Camera calibration technique
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The document discusses camera calibration techniques. It aims to determine intrinsic camera parameters like focal length and optical center, and extrinsic parameters like the camera's position and orientation in 3D space. Zhang's algorithm is described, which allows estimating these parameters using a planar calibration target. It formulates the camera projection model and shows how to estimate the homography H relating the target's 3D points to 2D image points. H is defined up to a scale factor, so the absolute scale of the scene cannot be determined from this calibration alone. Constraints are also described to impose orthonormality of the rotation vectors.
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Camera calibration technique
- 1. Camera calibration technique wprowadzenie teoretyczne Krzysztof Wegner Chair of Multimedia Telecommunications and Microelectronics Poznań University of Technology, Poland 1
- 2. Goal of the calibration Knowledge about Intrinsic camera parameters Focal length Optical center Extrinsic camera parameters - Position of the camera in 3D world Orientation of the camera Translation 2
- 3. Goal of the calibration Knowledge about Intrinsic camera parameters Focal length Optical center Extrinsic camera parameters - Position of the camera in 3D world Orientation of the camera Translation Common word coordinate system 3
- 4. Camera parameters Intrinsic camera parameters Focal length Optical center Extrinsic camera parameters - Position of the camera in 3D world Orientation of the camera Translation 4 𝑨 = 𝑓𝑢 𝛾 𝑜 𝑢 0 𝑓𝑣 𝑜 𝑣 0 0 1 𝑹 = 𝒓 𝟏 𝒓 𝟐 𝒓 𝟑 𝑻 = 𝑡 𝑥 𝑡 𝑦 𝑡 𝑧
- 5. Camera model Projection of a 3D point 𝑴 = 𝑋 𝑌 𝑍 1 𝑇 Onto a point 𝒎 = 𝑢 𝑣 1 𝑇 at image plane s is a scale – distance to the point 𝑴 5 𝑠 ∙ 𝑢 𝑣 1 = 𝐴 ∙ 𝑅 −𝑅 ∙ 𝑇 ∙ 𝑋 𝑌 𝑍 1
- 6. Zhang’s Algorithm Allows estimation of intrinsic parameters - 𝑨 matrix extrinsic parameters – rotation matrix 𝑹 and translation vector 𝒕 = −𝑹 ∙ 𝒕 Planar template 𝑍 = 0 6 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑠 ∙ 𝑢 𝑣 1 = 𝐴 ∙ 𝑟1 𝑟2 𝑟3 𝑡 ∙ 𝑋 𝑌 𝑍 1
- 7. Zhang’s algorithm Planar template 𝑍 = 0 7 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑯 = 𝑨 ∙ 𝒓 𝟏 𝒓2 𝒕 𝑠 ∙ 𝑢 𝑣 1 = 𝑨 ∙ 𝒓 𝟏 𝒓 𝟐 𝒓 𝟑 𝒕 ∙ 𝑋 𝑌 𝑍 1 = 𝑨 ∙ 𝒓 𝟏 𝒓 𝟐 𝒓 𝟑 𝒕 ∙ 𝑋 𝑌 0 1 = = 𝑨 ∙ 𝒓 𝟏 ∙ 𝑋 + 𝒓 𝟐 ∙ 𝑌 + 𝒓 𝟑 ∙ 0 + 𝒕 ∙ 1 = 𝑨 ∙ 𝒓 𝟏 ∙ 𝑋 + 𝒓 𝟐 ∙ 𝑌 + 𝒕 ∙ 1 = = 𝑨 ∙ 𝒓 𝟏 𝒓2 𝒕 ∙ 𝑋 𝑌 1 = 𝑯 ∙ 𝑋 𝑌 1
