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Answer the questions below, and provide as much of your work as necessary.
Export or scan your homework and store it as a PDF version before submitting online to UBLearns.
Image Formation (20 points)
Please name the possible shapes produced in an image under perspective of (1) a scene con- taining one sphere; (2) a scene containing one disk? You may assume the sphere and disk have a non-zero radius, and the disk is very thin? (5 points/ each)
Suppose you have a camera (unit aspect ratio, optical center at (0,0), no skew) that takes a picture here on earth. Now suppose we have a new world where everything is twice as large, including distances between objects. Assuming perspective projection, compare the two images. (10 points)
2D Transformation (15 points)
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University at Buffalo
Department of Computer Science and and Engineering CSE 473/573 – Computer Vision and Image Processing
Instructions
Spring 2021, TuTh 9:30AM-10:50AM
Homework #1
Due Date: 2/18/21, 11:59PM
There are two steps in the house transformation shown below to map a point (x,y) to (x′,y′): translation, and rotation. The transformation between the first and the third 2D coordinates is
x′ x
y′ = R·y, where R = Rrotation ·Rtranslation. Suppose P1 = (10,20) in the first coordinate and 11
θ = 30o, calculate the transformation matrix Rtranslation, Rrotation, R. (5 points/each)
CSE 473/573 Homework 1
3 3D Rotation (30 points)
Suppose we have a coordinate system A that can be mapped to a coordinate system B in two steps: 1) R1: rotate π around X axes; 2) R2: rotate π/2 around Z axes. The rotation follows right hand rule. The overall rotation R carries out this mapping from A to B.
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Give each 3×3 matrix R1, R2, R. (15 points, 5 points/each)
Given a point whose coordinate is [10, 0, 20] in A, calculate its coordinates B. (5 points) Give the 3×3 matrix R′ carries out mapping from B to A. (5 points)
Given a point whose coordinate is [10, 0, 20] in B, calculate its coordinates A. (5 points)
Projection Matrix (15 points)
Suppose we have the following intrinsic parameters of camera: focal length is 2, the coordinate of image center is [1, 1] . There is unit aspect ratio and no skew of the pixels.
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Determine the intrinsic matrix. (5 points)
If the coordinate of a point is [0,2,1] in the camera coordinate, calculate its coordinate on the image plane. (10 points)
Intrinsic Matrix (20 points)
The projection from world coordinates to the image plane, can be represented by intrinsic (camera) parameters and extrinsic (world) parameters. From world coordinates to the camera coordinates, the extrinsic parameters are
fx 0 ox The intrinsic matrix is given by Min = 0 fy oy .
001 The entire projection matrix can be expressed as
• Express each element of M using the extrinsic and intrinsic parameters. (10 points)
• Express the intrinsic parameters (fx , fy , ox , oy ) only in terms of each element of M . (10 points)