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2. Consider an image of a static scene acquired by a camera fixed on a tripod. Now the camera is rotated (butit remains fixed on the tripod without any translation) and another picture of the same scene is acquired. Let p1 and p2 be the pixel coordinates of the images of some physical point in the scene in the two images respectively. Note that p1 and p2 are in different coordinate systems. Derive a relation between p1 and p2 in terms of the matrix R which represents the rotational motion of the camera axes from the first position to the second, and the intrinsic parameter matrix K of the camera. Furthermore, if K is known, explain how will you determine R. [6 points]
3. In class, we have seen image formation on a flat screen (i.e. image plane) with a pinhole camera. Now supposethe screen was wrapped on the inner surface of a hemisphere and hence, the 3D points were projected onto a concave hemispherical surface. Derive a relationship between the coordinates of a 3D point P = (X,Y,Z) and its image on such a screen (both in camera coordinate system). If you had to calibrate this sort of a system, what are the additional intrinsic parameters of the camera as compared to the case of an image plane ? [6 points]
4. In this exercise, we will prove the orthocenter theorem pertaining to the vanishing points Q,R,S of three mutually perpendicular directions OQ,OR,OS, where O is the pinhole (origin of camera coordinate system). Let the image plane be Z = f without any loss of generality. Recall that two directions v1 and v2 are orthogonal if = 0. One can conclude that OS is orthogonal to OR − OQ (why?). Also the optical axis Oo (where o is the optical center) is orthogonal to OR − OQ (why?). Hence the plane formed by triangle OSo is orthogonal to OR−OQ and hence line oS is perpendicular to OR−OQ = QR (why?). Likewise oR and oQ are perpendicular to QS and RS. Hence we have proved that the altitudes of the triangle QRS are
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concurrent at the point o. QED. Now, in this proof, I considered the three perpendicular lines to be passing through O. What do you think will happen if the three lines did not pass through O? [6 points]
5. Consider two sets of corresponding points and . Assume that each pair of corresponding points is related as follows: p2i = αRp1i + t + ηi where R is an unknown rotation matrix, t is an unknown translation vector, α is an unknown scalar factor and ηi is a vector (unknown) representing noise. Explain how you will extend the method we studied in class for estimation of R to estimate α and t as well. Derive all necessary equations (do not merely guess the answers). [6 points]
6. You are given two datasets in the folder http://www.cse.iitb.ac.in/~ajitvr/CS763_Spring2016/HW1/ Calib_data. The file names are Features2D dataset1.mat, Features3D dataset1.mat, Features2D dataset2.mat and
Features3D dataset2.mat. Each dataset contains (1) the XYZ coordinates of N points marked out on a calibration object, and (2) the XY coordinates of their corresponding projections onto an image plane. Your job is to write a MATLAB program which will determine the 3×4 projection matrix M such that P1 = MP where P is a 4×N matrix containing the 3D object points (in homogeneous coordinates) and P1 is a 3×N matrix containing the image points (in homogeneous coordinates). Use the SVD method and print out the matrix M on screen (include it in your pdf file as well). Write a piece of code to verify that your computed M is correct. For any one dataset, repeat the computation of the matrix M after adding zero mean i.i.d. Gaussian noise of standard deviation σ = 0.05 × maxc (where maxc is the maximum absolute value of the X,Y,Z coordinate across all points) to every coordinate of P and P1 (leave the homogeneous coordinates unchanged). Comment on your results. Include these comments in your pdf file that you will submit. Tips: A mat file can be loaded into MATLAB memory using the ‘load’ command. To add Gaussian noise, use the command ‘randn’. [10 points]
(a) Apply the homography transformation in the file ‘Hmodel.mat’ to the image ‘goi1 downsampled.jpg’ using reverse warping to generate a warped image. Now estimate the homography that transforms the first image into its warped version. Apply the estimated transformation to the first image (using reverse warping) and display all three images side by side in your report. Also print the model and estimated homography matrices (make sure you normalize both so that H(3,3) = 1 in both cases).
(b) Determine the homography that transforms the image ‘goi1 downsampled.jpg’ to the second image ‘goi2 downsampled.jpg’. Warp the first image (using reverse warping) and compare it to the second. Display all three images side by side in your report. Also print the estimated homography matrix normalized so that H(3,3) = 1.
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