- 8. Zhang’s algorithm Planar template 𝑍 = 0 8 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑯 = 𝑨 ∙ 𝒓 𝟏 𝒓2 𝒕 𝑠 ∙ 𝑢 𝑣 1 = 𝑨 ∙ 𝒓 𝟏 𝒓 𝟐 𝒓 𝟑 𝒕 ∙ 𝑋 𝑌 𝑍 1 = 𝑨 ∙ 𝒓 𝟏 𝒓 𝟐 𝒓 𝟑 𝒕 ∙ 𝑋 𝑌 0 1 = = 𝑨 ∙ 𝒓 𝟏 ∙ 𝑋 + 𝒓 𝟐 ∙ 𝑌 + 𝒓 𝟑 ∙ 0 + 𝒕 ∙ 1 = 𝑨 ∙ 𝒓 𝟏 ∙ 𝑋 + 𝒓 𝟐 ∙ 𝑌 + 𝒕 ∙ 1 = = 𝑨 ∙ 𝒓 𝟏 𝒓2 𝒕 ∙ 𝑋 𝑌 1 = 𝑯 ∙ 𝑋 𝑌 1
- 9. Estimating Homography H We know position of the pattern’s feature points 𝑋, 𝑌 From registrated image we know 𝑢, 𝑣 So 9 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1 𝑠 ∙ 𝑢 𝑣 1 = 𝐻11 𝐻12 𝐻13 𝐻21 𝐻22 𝐻23 𝐻31 𝐻32 𝐻33 ∙ 𝑋 𝑌 1 𝑠 ∙ 𝑢 = 𝐻11 ∙ 𝑋 + 𝐻12 ∙ 𝑌 + 𝐻13 𝑠 ∙ 𝑣 = 𝐻21 ∙ 𝑋 + 𝐻22 ∙ 𝑌 + 𝐻23 𝑠 = 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33 𝑢 = 𝐻11 ∙ 𝑋 + 𝐻12 ∙ 𝑌 + 𝐻13 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33 𝑣 = 𝐻21 ∙ 𝑋 + 𝐻22 ∙ 𝑌 + 𝐻23 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33
- 10. Estimating Homography H We know position of the pattern’s feature points 𝑋, 𝑌 From registrated image we know 𝑢, 𝑣 10 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑢 = 𝐻11 ∙ 𝑋 + 𝐻12 ∙ 𝑌 + 𝐻13 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33 𝑣 = 𝐻21 ∙ 𝑋 + 𝐻22 ∙ 𝑌 + 𝐻23 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33 𝑢 ∙ 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33 = 𝐻11 ∙ 𝑋 + 𝐻12 ∙ 𝑌 + 𝐻13 𝑣 ∙ 𝐻31 ∙ 𝑋 + 𝐻32 ∙ 𝑌 + 𝐻33 = 𝐻21 ∙ 𝑋 + 𝐻22 ∙ 𝑌 + 𝐻23 𝐻31 ∙ 𝑢 ∙ 𝑋 + 𝐻32 ∙ 𝑢 ∙ 𝑌 + 𝑢 ∙ 𝐻33 − 𝐻11 ∙ 𝑋 − 𝐻12 ∙ 𝑌 − 𝐻13 = 0 𝐻31 ∙ 𝑣 ∙ 𝑋 + 𝐻32 ∙ 𝑣 ∙ 𝑌 + 𝑣 ∙ 𝐻33 − 𝐻21 ∙ 𝑋 − 𝐻22 ∙ 𝑌 − 𝐻23 = 0 𝐻11 ∙ 𝑋 + 𝐻12 ∙ 𝑌 + 𝐻13 − 𝐻31 ∙ 𝑢 ∙ 𝑋 − 𝐻32 ∙ 𝑢 ∙ 𝑌 − 𝑢 ∙ 𝐻33 = 0 𝐻21∙ 𝑋 + 𝐻22 ∙ 𝑌 + 𝐻23 − 𝐻31 ∙ 𝑣 ∙ 𝑋 − 𝐻32 ∙ 𝑣 ∙ 𝑌 − 𝑣 ∙ 𝐻33 = 0
- 11. Estimating Homography H Let’s assign We have 2 equations and 9 variables so we need at least 5 points to solve uniquely for ℎ 11 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝐻11 ∙ 𝑋 + 𝐻12 ∙ 𝑌 + 𝐻13 − 𝐻31 ∙ 𝑢 ∙ 𝑋 − 𝐻32 ∙ 𝑢 ∙ 𝑌 − 𝑢 ∙ 𝐻33 = 0 𝐻21∙ 𝑋 + 𝐻22 ∙ 𝑌 + 𝐻23 − 𝐻31 ∙ 𝑣 ∙ 𝑋 − 𝐻32 ∙ 𝑣 ∙ 𝑌 − 𝑣 ∙ 𝐻33 = 0 𝒄 𝒖 ∙ 𝒉 = 𝟎 𝒄 𝒗 ∙ 𝒉 = 𝟎 𝒄 𝒖 = 𝑋 𝑌 1 0 0 0 −𝑢 ∙ 𝑋 −𝑢 ∙ 𝑌 −𝑢 𝒄 𝒖 = 0 0 0 𝑋 𝑌 1 −𝑣 ∙ 𝑋 −𝑣 ∙ 𝑌 −𝑣
- 12. Estimating Homography H Defined up to a scale factor 𝜆 We know position of the pattern’s feature points 𝑋, 𝑌 From registrated image we know 𝑢, 𝑣 Multiplication of both side by 𝜆 don’t change known 𝑋, 𝑌, 𝑢, 𝑣 So we don’t know whether we obtain 𝑯 or 𝑯 So we don’t know scale of the scene 12 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1 𝜆 ∙ 𝑠 ∙ 𝑢 𝑣 1 = 𝝀 ∙ 𝑯 ∙ 𝑋 𝑌 1 𝜆 ∙ 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1
- 13. Homography H Defined up to a scale factor 𝜆 We know position of the pattern’s feature points 𝑋, 𝑌 From registrated image we know 𝑢, 𝑣 13 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1
