On the Effective Dimension of Light Transport
On the Effective Dimension of Light Transport Christian Lessig and Eugene Fiume Dynamic Graphics Project, Department of Computer Science, University of Toronto
2
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
3
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
4
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
5
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
6
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
7
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
Christian
Lessig∗
8
June 28, 2010 Problem statement
U
H2
S2
x
Z T (n(x) · ω) E(ω) dω
B(x) = H2
˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U U
L X
tl yl0 (n(x) · ω)
T (n(x) · ω) = l=0 L X
T (n(x) · ω) = [email protected]
tl Pl (n(x) · ω) l=0 Dynamic Graphics Project
University of Toronto
kB − BkU = kTE − TEkU ≈
Christian
Lessig∗
9
L X
June 2010 T (n(x) · ω) = statement tl yl0 (n(x)28, · ω) Problem l=0
L X
T (n(x) · ω) =
tl Pl (n(x) · ω) l=0 U H2 S 2 x l L X X = t˜l Z ylm (n(x)) ylm (ω) B(x) l=0 =m=−lT (n(x) · ω) E(ω) dω H2
L X l X ˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U elm ylm (ω) U E(ω) = l=0 m=−l 2 L SM X tl yl0 (n(x) · ω) T (n(x) · ω) = l=0 L X ˜ dim (T) dim (T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
l=0 Dynamic Graphics Project
University of Toronto
Christian Lessig∗ kB − BkU = kTE − TEkU ≈
Christian
June 28, 2010 T (n(x) · ω) = Problem statement
Lessig∗
10
L X
28, 2010 tl yJune · ω) l0 (n(x) l=0
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) Z l=0 U H2 S 2 x l L B(x) = T (n(x) · ω) E(ω) dω X X H2 = ylm (n(x)) ylm (ω) t˜l Z B(x) l=0 = m=−lT (n(x) · ω) E(ω) dω 2 2 ˜ ˜ kB − Bk H2 U = kTE − TEkU ≈
U
H2
S2
L X l X ˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U elm ylm (ω) U E(ω) = L X
l=0 m=−l
tl yl0 (n(x) · ω)
T (n(x) · ω) = l=0 L X
T (n(x) · ω) =
tl Pl (n(x) · ω)
2 L SM X T (n(x) · ω) = tl yl0 (n(x) · ω)
l=0
L X ˜ dim (T) dim (T) tl Pl (n(x) · ω) T (n(x) · ω) = l L X X l=0 [email protected] Graphics Project University of Toronto = ylmDynamic (n(x)) ylm (ω) t˜l l=0
Christian Lessig∗ kB − BkU = kTE − TEkU ≈
Formulas 11
Christian L Christian Lessig∗ X 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) statement l=0 June 28, 2010
June 28, 2010
Problem
Lessig∗
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) Z l=0 U H2 S 2 x l L B(x) = T (n(x) · ω) E(ω) dω X X U H2 S 2 x H2 = ylm (n(x)) ylm (ω) t˜l Z Z · ω) E(ω) dω B(x) l=0 = m=−lT (n(x) 2 2 ˜ ˜ kB − Bk H2 U = kTE − TEkU ≈ T (n(x) · ω) E(ω) B(x) = L X l H2 X 2 ˜ 2 ≈ ˜ = kTE − TEk kB − Bk U elm ylm (ω) U E(ω) = 2 2 ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = 2 L SM X l=0 T (n(x) · ω) = tl yl0X (n(x) · ω) L L X T (n(x)l=0 · ω) = tl yl0 (n(x) · ω) T (n(x) · ω) = tl Pl (n(x) · ω) L X l=0 ˜ dim (T) dim (T)l=0 t l Pl X (n(x) · ω) T (n(x) · ω) = L l L X X [email protected] Graphics Project of Toronto· ω) tl Pl (n(x) T (n(x)l=0 · ω) = University = ylmDynamic (n(x)) ylm (ω) t˜l
U
H2
S2
Christian Lessig∗ kB − BkU = kTE − TEkU ≈ Formulas ∗ 12 Christian Lessig Christian Lessig∗ June 28, 2010 ∗ L Christian X June Lessig 28, 2010 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) Problem statement l=0 June 28, 2010
