueshiba and F. Tomita, “Plane-based calibration algorithm for multi-camera systems via factorization of homography matrices,” in “ |

4. | T. Svoboda, D. Martinec, and T. Pajdla, “A convenient multi-camera selfcalibration for virtual environments,” Presence: Teleop. Virt. Environ. |

5. | G. Kurillo, Z. Li, and R. Bajcsy, “Wide-area external multi-camera calibration using vision graphs and virtual calibration object,” in “Proc. ACM/IEEE International Conference on Distributed Smart Cameras,” (2008), pp. 1–9. [CrossRef] |

6. | H. Medeiros, H. Iwaki, and J. Park, “Online distributed calibration of a large network of wireless cameras using dynamic clustering,” in “Proc. ACM/IEEE International Conference on Distributed Smart Cameras,” (2008), pp. 1–10. [CrossRef] |

7. | X. Chen, J. Davis, and P. Slusallek, “Wide area camera calibration using virtual calibration objects,” in “Proc. IEEE International Conference on Computer Vision and Pattern Recognition,” Hilton Head, SC, USA (2000), pp. 520–527. |

8. | S. N. Sinha and M. Pollefeys, “Camera network calibration and synchronization from silhouettes in archived video,” Int. J. Comput. Vision |

9. | E. Boyer, “On using silhouettes for camera calibration,” in “Proc. Asian Conference on Computer Vision,” Hyderabad, India (2006), pp. 1–10. |

10. | J. Kassebaum, N. Bulusu, and W.-C. Feng, “3-d target-based distributed smart camera network localization,” IEEE Trans. Image Process. |

11. | C. Tomasi and T. Kanade, “Shape and motion from image streams under orthography: a factorization method,” Int. J. Comput. Vision |

12. | P. Sturm and B. Triggs, “A factorization based algorithm for multi-image projective structure and motion,” in “Proc. European Conference on Computer Vision,” Cambridge, UK (1996), pp. 709–720. |

13. | D. Jacobs, “Linear fitting with missing data: Applications to structure from motion and to characterizing intensity images,” in “Proc. IEEE International Conference on Computer Vision and Pattern Recognition,” San Juan, Puerto Rico (1997), pp. 206–212. |

14. | D. Martinec and T. Pajdla, “Structure from many perspective images with occlusions,” in “Proc. European Conference on Computer Vision,” Copenhagen, Denmark, (2002), pp. 355–369. |

15. | P. Sturm, “Algorithms for plane-based pose estimation,” in “Proc. IEEE International Conference on Computer Vision and Pattern Recognition,” Hilton Head Island, SC, USA, (2000), pp. 706–711. |

16. | M. Wilczkowiak, P. Sturm, and E. Boyer, “Using geometric constraints through parallelepipeds for calibration and 3D modelling,” IEEE Trans. Pattern Anal. Mach. Intell. |

17. | O. Faugeras, |

18. | K. S. Arun, T. S. Huang, and S. D. Blosten, “Least-squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. |

19. | A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. |

20. | R. Hartley, “In defence of the 8-point algorithm,” in “Proc. International Conference on Computer Vision,” Sendai, Japan (1995), pp. 1064–1070. |

**OCIS Codes**

(150.0155) Machine vision : Machine vision optics

(150.1135) Machine vision : Algorithms

(150.1488) Machine vision : Calibration

**ToC Category:**

Machine Vision

**History**

Original Manuscript: August 10, 2012

Revised Manuscript: September 27, 2012

Manuscript Accepted: October 2, 2012

Published: October 22, 2012

**Citation**

Jae-Hean Kim and Bon-Ki Koo, "Convenient calibration method for unsynchronized camera networks using an inaccurate small reference object," Opt. Express **20**, 25292-25310 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25292

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### References

- R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd. ed. (Cambridge University Press, 2003).
- Z. Zhang, “A flexible new technique for camera calibration,” Tech. Rep. MSR-TR-98-71, Microsoft Corporation (1998).
- T. Ueshiba and F. Tomita, “Plane-based calibration algorithm for multi-camera systems via factorization of homography matrices,” in “Proc. IEEE International Conference on Computer Vision,” (Nice, France, 2003), pp. 966–973. [CrossRef]
- T. Svoboda, D. Martinec, and T. Pajdla, “A convenient multi-camera selfcalibration for virtual environments,” Presence: Teleop. Virt. Environ.14, 407–422 (2005). [CrossRef]
- G. Kurillo, Z. Li, and R. Bajcsy, “Wide-area external multi-camera calibration using vision graphs and virtual calibration object,” in “Proc. ACM/IEEE International Conference on Distributed Smart Cameras,” (2008), pp. 1–9. [CrossRef]
- H. Medeiros, H. Iwaki, and J. Park, “Online distributed calibration of a large network of wireless cameras using dynamic clustering,” in “Proc. ACM/IEEE International Conference on Distributed Smart Cameras,” (2008), pp. 1–10. [CrossRef]
- X. Chen, J. Davis, and P. Slusallek, “Wide area camera calibration using virtual calibration objects,” in “Proc. IEEE International Conference on Computer Vision and Pattern Recognition,” Hilton Head, SC, USA (2000), pp. 520–527.
- S. N. Sinha and M. Pollefeys, “Camera network calibration and synchronization from silhouettes in archived video,” Int. J. Comput. Vision87, 266–283 (2010). [CrossRef]
- E. Boyer, “On using silhouettes for camera calibration,” in “Proc. Asian Conference on Computer Vision,” Hyderabad, India (2006), pp. 1–10.
- J. Kassebaum, N. Bulusu, and W.-C. Feng, “3-d target-based distributed smart camera network localization,” IEEE Trans. Image Process.19, 2530–2539 (2010). [CrossRef] [PubMed]
- C. Tomasi and T. Kanade, “Shape and motion from image streams under orthography: a factorization method,” Int. J. Comput. Vision9, 137–154 (1992). [CrossRef]
- P. Sturm and B. Triggs, “A factorization based algorithm for multi-image projective structure and motion,” in “Proc. European Conference on Computer Vision,” Cambridge, UK (1996), pp. 709–720.
- D. Jacobs, “Linear fitting with missing data: Applications to structure from motion and to characterizing intensity images,” in “Proc. IEEE International Conference on Computer Vision and Pattern Recognition,” San Juan, Puerto Rico (1997), pp. 206–212.
- D. Martinec and T. Pajdla, “Structure from many perspective images with occlusions,” in “Proc. European Conference on Computer Vision,” Copenhagen, Denmark, (2002), pp. 355–369.
- P. Sturm, “Algorithms for plane-based pose estimation,” in “Proc. IEEE International Conference on Computer Vision and Pattern Recognition,” Hilton Head Island, SC, USA, (2000), pp. 706–711.
- M. Wilczkowiak, P. Sturm, and E. Boyer, “Using geometric constraints through parallelepipeds for calibration and 3D modelling,” IEEE Trans. Pattern Anal. Mach. Intell.27, 194–207 (2005). [CrossRef] [PubMed]
- O. Faugeras, Three-Dimensional Computer Vision (The MIT Press, 1993).
- K. S. Arun, T. S. Huang, and S. D. Blosten, “Least-squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-9, 698–700 (1987). [CrossRef]
- A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell.21, 476–480 (1999). [CrossRef]
- R. Hartley, “In defence of the 8-point algorithm,” in “Proc. International Conference on Computer Vision,” Sendai, Japan (1995), pp. 1064–1070.

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