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CALIBRATION

CALIBRATION. Presentation by Sumit Tandon Department of Electrical Engineering University of Texas at Arlington Course # EE6358 Computer Vision. Topics to be covered. Coordinate systems Rigid body transformations Rotation matrix Absolute orientation Relative orientation

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CALIBRATION

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  1. CALIBRATION Presentation by Sumit Tandon Department of Electrical Engineering University of Texas at Arlington Course # EE6358 Computer Vision

  2. Topics to be covered • Coordinate systems • Rigid body transformations • Rotation matrix • Absolute orientation • Relative orientation • Exterior orientation • Interior orientation • Camera calibration EE-6358 Computer Vision

  3. Co-ordinate Systems • Pixel Co-ordinate System • Image Co-ordinate System – it is a part of the Camera Co-ordinate System • Camera Co-ordinate System • Scene/Absolute Co-ordinate System EE-6358 Computer Vision

  4. Relation between Image Co-ordinate and Pixel Co-ordinates EE-6358 Computer Vision

  5. What do we calibrate • Absolute orientation: between two coordinate systems • Relative orientation: between two camera systems • Exterior orientation: between camera and scene systems • Interior orientation: within a camera, such as camera constant, principal point, lens distortion, etc EE-6358 Computer Vision

  6. Rigid Body Transformations • R = rotation matrix • p2 = co-ordinates of the same point in a different co-ordinate axes • p0 can be viewed to be the co-ordinates of the origin of the first co-ordinate system when viewed from the second co-ordinate system • To map/transform each point in the image from one co-ordinate system to the other we need the rotation matrix R and the translation vector p0 EE-6358 Computer Vision

  7. Rigid Body Transformation Contd. EE-6358 Computer Vision

  8. Rotation Matrix • Rotation matrix is orthonormal • The rotation matrix can be represented in three forms • Euler angles – Not very stable because a small change in angle can result in a large change in rotation EE-6358 Computer Vision

  9. Rotation Matrix Contd • Unit Quaternion Representation – Calculate the unit quaternions and the rotation matrix can be obtained • The rotation matrix can be represented as EE-6358 Computer Vision

  10. Rotation matrix contd • An important property of the quaternions is the representation of the rotated point r’ = qrq’ • where • q = quaternion representing rotation • r = quaternion representation of the point r • q’ = complex conjugate of q • Axis of rotation – Use a scalar value to denote the angle of rotation about unit vector. But it has same problems as the Euler angle EE-6358 Computer Vision

  11. Absolute Orientation • It is concerned with rigid body transformations between two co-ordinate systems • Simply put, we need to calculate the rotation matrix and the translation vector • Algorithm used • Find out the centroid of the cloud of the camera centered and the absolute co-ordinate system points EE-6358 Computer Vision

  12. Absolute Orientation contd. • Subtract the centroids from the actual points to get two bunches of rays originating at the centroids • To find the rotation matrix minimize the sum of the dot-products of the absolute co-ordinate rays and the rotated camera centered rays • Find the translation vector from the centroids and the rotation matrix EE-6358 Computer Vision

  13. Absolute Orientation contd. • We have EE-6358 Computer Vision

  14. Absolute Orientation Contd. • Where EE-6358 Computer Vision

  15. Absolute Orientation with Scale • If absolute coordinate system and camera coordinate system have different ratios, scale problem is introduced • Distances between points in the 3D scene are independent of the choice of the co-ordinate system • Scale factor can be calculated as • Now the transformation can be represented as EE-6358 Computer Vision

  16. Relative Orientation • The problem of Relative Orientation deals with the determination of the relation between two camera co-ordinate systems • The projection of the scene point on the left and right image planes is known • The baseline and the depth appear as ratio EE-6358 Computer Vision

  17. Relative Orientation contd. EE-6358 Computer Vision

  18. Relative Orientation contd. • Let and be the direction from camera centers to scene point, respectively • Since baseline of stereo system and the normal of epipolar plane are perpendicular, we have the condition • By Least Square Method, the minimization condition then becomes EE-6358 Computer Vision

  19. Relative Orientation contd. • Depth and translation (baseline) appear as a ratio • To solve for the matrices we put the constraint • Constrained minimization problem is • Where is the Lagrange multiplier • Solution of the above equation gives the unit vector corresponding to the baseline (translation) EE-6358 Computer Vision

  20. Relative Orientation contd. • A value of rotation was assumed for calculating the direction of baseline • Estimates for rotation and baseline can be refined using Iterative Methods • Change in the minimizing condition is • Where • As before we get an expression for minimization with the Lagrange multiplier subject to EE-6358 Computer Vision

  21. Relative Orientation contd. • Updated Baseline is given by the formula • Representing rotation by Unit Quaternion, the infinitesimal rotation is given by • Updated Rotation is given by EE-6358 Computer Vision

