1 Dr. Scott Schaefer Surface Parameterization. Parameterization and Texturing 2/30.
30
1 Dr. Scott Schaefer Surface Parameterization
-
date post
21-Dec-2015 -
Category
Documents
-
view
232 -
download
3
Transcript of 1 Dr. Scott Schaefer Surface Parameterization. Parameterization and Texturing 2/30.
1
Dr. Scott Schaefer
Surface Parameterization
Parameterization and Texturing
2/30
3/30
Mercator Projection
4/30
Mercator Projection
Image taken from http://idvux.spaces.live.com
Types of Distortion
Isometric MappingsPreserve lengths
Conformal MappingsPreserve angles
Equiareal MappingsPreserve area
5/30
Mathematics of Parameterization
6/30
Mathematics of Parameterization
7/30
12121 ˆˆˆ ppttp
Mathematics of Parameterization
8/30
12121 ˆˆˆ ppttp
21,
singular values
Mathematics of Parameterization
9/30
12121 ˆˆˆ ppttp
21
conformal
Mathematics of Parameterization
10/30
12121 ˆˆˆ ppttp
121
equiareal
Mathematics of Parameterization
11/30
12121 ˆˆˆ ppttp
121
isometric
Conformal and Harmonic Mappings
12/30
)),(),,((),(ˆ yxvyxuyxp
Conformal and Harmonic Mappings
13/30
)),(),,((),(ˆ yxvyxuyxp
y
v
x
u
x
v
y
u
Conformal and Harmonic Mappings
14/30
)),(),,((),(ˆ yxvyxuyxp
y
v
x
u
x
v
y
u
0u 0v
Conformal => Harmonic
Harmonic Maps
15/30
2,1i1,i
2,i
),( ii up
1,1i),( 00 up
01,12,1,12,0 )cot()cot()cot()cot()( uupUi
iii
iii
Harmonic Maps
16/30
2,1i1,i
2,i
1,1i
01,12,1,12,0 )cot()cot()cot()cot()( uupUi
iii
iii
B
U
C
CL T 0
0
),( ii up
),( 00 up
Mean Value Map
17/30
2,1i1,i
2,i
1,1i
),( ii up
),( 00 up1ii
0
1 )2/tan()2/tan(
ppw
i
iii
Mean Value Map
18/30
2,1i1,i
2,i
1,1i
),( ii up
),( 00 up1ii
B
U
C
CM T 0
0
ith row contains mean value weights for ith vertex
Comparison
19/30
Original Barycenter Harmonic Mean value
Conformal Maps
20/30
)),(),,((),(ˆ yxvyxuyxp
y
v
x
u
x
v
y
u
Conformal Maps
21/30
0 vu
Conformal Maps
22/30
iiii
vuvu
2
,min
Conformal Maps
23/30
),,( 000 vup ),,( 111 vup
),,( 222 vup
iiii
vuvu
2
,min
Conformal Maps
24/30
),,( 000 vup ),,( 111 vup
),,( 222 vup
x
),,(
),,(),,(),,()(
210
210102021
ppparea
uppxareauppxareauppxareaxu
),,(
),,(),,(),,()(
210
210102021
ppparea
vppxareavppxareavppxareaxv
iiii
vuvu
2
,min
Conformal Maps
25/30
),,( 000 vup ),,( 111 vup
),,( 222 vup
x
),,(
)()()()(
210
210102021
ppparea
uppuppuppxu
),,(
)()()()(
210
201120012
ppparea
vppvppvppxv
iiii
vuvu
2
,min
Conformal Maps
26/30
),,( 000 vup ),,( 111 vup
),,( 222 vup
x
210102021
210102021
210 )()()(
)()()(
),,(
1)()(
vppvppvpp
uppuppupp
pppareaxvxu
iiii
vuvu
2
,min
Conformal Maps
27/30
),,( 000 vup ),,( 111 vup
),,( 222 vup
x
2102110221
21021102210
2
212
0 )()()()(
)()()()(1)()(
vppppvpppp
uppppuppppuppxvxu
u
iiii
vuvu
2
,min
Conformal Maps
28/30
),,( 000 vup ),,( 111 vup
),,( 222 vup
x
2102110221
21021102210
2
212
0 )()()()(
)()()()(1)()(
vppppvpppp
uppppuppppuppxvxu
u
iiii
vuvu
2
,min
Sum over all trianglesLinear system of equations!
Iterative Optimization
29/30
Sti
tch g
lobally
Input 3D mesh
Output 2D parameterizationImage taken from A Local/Global Approach to Mesh Parameterization
Conclusions
Many, many, many methods exist for parameterization
Not all guarantee 1-to-1 mappings (fold-overs)
Most measure some degree of conformality
Need a balance between distortions
30/30
