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 #5
Due Date: 5/7/21, 11:59PM
• 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.
1 Camera Calibration (60 points)
The projection from world coordinates to the image plane, can be represented as a combination of intrinsic (camera) parameters and extrinsic (world) parameters. The first step is from the world coordinate (w) to the camera coordinate (c). The extrinsic parameters include one rotation matrix
Xc r11 r12 r13 Xw Tx
and one translation vector: Yc = r21 r22 r23·Yw +Ty. The second step is from camera
Zc r31r32r33 Zw Tz
xh fx 0 ox Xc coordinate (c) to the image coordinate (i) via intrinsic parameters, yh = 0 fy oy · Yc .
w 0 0 1 Zc From world coordinates to the camera coordinates, we can incorporate the intrinsic and extrinsic
m11 m12 m13 m14 xh m11 m12 m13 m14 Xw
parameters to get M = m21 m22 m23 m24 , where yh = m21 m22 m23 m24 m31 m32 m33 m34 w m31 m32 m33 m34
·Yw, Zw
1 (xi,yi) = (xh/w,yh/w) is the image coordinate, and (Xw,Yw,Zw) is the world coordinate of the corresponding point. Suppose there are N points and the n-th point is indicated by its image
coordinate (xni , yin) and world coordinate (Xwn, Ywn, Zwn).
• Express M using the extrinsic parameters and intrinsic parameters. (10 points)
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CSE 473/573 Homework 5 Spring 2021
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Calculate m231 + m232 + m233. (10 points)
We can build a linear equation: A·m = 0, where A is a 2N ×12 matrix and m = m11 m12 m13 m14 … m31 m32 m33 m34T is a 12 × 1 matrix. Express A only in terms of {xni , yin, Xwn, Ywn, Zwn}, n = 1…N. (20 points)
Since A·x = 0 equals A·(λx) = 0, we have infinite solutions for A·x = 0. Thus weneedtwostepstogetm: 1)SolvexwhereA·x=0and||x||=1;2)Solveλ to get m = λ · x. Suppose we have used single value decomposition (SVD) to get x =
x11 x12 x13 x14 … x31 x32 x33 x34T . Express λ and m only in terms of the 12 ele- ments of x. (20 points)
Image Rectification (20 points)
L and R are two images of one object taken from different views. The relationship between the two f11 f12 f13
image coordinates can be represented by the fundamental matrix F: xl yl 1 · f21 f22 f23 · f31 f32 f33
xr
yr = 0 , where [xl,yl] and [xr,yr] are the coordinates of the corresponding points on the left and
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right images. We use [xnl , yln] and [xnr , yrn] to denote the left/right coordinate of the n-th point.
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We can build a linear system: A · f = 0, where A is a N × 9 matrix and
f = f11 f12 f13 f21 f22 f23 f31 f32 of{xnl ,yln,xnr,yrn},n=1…N.
Morphology (20 points)
0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0
0 1 1 1 1 0 0 0 0
f33T is a 9×1 matrix. Express A only in terms
0 0 0 1 1 1 0 0 0
SupposeI=0 0 0 1 1 1 1 0 0,andweconsider3×3squarestructureelement.
0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 000000000
• What is the result applying dilation on I? (10 points) • What is the result applying erosion on I? (10 points)
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