- 14. Homography H Defined up to a scale factor 𝜆 We know position of the pattern’s feature points 𝑋, 𝑌 From registrated image we know 𝑢, 𝑣 Multiplication of both side by 𝜆 don’t change known 𝑋, 𝑌, 𝑢, 𝑣 14 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1 𝜆 ∙ 𝑠 ∙ 𝑢 𝑣 1 = 𝝀 ∙ 𝑯 ∙ 𝑋 𝑌 1 𝜆 ∙ 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1
- 15. Homography H Defined up to a scale factor 𝜆 Multiplication of both side by 𝜆 don’t change known 𝑋, 𝑌, 𝑢, 𝑣 So we don’t know whether we obtain 𝑯 or 𝑯 So we don’t know Z scale of the scene 15 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1 𝜆 ∙ 𝑠 ∙ 𝑢 𝑣 1 = 𝝀 ∙ 𝑯 ∙ 𝑋 𝑌 1 𝜆 ∙ 𝑠 ∙ 𝑢 𝑣 1 = 𝑯 ∙ 𝑋 𝑌 1
- 16. Constraints on the intrinsic Vectors 𝒓 𝟏, 𝒓 𝟐 are orthonormal so taking dot product gives and length of 𝒓 𝟏, 𝒓 𝟐 should be the same For we have and 16 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑯 = 𝒉 𝟏 𝒉 𝟐 𝒉 𝟑 = 𝜆 ∙ 𝑨 ∙ 𝒓 𝟏 𝒓2 𝒕 𝒓 𝟏 𝑻 ∙ 𝒓 𝟐 = 𝟎 𝒓 𝟏 𝟐 = 𝒓 𝟐 𝟐 ⟹ 𝒓 𝟏 𝑻 ∙ 𝒓 𝟏 = 𝒓 𝟐 𝑻 ∙ 𝒓 𝟐 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝒉 𝟐 𝒉 𝟑 = 𝒓 𝟏 𝒓2 𝒕 𝒓 𝟏 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝒓2 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒕 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟑
- 17. Constraints on the intrinsic Puting all together 17 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝒓 𝟏 𝑻 ∙ 𝒓 𝟐 = 𝟎 𝒓 𝟏 𝑻 ∙ 𝒓 𝟏 = 𝒓 𝟐 𝑻 ∙ 𝒓 𝟐 𝒓 𝟏 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝒓2 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒕 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟑 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝑻 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝜆−1∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝑻 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝑻 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝜆−1 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑨−𝑻 ∙ 𝜆−1 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐
- 18. Constraints on the intrinsic Puting all together 18 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝒓 𝟏 𝑻 ∙ 𝒓 𝟐 = 𝟎 𝒓 𝟏 𝑻 ∙ 𝒓 𝟏 = 𝒓 𝟐 𝑻 ∙ 𝒓 𝟐 𝒓 𝟏 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝒓2 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒕 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟑 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝑻 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝜆−1∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝑻 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝑻 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝜆−1 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑨−𝑻 ∙ 𝜆−1 ∙ 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐
- 19. Closed form solution Try to solve Intrinsic matrix Let’s assign 19 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑨 = 𝑓𝑢 𝛾 𝑜 𝑢 0 𝑓𝑣 𝑜 𝑣 0 0 1 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝑩 = 𝑨−𝑻 ∙ 𝑨−𝟏 𝑩 = 1 𝑓𝑢 2 −𝛾 𝑓𝑢 2 𝑓𝑣 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 −𝛾 𝑓𝑢 2 𝑓𝑣 1 𝑓𝑣 2 + 𝛾2 𝑓𝑢 2 𝑓𝑣 2 −𝑜 𝑣 𝑓𝑣 2 − 𝛾 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 2 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 −𝑜 𝑣 𝑓𝑣 2 − 𝛾 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 2 1 + 𝑜 𝑣 2 𝑓𝑣 2 + 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 2 𝑓𝑢 2 𝑓𝑣 2 = 𝐵11 𝐵12 𝐵13 𝐵12 𝐵22 𝐵23 𝐵13 𝐵23 𝐵33
- 20. Closed form solution Let’s assign 𝑩 is symetrical Let’s assign 20 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑩 = 𝑨−𝑻 ∙ 𝑨−𝟏 𝑩 = 1 𝑓𝑢 2 −𝛾 𝑓𝑢 2 𝑓𝑣 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 −𝛾 𝑓𝑢 2 𝑓𝑣 1 𝑓𝑣 2 + 𝛾2 𝑓𝑢 2 𝑓𝑣 2 −𝑜 𝑣 𝑓𝑣 2 − 𝛾 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 2 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 −𝑜 𝑣 𝑓𝑣 2 − 𝛾 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 2 1 + 𝑜 𝑣 2 𝑓𝑣 2 + 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 2 𝑓𝑢 2 𝑓𝑣 2 = 𝐵11 𝐵12 𝐵13 𝐵12 𝐵22 𝐵23 𝐵13 𝐵23 𝐵33 𝒃 = 𝐵11 𝐵12 𝐵13 𝐵22 𝐵23 𝐵33
- 21. Closed form solution We try to solve Let’s see pattern in the equations 21 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑨−𝑻 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝑩 = 𝑨−𝑻 ∙ 𝑨−𝟏 𝒉 𝟏 𝑻 ∙ 𝑩 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑩 ∙ 𝒉 𝟏 = 𝒉 𝟐 𝑻 ∙ 𝑩 ∙ 𝒉 𝟐 𝒉𝒊 𝑻 ∙ 𝑩 ∙ 𝒉𝒋 = 𝒗𝒊𝒋 𝑻 ∙ 𝒃 𝒉𝒊 𝑻 ∙ 𝑩 ∙ 𝒉𝒋 = 𝒉𝒊 𝑻 ∙ 𝐵11 𝐵12 𝐵13 𝐵12 𝐵22 𝐵23 𝐵13 𝐵23 𝐵33 ∙ 𝒉𝒋=𝒉𝒊 𝑻 ∙ 𝐵11 ∙ ℎ𝑗1 + 𝐵12 ∙ ℎ𝑗2 + 𝐵13 ∙ ℎ𝑗3 𝐵12 ∙ ℎ𝑗1 + 𝐵22 ∙ ℎ𝑗2 + 𝐵23 ∙ ℎ𝑗3 𝐵13 ∙ ℎ𝑗1 + 𝐵23 ∙ ℎ𝑗2 + 𝐵33 ∙ ℎ𝑗3 = = 𝐵11 ∙ ℎ𝑖1 ∙ ℎ𝑗1 + 𝐵12 ∙ ℎ𝑖1 ∙ ℎ𝑗2 + 𝐵13 ∙ ℎ𝑖1 ∙ ℎ𝑗3 + = 𝒗𝒊𝒋 𝑻 ∙ 𝒃 +𝐵12 ∙ ℎ𝑖2 ∙ ℎ𝑗1 + 𝐵22 ∙ ℎ𝑖2 ∙ ℎ𝑗2 + 𝐵23 ∙ ℎ𝑖2 ∙ ℎ𝑗3 + +𝐵13 ∙ ℎ𝑖3 ∙ ℎ𝑗1 + 𝐵23 ∙ ℎ𝑖3 ∙ ℎ𝑗2 + 𝐵33 ∙ ℎ𝑖3 ∙ ℎ𝑗3 𝒗𝒊𝒋 𝑻 = ℎ𝑖1 ∙ ℎ𝑗1 ℎ𝑖1 ∙ ℎ𝑗2 + ℎ𝑖2 ∙ ℎ𝑗1 ℎ𝑖1 ∙ ℎ𝑗3 + ℎ𝑖3 ∙ ℎ𝑗1 ℎ𝑖2 ∙ ℎ𝑗2 ℎ𝑖2 ∙ ℎ𝑗3 + ℎ𝑖3 ∙ ℎ𝑗2 ℎ𝑖3 ∙ ℎ𝑗3
- 22. Closed form solution We try to solve so 22 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝒉 𝟏 𝑻 ∙ 𝑩 ∙ 𝒉 𝟐 = 𝟎 𝒉 𝟏 𝑻 ∙ 𝑩 ∙ 𝒉 𝟏 − 𝒉 𝟐 𝑻 ∙ 𝑩 ∙ 𝒉 𝟐 = 𝟎 𝒉𝒊 𝑻 ∙ 𝑩 ∙ 𝒉𝒋 = 𝒗𝒊𝒋 𝑻 ∙ 𝒃 𝒗 𝟏𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟏 𝑻 ∙ 𝒃 − 𝒗 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟏 𝑻 − 𝒗 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟐 𝑻 𝒗 𝟏𝟏 − 𝒗 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎
