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Active Calibration of Cameras: Theory and Implementation Anup Basu

Active Calibration of Cameras: Theory and Implementation Anup Basu. Sung Huh CPSC 643 Individual Presentation II March 4 th , 2009. Outline. Introduction Theoretical Derivation Strategies for Active Calibration Theoretical Error Analysis Experimental Result Conclusion and Future Work.

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Active Calibration of Cameras: Theory and Implementation Anup Basu

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  1. Active Calibration of Cameras:Theory and ImplementationAnupBasu Sung Huh CPSC 643 Individual Presentation II March 4th, 2009

  2. Outline • Introduction • Theoretical Derivation • Strategies for Active Calibration • Theoretical Error Analysis • Experimental Result • Conclusion and Future Work

  3. Outline • Introduction • Theoretical Derivation • Strategies for Active Calibration • Theoretical Error Analysis • Experimental Result • Conclusion and Future Work

  4. Introduction • Important step for a 2D image to relates to the 3D world • Involves relating the optical features of a lens to the sensing device • Pose estimation, 3D motion estimation, automated assembly • Parameters: image center and focal length • Expressed in terms of image pixels • Linear vs. Nonlinear, Lens distortion consideration vs. w/o consideration

  5. Linear Nonlinear • Simpler to implement • Most cannot model camera distortions • Capable to consider complicated imaging model with many parameters • Computationally expensive search procedure • Reasonable good initial guess for convergence of the solutions Technique: Linearity

  6. Major Drawback of existing algorithm • Calibrate with predefined pattern • Relating image projections to the camera parameters • Recent algorithms suffer from the same limitation • New discovery: Active Calibration

  7. Active Calibration • Camera capable of panning and tilitng can automatically calibrate itself • Modeled from eye movement • Active machines can keep track of object of interest • Facilitate region-of-interest process

  8. Active Calibration – How different? • Does not need a starting estimate for focal length and image center • Does not need prior information about focal length • Does not need to match points or feature b/w images • Reasonably accurate localization of contour • Estimate of center (Not too far from true value)

  9. Method of Calibration • Using perspective distortion to measure calibration parameters • Without using perspective distortion

  10. Outline • Introduction • Theoretical Derivation • Strategies for Active Calibration • Theoretical Error Analysis • Experimental Result • Conclusion and Future Work

  11. Theoretical Derivation

  12. Theoretical DerivationLemma 1 • Camera rotates by R and translate by T • New image contours

  13. Theoretical DerivationLemma 1 - Proof • Use two set of equation • ,

  14. Theoretical DerivationProposition 1 • Depth (Z) is larger than ΔX, ΔY, ΔZ • Camera moves by small tilt angle

  15. Theoretical DerivationProposition 1 – Proof • Rotation matrix R at small tilt angle • are negligible • From Lemma 1

  16. Theoretical DerivationProposition 1 – Proof • Expand right side of equation with Taylor series, because of small θt • With the same assumption, if camera moves by small pan angle θp

  17. Outline • Introduction • Theoretical Derivation • Strategies for Active Calibration • Theoretical Error Analysis • Experimental Result • Conclusion and Future Work

  18. Strategy for Active Calibration • Want – A relation b/w lens parameters and image information w/ given image contours before & after camera motion • Relate focal length to other camera parameters and the pan/tilt angles

  19. Strategy for Active CalibrationProposition 2 • Similar assumption as Proposition 1 • Center of the lens is estimated with a small error (δx, δy)

  20. Strategy for Active CalibrationProposition 2 – Proof • From Proposition 1 • Estimate image Center with error (δx, δy) • Ignore δxδy term

  21. Strategy for Active CalibrationProposition 3 (Plan A) • Using tilt (or pan) movement and considering three independent static contours, two linear equation in δxδy can be obtained if negligible terms are ignored

  22. Strategy for Active CalibrationProposition 3 – Proof • Two different contour, C1 and C2 • Point lying on C1 & C2, (x(1),y(1)) and (x(2),y(2)) • From Proposition 2

  23. Strategy for Active CalibrationProposition 3 – Proof • Equate right side equations and simplify where

  24. Strategy for Active CalibrationProposition 3 – Proof • Third contour, C3 • Point on C3, (x(3),y(3)) where

  25. Strategy for Active CalibrationProposition 3 – Proof • Finding fx and fy with estimated center • “e” denote the estimate of a certain parameter

  26. Strategy for Active Calibration Procedure Summary for Plan A • Estimate δx and δy using (3) and (4) with three distinct image contour • Obtain estimate for fx and fy by substituting resulting estimate into (5) and (6) • Term and make (5) and (6) unstable

  27. Strategy for Active Calibration Procedure Summary for Plan A • Variation in x-coord. for any point is due to change in perspective distortion (tilt) • Little change in the image y-coord. corresponding to a given 3-d point (pan) • and are small (few pixel) • Relative error can be large • presence of noise and inaccuracies in localization of a contour • Estimate in (5) & (6) are often unreliable

