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![Page 1: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/1.jpg)
Optimization of Inverse Snyder Polyhedral Projection
Erika [email protected]
Faramarz [email protected]
Dept. of Computer Science
University of Calgary
Calgary, Canada
October 5, 2011
![Page 2: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/2.jpg)
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Inverse Snyder Optimization
• What is Snyder?
• Inversion Issues
• Optimizations:– Operation Reduction– Iteration Reduction– Iteration Elimination
• Results
• Summary (NASA, 2000)
![Page 3: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/3.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
![Page 4: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/4.jpg)
4
What is Snyder?
• Projecting Spherical Earth to Planar Map
F-1(p)
F(p)
(Google Maps, 2010 – Mercator Projection)(NASA, 2000)
![Page 5: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/5.jpg)
5
What is Snyder?
• Projecting Spherical Earth to Planar Map
Mercator Projection
![Page 6: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/6.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
![Page 7: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/7.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
(NASA, 2000-2006)
Mollweide
![Page 8: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/8.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
Mollweide
Lambert Azimuthal Equal-Area
(NASA, 2000-2006)
![Page 9: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/9.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
MollweideWerner
Lambert Azimuthal Equal-Area
(NASA, 2000-2006)
![Page 10: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/10.jpg)
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What is Snyder?
P-1
P
• Projecting Spherical Earth to Planar Map
• Preserve Area
![Page 11: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/11.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
(Snyder, 1992)
![Page 12: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/12.jpg)
12
What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
PYXIS Innovation
• Industrial applications in virtual worlds
Truncated Icosahedron:• Angular Deformation: < 3.75o
• Scale Variation: < 3.3%
Icosahedron:• Angular Deformation: < 17.27o
• Scale Variation: < 16.3%
![Page 13: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/13.jpg)
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What is Snyder?
• Projecting Spherical Earth to Planar Map
• Preserve Area
• Industrial applications in virtual worlds
Truncated Icosahedron:• Angular Deformation: < 3.75o
• Scale Variation: < 3.3%
Icosahedron:• Angular Deformation: < 17.27o
• Scale Variation: < 16.3%
Snyder’s Icosahedron Face
![Page 14: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/14.jpg)
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Constructing the Projection
• Identify Symmetric Region
![Page 15: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/15.jpg)
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Constructing the Projection
A’
B’
C’
G g
![Page 16: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/16.jpg)
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A’
B’
C’
G g
Constructing the Projection
A’
B’
C’
G gD’
AzH
![Page 17: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/17.jpg)
17
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
![Page 18: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/18.jpg)
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• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
![Page 19: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/19.jpg)
19
• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
![Page 20: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/20.jpg)
20
• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
linear
non-linear, trigonometric functions
![Page 21: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/21.jpg)
21
• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
![Page 22: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/22.jpg)
22
• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
non-linear with
inverse trig. funcs
![Page 23: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/23.jpg)
23
• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
non-linear with
inverse trig. funcs
q d’
![Page 24: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/24.jpg)
24
• Find planar azimuth: Az’
• Position P’ based on d’ from q
• Unwrap azimuth
A’
B’
C’
G g
A
B
C
Constructing the Projection
A’
B’
C’
G gD’
AzD
Az’H
g
non-linear with
inverse trig. funcs
q d’
![Page 25: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/25.jpg)
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A
B
C
Inverse Projection
g
![Page 26: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/26.jpg)
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A
B
C
DAz’
d’
g
A
B
C
g
Inverse Projection
![Page 27: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/27.jpg)
27
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
Inverse Projection
![Page 28: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/28.jpg)
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Inversion Issues
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
• Find spherical azimuth: Az
![Page 29: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/29.jpg)
29
Inversion Issues
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
• Find spherical azimuth: Azlinear
non-linear, trigonometric functions
![Page 30: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/30.jpg)
30
Inversion Issues
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
• Find spherical azimuth: Azlinear
non-linear, trigonometric functions
non-linear, inverse trig. functions
![Page 31: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/31.jpg)
31
non-linear, trigonometric functions
Inversion Issues
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
• Find spherical azimuth: Azlinear
non-linear, inverse trig. functions
Use IterativeMethod to Solve
For Az
![Page 32: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/32.jpg)
32
Inversion Issues
• Frequently called!
