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Unformatted text preview: dinates can
Perspective projection of homogenous coordinates can also be done
also be done by matrix multiplication:!
by matrix multiplication:
0
1
10 0 0
B0 1 0 0C
C
Mp = B
@0 0 1 0A
0 0 1/f 0
01 0
1
x
x
By C B y C
C
Mp B C = B
@z A @ z A
1
z/f Graphics Lecture 2: Slide 20 !
19 / 45 Perspective Projection Matrix: Normalisation
Perspective Projection Matrix: Normalisation Perspective projection matrix: Normalisation !
• Remember we can normalise homogeneous coordinates,
so !
Remember we can normalise homogeneous coordinates, so
Remember we can normalise homogeneous coordinates, so
01 0 1
1
01 0
x
x
x
x
By C B y C
By C B y C
C hich
M B C=B
the same as
Mpp B zC = B z C wwhich is the same as !
@ A @ z A which isis the same as
A@
A
@z
!
1
z/f
1
z/f as required.
as required. ! 0
1
0
1
xf /z
xf /z
Byf /z C
Byf /z C
B
C
B
@ ff C
A
@
A
11 • as required. !
! Graphics Lecture 2: Slide 21 !
20 45
20 / / 45 Projection matrices are singular
Projection matrices are singular !
• Notice that both projection matrices are singular (i.e.
‘noninvertible’, zerodeterminant, …)!
Notice that both projection matrices are singular 2
0 1
B0
Mp = B
@0
0 00
10
01
0 1/f 1
0
0C
C
0A
0 0
1
B0
Mo = B
@0
0 0
1
0
0 0
0
0
0 1
0
0C
C
0A
1 • This is because a projection transformation cannot be
This is because a projection transformation cannot be inverted.
inverted. !
• Given 2D image, we in general reconstruct the original
Given a 2Daimage, we cannotcannot in general reconstruct the
3D original 3D scene. !
scene.
Graphics Lecture 2: Slide 22 !
2 A.K.A ‘noninvertible’, zerodeterminant, . . . Homogenous Coordinates as Vectors Homogenous coordinates as vectors !
We now take a second look at homogeneous coordinates, and their
• We now take a second look at homogeneous
relation to vectors. coordinates, and their relation to vectors. !
• In the previous lecture we described the fourth ordinate
In the previous lecture we described the fourth ordinate as a scale
as
factor.a scale factor. !
!
Homogeneous
Cartesian
Homogeneous
Cartesian !
01
x
By C
BC
@z A
s ! 0
1
x/s
@y/s...
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This document was uploaded on 03/26/2014.
 Spring '14

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