  28. Strategy for Active Calibration Proposition 4 (Plan B) • Using a single contour and pan/tilt camera movements fx and fy can be obtained if negligible terms are ignored

  29. Strategy for Active Calibration Proposition 4 – Proof • δx and δy are non-zero in the second equation in Proposition 1 • Simplify • The last three terms are negligible even if δx and δy are large

  30. Strategy for Active Calibration Proposition 4 – Proof • Simplifying eq (7) • fx can be obtained with similar way

  31. Strategy for Active Calibration Proposition 4 – Corollary • Given two independent contours, pan/tilt camera movements, and estimate of fx and fy given by and respectively, δx and δy can be obtained by solving • Considering from two independent contour from Proposition 2

  32. Strategy for Active Calibration Proposition 4 – Proof • Consider (8) • Most practical system • y < 300, fy > 500 • (8) is in form • A = 1, B < 0, C is small compare to B

  33. Strategy for Active Calibration Procedure Summary for Plan B • Estimate fx and fy from (12) and (13) using a single image contour • Solve for δx and δy by substituting resulting estimates into (10) and (11) and using another independent contour

  34. Strategy for Active Calibration Proposition 5 • When there is error in contour localization after pan/tilt movements, the ratio of the error in Plan A compared to Plan B for estimating fx(fy)is approximately

  35. Strategy for Active Calibration Proposition 5 – Proof • Introduce similar error term in and in (13) and (5) respectively • Simplify the expressions and consider the approximate magnitude of error in both the expressions • Take the ratio of these two terms

  36. Strategy for Active Calibration Proposition 5 – Implication • Error in Plan A can be as large as 30 times that of Plan B, for estimating focal lengths • Plan A is theoretically more precise, but not reliable for noisy real scenes

  37. Outline • Introduction • Theoretical Derivation • Strategies for Active Calibration • Theoretical Error Analysis • Experimental Result • Conclusion and Future Work

  38. Theoretical Error Analysis • Effect of errors from various sources on the estimation of different parameters • Errors in measurements of pan/tilt angles • Effect of noise in the extraction of image contours

  39. Theoretical Error Analysis • Remark 1 • Error in measurement of the pan (tilt) angle generates a proportional error in the estimate of fx(fy) • Proof • Consider(5) • fx is proportional to the pan angle • Any error in the measurement translates to a corresponding error in fx • Any error from tilt angle generate a proportional error in fy

  40. Theoretical Error Analysis • Remark 2 • Errors in measurement of the pan & tilt angles do not affect the estimate of the lens center • independent contours from the same image are considered • Proof • Linear equations in δx and δy are obtained by equating the right hand sides of two equations

  41. Theoretical Error Analysis • Consider (1) & (2) • Denote ε1: error in tilt angle • Contour extracted from same image • Then (θt+ε1) of (3) cancels out from both sides • Error in pan/tilt angle do not affect the estimate of lens center

  42. Theoretical Error Analysis • Consider two independent images generating the contours in (1) & (2) • K1 in (3) modifies to • is not equal to 1 in general • Errors in angle can change the estimate of the lens center if contours from independent images are considered

  43. Theoretical Error Analysis • Remark 3 • The coefficients of the linear (3)-(6) are unbiased in the presence of uncorrelated noise with zero mean • Coefficients involve a linear combination of terms • These terms are unbiased in the presence of uncorrelated noise with zero mean

  44. Theoretical Error Analysis • Remark 4 • The variance of the coefficients of (3)-(6) is inversely proportional to the number of points on a contour • Uncorrelated noise with zero mean is considered • Form of variances • Inversely proportional to the number of points for which the averages were computed

  45. Outline • Introduction • Theoretical Derivation • Strategies for Active Calibration • Theoretical Error Analysis • Experimental Result • Conclusion and Future Work

  46. Experimental ResultSimulation – Validity of Algorithms • Synthetic data used • Three independent contour represented by three sets of 3D points • Points projected onto the image plane • Values quantized to the nearest integer • Without noise, A produced more accurate estimate • Less than 1 percent relative error in focal length estimate

  47. Experimental ResultSimulation – Variation of error in focal length estimate • Change fx and fy from 100 to 1000 with interval 100 • Keep other parameters fixed • Discretization error influence A more when focal length was small • A is not very robust to noise • Larger the focal length, smaller the error relative to the focal length • Better estimate production with A

  48. Experimental ResultSimulation – Variation of error in focal length estimate • Error estimate from B does not drop off rapidly • B is theoretically less accurate than A

  49. Experimental ResultSimulation – Gaussian Noise Added • Poor performance with A • 20, 28, and 40 percent error with 3, 4, and 5 noise standard deviation • High robustness with B

  50. Experimental ResultTracking Contour • Match contours of interest during pan/tilt • For automatic calibration • Edges in the original image was thickened using the morphological operation of “dilation” • Edges after pan/tilt was AND-ed with the dilated image to extract corresponding contours after camera rotation

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