Massachusetts Institute of Technology Department of Computer Science and Electrical Engineering 6.801/6.866 Machine Vision QUIZ II

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1 Massachusetts Institute of Technology Department of Computer Science and Electrical Engineering 6.801/6.866 Machine Vision QUIZ II Handed out: 001 Nov. 30th Due on: 001 Dec. 10th Problem 1: (a (b Interior orientation camera calibration. How many degrees of freedom are there in interior orientation? How many degrees of freedom are there in exterior orientation? What is the minimum number of correspondences between target coordinates and image coordinates needed to fully constrain the camera calibration problem? Assume radial distortion is insignificant. Does the number of correspondences depend on whether one allows for an unknown horizontal scale factor? Note: answer part (a simply by matching the number of constraints to the number of unknowns, without reference to any particular method for estimating the unknown parameters. In Tsai s method for a non-planar target, when recovering a subset of the rotation and translation parameters using linear least squares, the following equation (see top of page 6 of the hand out is used (c (d (x S y I sr 11 + (y S y I sr 1 + (z S y I sr 13 + y I st x (x S x I r 1 (y S x I r (z S x I r 3 x I t y = 0 What is the minimum number of correspondences between target coordinates (x S,y S,z S T and image coordinates (x I,y I T needed to recover the unknowns? Note: keep in mind that the equation is homogeneous. In Tsai s method for a planar target, when recovering a subset of the rotation and translation parameters using linear least squares, the following equation (see top of page 8 of the hand out is used (x S y I r 11 + (y S y I r 1 + y I t x (x S x I r 1 (y S x I r x I t y = 0 What is the minimum number of correspondences between target coordinates (x S,y S,z S T and image coordinates (x I,y I T needed to recover the unknowns? Note: keep in mind that the equation is homogeneous. In Tsai s method, when recovering the principle distance f and the offset t z in the z direction, the following equations are used: s(r 11 x S + r 1 y S + r 13 z S + t x f x I t z = (r 31 x S + r 3 y S + r 33 z S x I (r 1 x S + r y S + r 3 z S + t x f y I t z = (r 31 x S + r 3 y S + r 33 z S y I

2 6.801/6.866 Quiz # (see top of page 10 of the hand out. What is the minimum number of correspondences between target coordinates (x S,y S,z S T and image coordinates (x I,y I T needed to recover the unknowns f and t z. (e (f Explain why it is bad for target points to be all at more or less the same distance along the optical axis from the camera. Is this a problem just for Tsai s method or inherent in the camera calibration problem? Find the error(s in the analysis at the top of page 9 down to the phrase which can be solved easily. Problem a: Least Squares Image Adjustment. In the classical least squares approach to photogrammetry, one minimizes the sum of squares of differences N (x I i x P i + (y I i y P i i=1 between observed image positions (x I,y I and predicted image positions (x P,y P based on scene coordinates and camera parameters. Perspective projection gives us x P i x o f = x c z C and y P i y o f = y c z C This approach is used whether one is trying to recover the parameters of the imaging situation (interior orientation, exterior orientation, and so on, or recover coordinates of points in the environment. Given image measurements of a feature point (x I 1,y I 1 determine the best fit coordinates of the corresponding point (x s 1,y s 1,z s 1 in the scene by minimizing the sum of squares of errors. Comment on the result. What happens when N>1? Problem b: In the case of (i absolute orientation, (ii exterior orientation, and (iii interior orientation, we used two-dimensional versions of the three-dimensional problems to get some insight. We didn t do this for relative orientation. In the case of linear cameras operating in the plane, what is the minimum number of correspondences between rays from the left camera and rays from the right camera that are needed to fully constrain the relative position and orientation of the right camera with respect to the left camera?

3 6.801/6.866 Quiz # 3 Problem 3: (a In each of the following six motion fields exactly one of the components of the six motion parameters was non-zero. For each of the six patterns state which parameter was non-zero

4 /6.866 Quiz # (b If possible, estimate the field of view θ of the camera in degrees (the ratio of one half the width of the image to the principle distance is tan θ/. If it is not possible to recover the width of the field of view, explain why. Problem 4: This problem address a possible ambiguity in interpreting motion fields. By differentiating the perspective projection equation 1 f r = R R ẑ with respect to time t, we get an equation for the motion field ṙ = dr/dt. Next, in rigid body motion we have for the velocity of a point in space relative to the camera: Ṙ = t ω r where t is the instantaneous translational velocity of the camera, while ω is the instantaneous rotational velocity. Now consider a camera looking at a planar surface. Suppose R 0 is an arbitrary point in the surface. Lines connecting points in the surface, such as (R R 0, are all perpendicular to the surface normal n of the plane. That is, (R R 0 n = 0 Show that the motion field when t = a, n = b, and ω = c is the same as when t = b, n = a, and ω = c + k a b for suitable choice of the constant k. What is the value of k? Draw a diagram showing an example of this kind of ambiguity where two different motions and two different surfaces give rise to the same motion field. (Note: In order to make things simpler, you may want to pick c = 0. Problem 5: Here we attempt to discover small systematic errors that may arise when using the simple formulae for estimating sub-pixel motion, when motion is considered constant over a patch. E i,j,k is the brightness in frame k at the pixel in row i and column j. The interval between frames is δt, and the pixel spacing is δx and δy. Then a least squares method based on the brightness change constraint equation yields:

5 6.801/6.866 Quiz # 5 ( ū = v ( i i j Ex i j E y E x i 1 ( j E x E y j Ey i i j E x E t j E y E t for the estimated velocity (ū, v We estimate the gradient components (see figure 1-7 in Robot Vision using: (E x i+1/,j+1/,k+1/ 1 (E i+1,j,k + E i+1,j,k+1 + E i+1,j+1,k + E i+1,j+1,k+1 4 δx (E i,j,k + E i,j,k+1 + E i,j+1,k + E i,j+1,k+1, (E y i+1/,j+1/,k+1/ 1 4 δy (E t i+1/,j+1/,k+1/ 1 4 δt 4δx (E i,j+1,k + E i,j+1,k+1 + E i+1,j+1,k + E i+1,j+1,k+1 (E i,j,k + E i,j,k+1 + E i+1,j,k + E i+1,j,k+1, 4 δy (E i,j,k+1 + E i,j+1,k+1 + E i+1,j,k+1 + E i+1,j+1,k+1 (E i,j,k + E i,j+1,k + E i+1,j,k + E i+1,j+1,k. 4 δt Now consider a simple sinusoidal pattern E i,j,k = E 0 + A cos(ωi uk Show that (E x i,j,k = A sin ( ω(i + 1 u/ sin(ω/ cos(ωu/ (E y i,j,k = 0 (E t i,j,k =+A sin ( ω(i + 1 u/ cos(ω/ sin(ωu/ Also show that in this case ū = / E x E t Ex i j i j and hence ū = tan(ωu/ /tan(ω/ Note that this is independent of what image patch the sum is over. Conclude that for small ω, the motion estimate is approximately correct.