(PYXIS, 2011)
![Page 33: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/33.jpg)
33
Optimizations
1. Operation Reduction
A
B
C
G g
A’
B’
C’
DAz
D’Az’
q
d’
H P
P’
![Page 34: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/34.jpg)
34
Optimizations
1. Operation Reduction
2. Iteration ReductionA
B
C
G g
A’
B’
C’
DAz
D’Az’
q
d’
H P
P’
![Page 35: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/35.jpg)
35
Optimizations
1. Operation Reduction
2. Iteration Reduction
3. Iteration Avoidance
A
B
C
G g
A’
B’
C’
DAz
D’Az’
q
d’
H P
P’
![Page 36: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/36.jpg)
36
Operation Reduction
• Reduce repetitive calls– 2π, H– cos and sin calls
• Pre-computation of values
• Trigonometric calls (eg. sincos)
![Page 37: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/37.jpg)
37
Operation Reduction
• Reduce repetitive calls– 2π, H– cos and sin calls
• Pre-computation of values
• Trigonometric calls (eg. sincos)
• Nominal speed up
• Note: No look-up table for cos and sin (would increase error)
![Page 38: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/38.jpg)
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Iteration Reduction
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
linear
non-linear, trigonometric functions
non-linear, inverse trig. functions
• Recall finding spherical azimuth: Az
![Page 39: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/39.jpg)
39
Iteration Reduction
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
linear
non-linear, trigonometric functions
non-linear, inverse trig. functions
• Recall finding spherical azimuth: Az
![Page 40: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/40.jpg)
40
Iteration Reduction
A’
B’
C’
G g
A
B
C
D’Az
DAz’q d’
Hg
A
B
C
g
linear
non-linear, trigonometric functions
non-linear, inverse trig. functions
• Recall finding spherical azimuth: Az
![Page 41: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/41.jpg)
41
Iteration ReductionNewton Raphson: Iterative Solution Finding
![Page 42: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/42.jpg)
42
Iteration ReductionNewton Raphson: Iterative Solution Finding
• Idea: Consider treating the iterative solution as a one-dimensional function
![Page 43: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/43.jpg)
43
Iteration Reduction
• Idea: Consider treating the iterative solution as a one-dimensional function
![Page 44: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/44.jpg)
44
Iteration Reduction
PolynomialSum of Squares of
ResidualsVariance ofResiduals
Degree 1 1.19E+00 2.02E-04
Degree 2 9.72E-01 1.66E-04
Degree 3 2.30E-04 3.92E-08
Degree 4 2.06E-04 3.51E-08
Degree 5 2.51E-05 4.28E-09
Degree 6 2.20E-05 3.75E-09
Degree 7 9.06E-07 1.55E-10
Polynomial Approximating Azimuthal Shift
![Page 45: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/45.jpg)
45
Iteration Reduction
• Use polynomial for improved initial estimate of Newton-Raphson
![Page 46: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/46.jpg)
46
Iteration Elimination
• Idea: Skip the iteration entirely, using this approximating function!
• Note: Will need to evaluate error
![Page 47: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/47.jpg)
47
Results
Need to:
• Determine Runtime Improvements
• Contrast Original with Iteration Reduction especially regarding iteration drop
• Establish Error for Elimination Approach
![Page 48: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/48.jpg)
48
Results: Runtime Improvements
• Approach: – Profile inverse Snyder method using gprof– Ran 100 times, against four (4) quality levels
• Quality:
Quality 10 - Flat & Projected
![Page 49: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/49.jpg)
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Results: Runtime Improvements
Profiling Time
0
0.02
0.04
0.06
0.08
0.1
0.12
25 50 75 100
Quality
Ave
rag
e T
ime
(s)
Original
Reduction
Elimination
![Page 50: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/50.jpg)
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Results: Iteration Reduction
• Original vs ReducedAverage Iterations
0
0.5
1
1.5
2
2.5
3
3.5
4
25 50 75 100
Quality
Ave
rag
e It
erat
ion
s
Original
Reduction
![Page 51: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/51.jpg)
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• Original vs Reduced
• Recall: Skip has no iterations
Results: Iteration Reduction
Average Iterations
0
0.5
1
1.5
2
2.5
3
3.5
4
25 50 75 100
Quality
Ave
rag
e It
erat
ion
s
Original
Reduction
![Page 52: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/52.jpg)
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Results: Iteration ReductionQuality 10 Quality 30 Quality 60 Quality 100
Original
- 4 iterations - 3 Iterations - 2 Iterations - 0 or 1 Iterations
![Page 53: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/53.jpg)
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Results: Iteration Reduction
Original
Reduced
Quality 10 Quality 30 Quality 60 Quality 100
- 4 iterations - 3 Iterations - 2 Iterations - 0 or 1 Iterations
![Page 54: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/54.jpg)
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Results: Iteration Reduction
Original Reduced
- 4 iterations - 3 Iterations - 2 Iterations - 0 or 1 Iterations
![Page 55: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/55.jpg)
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Results: Error Check• Original vs Eliminated Approach
QualityAvg. Dist.
ErrorMax Dist.
ErrorAvg. AreaError (%)
Max AreaError (%)
25 9.38e-05 6.19e-04 1.98e-05 1.60e-02
50 9.44e-05 6.19e-04 9.22e-06 2.20e-02
75 9.46e-05 6.19e-04 3.33e-06 3.50e-02
100 9.49e-05 6.19e-04 1.76e-06 5.07e-02
![Page 56: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/56.jpg)
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Results: Error Check• Original vs Eliminated Approach
QualityAvg. Dist.
ErrorMax Dist.
ErrorAvg. AreaError (%)
Max AreaError (%)
25 9.38e-05 6.19e-04 1.98e-05 1.60e-02
50 9.44e-05 6.19e-04 9.22e-06 2.20e-02
75 9.46e-05 6.19e-04 3.33e-06 3.50e-02
100 9.49e-05 6.19e-04 1.76e-06 5.07e-02
On the Earth:Average Distance Error: 5.90mAverage Areal Error: 714 m2
![Page 57: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/57.jpg)
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Results: Error CheckDistance Error Areal Error (Percent)
Displacement in: - No error- Increase- Decrease
- X- Y- Z
- Uniform- None
![Page 58: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/58.jpg)
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Summary
• Definite Improvement– 45% time reduction when skipping iteration– 25% iteration reduction
• Nominal Error on Elimination Approach– 8.5 x 10-6% (714 m2) average areal change– 5.9 m distance error– Useful at coarse resolutions
![Page 59: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/59.jpg)
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Conclusion
Result: Determined an effective association of planar data to a spherical world
![Page 60: Optimization of Inverse Snyder Polyhedral Projection Erika Harrison eharris@ucalgary.ca Ali Mahdavi-Amiri amahdavi@ucalgary.ca Faramarz Samavati samavati@ucalgary.ca.](https://reader034.fdocuments.net/reader034/viewer/2022050714/56649ede5503460f94beeab5/html5/thumbnails/60.jpg)
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Questions?
Result: Determined an effective association of planar data to a spherical world