  22. Exterior Orientation • The Exterior Orientation problem is to determine the relation between the image plane co-ordinates (X,Y) and the co-ordinates of the scene point in absolute co-ordinate system (x,y,z) • The relation between the absolute and camera co-ordinates is known • The relation between the camera and image co-ordinates is known EE-6358 Computer Vision

  23. Exterior Orientation • Co-ordinates of a point in absolute co-ordinate system be • Co-ordinates in camera co-ordinate system be • Thus we can write EE-6358 Computer Vision

  24. Exterior Orientation contd. • Relation between the image co-ordinates and camera co-ordinates is given by • The relation for the calibration point is then given by EE-6358 Computer Vision

  25. Exterior Orientation contd. • For each calibration point (x’, y’) and (xa,ya,za) are known • 6 points are required to solve for 12 unknowns • If the condition for orthonormal rotation matrix is also considered then only 4 points are required for calibration EE-6358 Computer Vision

  26. Interior Orientation • This is the problem related to internal geometry of the camera • Parameters are • Camera constant – distance of the image plane from the center of projection • Principal point – location of the origin of image plane co-ordinate system • Lens distortion coefficients – changes in the image plane coordinates caused by the optical imperfections in the camera • Scale factors – for the distance between the rows and columns of the image EE-6358 Computer Vision

  27. Radial Distortion EE-6358 Computer Vision

  28. Interior Orientation contd. • Radial distortion is modeled as a polynomial of even powers of the radius • Let  estimate of the location of principal point • Therefore uncorrected image coordinates are • True image plane coordinates can be represented as • Corrections for radial lens distortion are modeled as • Where EE-6358 Computer Vision

  29. Interior Orientation contd. • Choice of calibration target • The target chosen generally consists of several straight lines at different positions and orientations • If exterior orientation calibration is not considered, any orientation of the lines can be taken • A straight line in the scene will be a straight line in the image too • The straight lines will appear as edges in the image – these edges are grouped together using Hough Transform EE-6358 Computer Vision

  30. Interior Orientation contd. EE-6358 Computer Vision

  31. Interior orientation contd. • The parametric equation of the lines is • Substituting the calibration model • We get a function • Least square method is then used to find the parameters EE-6358 Computer Vision

  32. Camera calibration • Camera calibration problem involves relating the image points with the scene points • It combines both Interior and Exterior Orientation problems • Methods for camera calibration • Tsai’s camera calibration algorithm • Affine method for camera calibration • Non-linear method for camera calibration EE-6358 Computer Vision

  33. Tsai’s method of camera calibration • Simple algorithm that decouples exterior and interior orientation problems • Calculates 11 parameters • Five interior parameters – focal length of camera, coefficient for radial lens distortion, center of projection, scale factor • Six exterior parameters – three for rotation and three for translation EE-6358 Computer Vision

  34. Affine method of camera calibration • This method considers the exterior and interior calibrations together • Can take care of • Scale error – due to inaccurate value of camera constant • Translation error – due to inaccurate estimate of principal point • Rotation of image sensor about optical axis • Skew error – due to non-orthogonal camera axes • Differential scaling – due to unequal spacing between rows and columns of image sensor • Cannot take care of errors due to lens distortion EE-6358 Computer Vision

  35. Affine method contd. • Affine transformation is defined as • Since is an arbitrary matrix, the transformation includes rotation, translation, scaling , skew transform. EE-6358 Computer Vision

  36. Affine method contd • Using the projection model, affine transformation becomes EE-6358 Computer Vision

  37. Affine method contd. • From exterior orientation problem • Substituting these values in the transformation model we get EE-6358 Computer Vision

  38. Affine method contd. • Equations to be solved EE-6358 Computer Vision

  39. Non-linear method for camera calibration • This method combines the exterior and interior orientation problems • This method also accounts for the lens distortions • From exterior orientation problem we have • From interior orientation problem we have EE-6358 Computer Vision

  40. Non-linear method for camera calibration contd. • Combining the two sets of equations we get • Rotation is expressed in Euler angles • With a set of calibration points we can solve for the unknowns EE-6358 Computer Vision

  41. Non-linear method for camera calibration contd. • Solution is obtained iteratively • Initial conditions assumed are • Angles – • Camera location in absolute coordinate system - • Nominal value of the focal length of camera - • Principal point – • Lens distortion coefficients - EE-6358 Computer Vision

  42. References • Machine Vision – Ramesh Jain, Rangachar Kasturi, Brian G Schunck, McGraw-Hill, 1995 • Robot Vision – Berthold Klaus Paul Horn, The MIT Press, 1986 • A Versatile Camera Calibration Technique for High-Accuracy 3D Machine Vision Metrology Using Off-the-shelf TV Cameras and Lenses – Roger Y Tsai, IEEE Journal of Robotics and Automation, 1987 EE-6358 Computer Vision

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