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Estimation of the camera projection matrix_Assignment 2
Assignment 2
Problems
1. Programming: Estimation of the camera projection matrix (45 points)
(a) Linear estimation (15 points) Download input data from the course website. The file hw2_points3D.txt contains the coordinates of 50 scene points in 3D (each line of the file gives the ˜ Xi, ˜ Yi, and ˜ Zi inhomogeneous coordinates of a point). The file hw2_points2D.txt contains the coordinates of the 50 corresponding image points in 2D (each line of the file gives the ˜ xi and ˜ yi inhomogeneous coordinates of a point). The scene points have been randomly generated and projected to image points under a camera projection matrix (i.e., xi = PXi), then noise has been added to the image point coordinates. Estimate the camera projection matrix PDLT using the direct linear transformation (DLT) algorithm (with data normalization). You must express xi = PXi as [xi]⊥PXi = 0 (not xi ×PXi = 0), where [xi]⊥xi = 0, when forming the solution. Include the numerical values of the resulting PDLT, scaled such thatkPDLTkFro = 1, in your report with sufficient precision such that it can be evaluated (hint: use format shortg in MATLAB prior to displaying your results). Following is an example of the expected precision. PDLT = 0.0061695 −0.00467 0.0088168 0.85179 0.0089068 −0.0024049 −0.0062232 0.523634 .8924×10−6 4.3809×10−6 2.4664×10−6 0.0012341 (b) Nonlinear estimation (30 points) Use PDLT as an initial estimate to an iterative estimation method, specifically the Levenberg-Marquardt algorithm, to determine the Maximum Likelihood estimate of the camera projection matrix that minimizes the projection error. You must parameterize the camera projection matrix as a parameterization of the homogeneous vector p = vec(P). It is highly recommended to implement a parameterization of homogeneous vector method where the homogeneous vector is of arbitrary length, as this will be used in following assignments (see section A6.9.2 (page 624) of the textbook, and the corrections and errata). In your report, show the initial cost (i.e., the cost at iteration 0) and the cost at the end of each successive iteration. Show the numerical values for the final estimate of the camera projection matrix PLM, scaled such that kPLMkFro = 1, in your report with sufficient precision such that it can be evaluated
Problems
1. Programming: Estimation of the camera projection matrix (45 points)
(a) Linear estimation (15 points) Download input data from the course website. The file hw2_points3D.txt contains the coordinates of 50 scene points in 3D (each line of the file gives the ˜ Xi, ˜ Yi, and ˜ Zi inhomogeneous coordinates of a point). The file hw2_points2D.txt contains the coordinates of the 50 corresponding image points in 2D (each line of the file gives the ˜ xi and ˜ yi inhomogeneous coordinates of a point). The scene points have been randomly generated and projected to image points under a camera projection matrix (i.e., xi = PXi), then noise has been added to the image point coordinates. Estimate the camera projection matrix PDLT using the direct linear transformation (DLT) algorithm (with data normalization). You must express xi = PXi as [xi]⊥PXi = 0 (not xi ×PXi = 0), where [xi]⊥xi = 0, when forming the solution. Include the numerical values of the resulting PDLT, scaled such thatkPDLTkFro = 1, in your report with sufficient precision such that it can be evaluated (hint: use format shortg in MATLAB prior to displaying your results). Following is an example of the expected precision. PDLT = 0.0061695 −0.00467 0.0088168 0.85179 0.0089068 −0.0024049 −0.0062232 0.523634 .8924×10−6 4.3809×10−6 2.4664×10−6 0.0012341 (b) Nonlinear estimation (30 points) Use PDLT as an initial estimate to an iterative estimation method, specifically the Levenberg-Marquardt algorithm, to determine the Maximum Likelihood estimate of the camera projection matrix that minimizes the projection error. You must parameterize the camera projection matrix as a parameterization of the homogeneous vector p = vec(P). It is highly recommended to implement a parameterization of homogeneous vector method where the homogeneous vector is of arbitrary length, as this will be used in following assignments (see section A6.9.2 (page 624) of the textbook, and the corrections and errata). In your report, show the initial cost (i.e., the cost at iteration 0) and the cost at the end of each successive iteration. Show the numerical values for the final estimate of the camera projection matrix PLM, scaled such that kPLMkFro = 1, in your report with sufficient precision such that it can be evaluated
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