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Category: EECS 442

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Problem 1

(a) Derive the combined rotation and translation needed to transform world coordinate W into camera

coordinate C as illustrated in figure 1. Notice that Cz and Cx belong to the plane defined by Wz and

Wx.

(b) Consider a square in the world coordinate system defined by the points a,b,c,d. Assume such a square

has unit area. Show that the same square in the camera reference system has still unit area.

(c) Are parallel lines in the world reference system still parallel in the camera reference system? Justify

your answer.

(d) Does the vector defined by a and b have the same orientation in both reference system? Justify your

answer.

Figure 1

1

Problem 2

Consider a perspective projection where a point

P =

x

y

z

is projected onto an image plane Π′

represented by k = f

′ as shown in figure 2. The first, second and third

coordinate axes are denoted by i,j, and k, respectively. Consider the projection of an infinitely long line

Q =

1

1

0

+ t

0

0

1

in the world coordinate system where −∞ ≤ t ≤ −1. Calculate its two endpoints.

Figure 2

Problem 3

Two points x1 = (1, 3)⊤, x2 = (3, 1)⊤ in R

2 are transformed to x

′

1

, x

′

2 by a planar projective transformation

H

H =

1.520 −1.902 1.000

3.300 23.490 3.000

1.000 3.000 1.000

(a) Find the line l that passes through x1 and x2.

(b) Find the line l

′

that passes through x

′

1 and x

′

2

. You can use MATLAB to help with your computation.

(c) Derive an analytical expression that relates l with l

′

through H.

2

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