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 U H S x 2 2 Z l=0 U H S x l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x H2 = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) 2 2 ˜ ˜ kB − Bk H2 U = kTE − TEkU ≈ T (n(x) · ω) E(ω) B(x) = ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ = kTE − TEk kB − Bk U elm ylm (ω) U ≈ E(ω) = 2 2 ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L X 2 L X · ω) = l=0 M TS(n(x) tl yl0 (n(x) · ω T (n(x) · ω) = tl yl0X (n(x) · ω) L L X l=0 T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) = tl Pl (n(x) · ω) X L X l=0 ˜ T tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) t l Pl X (n(x) · ω) T (n(x) · ω) = L l L X X l=0 [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University = ylmDynamic (n(x)) ylm (ω) t˜l l
U
H2
S2
Formulas Christian Lessig∗ kB − BkU = kTE − TEkU ≈ Formulas ∗ 13 Christian Lessig ∗ ∗ Christian Lessig June 28, 2010 Christian Lessig ∗ L Christian X June Lessig 28, 2010 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) June 28, 2010 Problem statement l=0 June 28, 2010
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 U H S x 2 2 Z l=0 U H S x U H2 S 2 x l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) Z H2 = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) B(x) = T (n(x) · ω) E(ω) dω 2 2 ˜ ˜ 2 = kTE − TEk ≈ kB − Bk H2 U U Objective: H T (n(x) · ω) E(ω) B(x) = ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ 2 2 = kTE − TEk kB − Bk ˜ ˜ U elm ylm (ω) U ≈ = kB − BkU = kTE − TEkU ≈E(ω) 2 2 ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L X 2 L X · ω) = l=0 L M TS(n(x) tl yl0 (n(x) · ω X tl yl0X (n(x) · ω) L L t y (n(x) · ω) T (n(x) · ω) = T (n(x) · ω) = X l=0 l l0 T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) =l=0 tl Pl (n(x) · ω) X L X l=0 ˜ T L tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) X t l Pl X (n(x) · ω) T (n(x) · ω) = L l L t PX T (n(x) · ω) = X l=0 l l (n(x) · ω) [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University =l=0 t˜l ylmDynamic (n(x)) ylm (ω) l
U
H2
S2
∗ kB − BkU = kTE − TEkU ≈ Formulas Christian Lessig L X Formulas ∗ 14 Christian Lessig T (n(x) · ω) = tl yl0 (n(x) · ω) ∗ ∗ Christian Lessig June 28, 2010 Christian Lessig l=0 ∗ L Christian Lessig X June 28, 2010 L X 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) June 28, 2010 Problem statement tl Pl (n(x) · ω) T (n(x) · ω) = l=0 June 28, 2010 l=0
L 2 2 X U H S x l L X X Shading˜equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 = ylm (n(x)) ylm (ω) tl U H S x 2 2 Z l=0 2 U H S x U Hm=−l S2 x l=0 l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x H2 = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) ZX L X l = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) B(x) = T (n(x) · ω) E(ω) dω e y (ω) E(ω) = 2 2 lm lm ˜ ˜ 2 = kTE − TEk ≈ kB − Bk H2 U U Objective: H m=−l T (n(x) · ω) E(ω) B(x) = l=0 ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ 2 2 = kTE − TEk kB − Bk ˜ ˜ U 2 elm ylm (ω) U ≈ = kB − BkU =SkTE − TEkU ≈E(ω) 2 2 M ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L with X 2 L S(n(x) X · ω) = l=0 L M T tl yl0 (n(x) · ω X ˜ T) dim (T) T (n(x) · ω) = tl yl0X (n(x) · ω) L L t T (n(x) · ω)dim = (X l=0 l yl0 (n(x) · ω) T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) =l=0 tl Pl (n(x) · ω) X L X l=0 ˜ T L tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) X t l Pl X (n(x) · ω) T (n(x) · ω) = L L t PX l T (n(x) · ω) = X (n(x) · ω) l=0 l l ˜ T˜ T [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University =l=0 tl ylmDynamic (n(x)) ylm (ω) l