- 23. Solving for b Two equation are defined but 𝒃 have 6 unknowns. So at least 3 images are required to uniquly solve for 𝒃 Because all images are captured with the same camera we can stack equation for 𝒃 together 23 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝒗 𝟏𝟐 𝑻 𝒗 𝟏𝟏 − 𝒗 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟐 ′𝑻 𝒗′ 𝟏𝟏 − 𝒗′ 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗′′ 𝟏𝟐 𝑻 𝒗′′ 𝟏𝟏 − 𝒗′′ 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟐 𝑻 𝒗 𝟏𝟏 − 𝒗 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝒗 𝟏𝟐 𝑻 𝒗 𝟏𝟏 − 𝒗 𝟐𝟐 𝑻 𝒗 𝟏𝟐 ′𝑻 𝒗′ 𝟏𝟏 − 𝒗′ 𝟐𝟐 𝑻 𝒗′′ 𝟏𝟐 𝑻 𝒗′′ 𝟏𝟏 − 𝒗′′ 𝟐𝟐 𝑻 ∙ 𝒃 = 𝟎 𝑽 = 𝒗 𝟏𝟐 𝑻 𝒗 𝟏𝟏 − 𝒗 𝟐𝟐 𝑻 𝒗 𝟏𝟐 ′𝑻 𝒗′ 𝟏𝟏 − 𝒗′ 𝟐𝟐 𝑻 𝒗′′ 𝟏𝟐 𝑻 𝒗′′ 𝟏𝟏 − 𝒗′′ 𝟐𝟐 𝑻
- 24. Solving for b There is trivia solution 𝒃 = 𝟎 But we look for non trivial solution so 𝒃 ≠ 𝟎 Such solution is given by eigenvector of asociated with the smallest eigenvalue (right singulart vector of 𝑽) 24 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑽 ∙ 𝒃 = 𝟎 𝑽 𝑻 ∙ 𝑽
- 25. Retriving intrinsic parameters Once we have We can calculate intrinsic parameters from 25 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝑩 = 1 𝑓𝑢 2 −𝛾 𝑓𝑢 2 𝑓𝑣 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 −𝛾 𝑓𝑢 2 𝑓𝑣 1 𝑓𝑣 2 + 𝛾2 𝑓𝑢 2 𝑓𝑣 2 −𝑜 𝑣 𝑓𝑣 2 − 𝛾 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 2 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 −𝑜 𝑣 𝑓𝑣 2 − 𝛾 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 𝑓𝑢 2 𝑓𝑣 2 1 + 𝑜 𝑣 2 𝑓𝑣 2 + 𝑜 𝑣 𝛾 − 𝑓𝑣 𝑜 𝑢 2 𝑓𝑢 2 𝑓𝑣 2 = 𝐵11 𝐵12 𝐵13 𝐵12 𝐵22 𝐵23 𝐵13 𝐵23 𝐵33 𝒃 = 𝐵11 𝐵12 𝐵13 𝐵22 𝐵23 𝐵33 𝑜 𝑣 = 𝐵12 ∙ 𝐵13 − 𝐵11 ∙ 𝐵23 𝐵11 ∙ 𝐵22 − 𝐵12 2 𝑓𝑢 = 𝜆 𝐵11 𝜆 = 𝐵33 − 𝐵13 2 + 𝑜 𝑣 𝐵12 ∙ 𝐵13 − 𝐵11 ∙ 𝐵23 𝐵11 𝑓𝑣 = 𝜆 ∙ 𝐵11 𝐵11 ∙ 𝐵22 − 𝐵12 2 𝛾 = −𝐵12 ∙ 𝑓𝑢 2 ∙ 𝑓𝑣 𝜆 𝑜 𝑢 = 𝛾 ∙ 𝑜 𝑣 𝑓𝑣 − 𝐵13 𝑓𝑢 2 𝜆
- 26. Retriving position We know constraints To complete rotation matrix we calculater third column Finally we ortonormalize rotation matrix 26 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330–1334, 2000 𝒓 𝟏 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟏 𝒓2 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟐 𝒕 = 𝜆−1 ∙ 𝑨−𝟏 ∙ 𝒉 𝟑 𝒓 𝟑 = 𝒓 𝟏 × 𝒓 𝟐 𝒓 𝟏 = 𝒓 𝟐 = 𝒓 𝟑 = 𝟏
- 27. Summary Camera parameters requires at least 5 point pattern Intrinsic camera parameters estimation requires at least 3 images at different orientation All parameters are defined up to a unknown scale M37232, October 2015, Geneve 27