∗ kB − BkU = kTE − TEkU ≈ Formulas Christian Lessig L X Formulas ∗ 15 Christian Lessig T (n(x) · ω) = tl yl0 (n(x) · ω) ∗ ∗ Christian Lessig June 28, 2010 Christian Lessig l=0 ∗ L Christian Lessig X June 28, 2010 L X 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) June 28, 2010 Problem statement tl Pl (n(x) · ω) T (n(x) · ω) = l=0 June 28, 2010 l=0
L 2 2 X U H S x l L X X Shading˜equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 = ylm (n(x)) ylm (ω) tl U H S x 2 2 Z l=0 2 U H S x U Hm=−l S2 x l=0 l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x H2 = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) ZX L X l = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) B(x) = T (n(x) · ω) E(ω) dω e y (ω) E(ω) = 2 2 lm lm ˜ ˜ 2 = kTE − TEk ≈ kB − Bk H2 U U Objective: H m=−l T (n(x) · ω) E(ω) B(x) = l=0 ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ 2 2 = kTE − TEk kB − Bk ˜ ˜ U 2 elm ylm (ω) U ≈ = kB − BkU =SkTE − TEkU ≈E(ω) 2 2 M ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L with X 2 L S(n(x) X · ω) = l=0 L M T tl yl0 (n(x) · ω X ˜ T) dim (T) T (n(x) · ω) = tl yl0X (n(x) · ω) L L t T (n(x) · ω)dim = (X l=0 l yl0 (n(x) · ω) T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) =l=0 tl Pl (n(x) · ω) X effective dimension L X l=0 ˜ T L tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) X t l Pl X (n(x) · ω) T (n(x) · ω) = L l L t PX T (n(x) · ω) = X (n(x) · ω) l=0 l l ˜ T˜ T [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University =l=0 tl ylmDynamic (n(x)) ylm (ω) l
2
˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U U
Formulas Christian Lessig∗ Lessig∗
16
Christian ∗ L Christian Lessig June 28, 2010 X 28, 2010 SimplificationT (n(x) · ω) = tl yJune l0 (n(x) · ω) June 28, 2010 l=0 Z
T (n(x) ·Tω) E(ω)· ω) dω = (n(x)
B(x) =
L X
tl Pl (n(x) · ω) U H2 S 2 x Z U H2 S 2 x l=0 2 2 l L Z T (n(x) · ω) E(ω) dω X B(x) = U H S x X Z S2 = ylm (n(x)) yTlm (ω) · ω) E t˜l B(x) = (n(x) Z X X Z ·H ω)2 E(ω) dω B(x) = m=−lT (n(x) l=0 ˜ tl ylm (n(x)) ylm (ω) = el0Hm20 yl0 m0 (ω) dω T (n(x) · ω) E(ω) B(x) = S 2 l,m l0 ,m0 ˜2 2 = kTE − TEk ˜ H kB − Bk L l Z U 2 2 X X ˜ ˜ XX kB − Bk U = kTE − TEkU ≈ y0 (ω) E(ω) = ylm (ω) yellm 0˜ t˜l el0 m0 ylm (n(x)) dω = 2lm (ω) m ˜ 2 ≈ = kTE − TEk kB − Bk U U S 2l=0 m=−l l,m l0 ,m0 Z L X XX L 2X ˜ S T (n(x) ω)0 (ω) = dω tl yl0 (n(x) · ω tl el0 m0 ylm (n(x)) ylm (ω) = M yl·0 m T (n(x) S· 2ω) = tl yl0X (n(x) · ω) L l=0 l,m l0 ,m0 | {z } l=0 T (n(x)δll·0 ω) tlLyl0 (n(x) · ω) δmm= 0 X L X X · ω)l=0 tl Pl (n(x) · ω T (n(x) = ˜ dim· ω) (T)= dim (T) = t˜l elm ylm (n(x))T (n(x) t l Pl X (n(x) · ω) L l=0 [email protected] l,m Dynamic Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University l H2
17
2
Simplification Z T (n(x) · ω) E(ω) dω
B(x) = H2
Z
T (n(x) · ω) E(ω) dω
B(x) = S2
Z
X
X t˜l ylm (n(x)) ylm (ω)
= S 2 l,m
el0 m0 yl0 m0 (ω) dω
Z
XX
l0 ,m0
t˜l el0 m0 ylm (n(x))
=
S2
l,m l0 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
=
ylm (ω) yl0 m0 (ω) dω
l,m l0 ,m0
ylm (ω) yl0 m0 (ω) dω S2 | {z } δll0 δmm0
X = [email protected] l,m
t˜l elm ylm (n(x)) Dynamic Graphics Project
University of Toronto
2 18
Z
Simplification T (n(x) · ω) E(ω) dω B(x) = H2
Z
T (n(x) · ω) E(ω) dω
B(x) = S2
Z
X
X t˜l ylm (n(x)) ylm (ω)
= S 2 l,m
el0 m0 yl0 m0 (ω) dω
Z
XX
l0 ,m0
t˜l el0 m0 ylm (n(x))
=
S2
l,m l0 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
=
ylm (ω) yl0 m0 (ω) dω
l,m l0 ,m0
ylm (ω) yl0 m0 (ω) dω 2 |S {z } δll0 δmm0
X t˜l elm ylm (n(x))
= l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
2 19
Z
Simplification T (n(x) · ω) E(ω) dω B(x) = H2
Z
T (n(x) · ω) E(ω) dω
B(x) = S2
Z
X
X t˜l ylm (n(x)) ylm (ω)
= S 2 l,m
el0 m0 yl0 m0 (ω) dω
Z
XX
l0 ,m0
t˜l el0 m0 ylm (n(x))
=
S2
l,m l0 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
=
ylm (ω) yl0 m0 (ω) dω
l,m l0 ,m0
ylm (ω) yl0 m0 (ω) dω 2 |S {z } δll0 δmm0
X t˜l elm ylm (n(x))
= l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
2 Z 20
T (n(x) · ω) E(ω) dω
B(x) = 2
ZH Z
T (n(x) · ω) E(ω) dω B(x) = Simplification B(x) = S T (n(x) · ω) E(ω) dω 2 H2
Z Z
X X = el0 m0 yl0 m0 (ω) dω t˜l ylm (n(x)) ylm (ω) B(x) = S 2 T (n(x) · ω) E(ω) dω l0 ,m0 S 2 l,m Z Z X XX 0 ylm (n(x)) = (n(x)) ylm (ω) ylme(ω) t˜t˜llyelm l0 m0 yl0 m0 (ω) dω l0 m S 2 ll,m 0 ,m0 l,m
Sl20 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
ylm (ω) yl0 m0 (ω) dω S2 l,m l0 ,m0 | {z } Z XX δll0 δmm0 t˜l el0 m0 ylm (n(x)) ylm (ω) yl0 m0 (ω) dω =X 2 = l,m tl˜0l,m elm ylm (n(x)) 0 |S {z } =
l,m
δll0 δmm0
X =
t˜l elm ylm (n(x)) X l,m t˜l elm ylm (n(x)) B(x) = l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
2 Z 21
T (n(x) · ω) E(ω) dω
B(x) = 2
ZH Z
T (n(x) · ω) E(ω) dω B(x) = Simplification B(x) = S T (n(x) · ω) E(ω) dω 2 H2
Z Z
X X = el0 m0 yl0 m0 (ω) dω t˜l ylm (n(x)) ylm (ω) B(x) = S 2 T (n(x) · ω) E(ω) dω l0 ,m0 S 2 l,m Z Z X XX 0 ylm (n(x)) = (n(x)) ylm (ω) ylme(ω) t˜t˜llyelm l0 m0 yl0 m0 (ω) dω l0 m S 2 ll,m 0 ,m0 l,m
Sl20 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
ylm (ω) yl0 m0 (ω) dω S2 l,m l0 ,m0 | {z } Z XX δll0 δmm0 t˜l el0 m0 ylm (n(x)) ylm (ω) yl0 m0 (ω) dω =X 2 = l,m tl˜0l,m elm ylm (n(x)) 0 |S {z } =
l,m
δll0 δmm0
X =
t˜l elm ylm (n(x)) X l,m t˜l elm ylm (n(x)) B(x) = l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
l,m
|
l0 ,m0
S2
˜l el0 m0 ylm (n Formulas = t 2 2 ˜ ˜ kB{z − BkU = kTE } − TEkU0 ≈ 0
X t˜l elm ylm (n(x))
= l,m
l,m l ,m
δll0 δmm0
SimplificationT (n(x) · ω) = X
22
∗ Christian Lessig X L X
= t˜l elm ylm (n(x)) 28,· ω) 2010 tl yJune l0 (n(x)l,m
l=0
L X X ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t l,m 2 2 U H S x l=0 l,m L X
2 n(x) w ω ∈ SM
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
H2
L X l2 2 X ˜ ˜ kB − Bk = kTE − TEk X U U ≈ elm ylm (ω) X t˜l elm ylm (ω) E(ω) = B(ω) = ˜l l=0 m=−l B(ω) = t l,m l,m X L 2X X SM blm ylm (ω) = b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
l,m
l0 ,m0
X = X t˜l elm ylm (n(x)) = l,m t˜l elm ylm (n(x)) l,m
|
S2
δll0 δ− ˜0 2 mm kB Bk {z U δll0 δmm0
Simplification T (n(x) · ω) = X
˜l el m Formulas t = 2 ˜ = kTE } − TEk ≈
L X
0
U l,m l0 ,m0
0
ylm (n
23
∗ Christian Lessig X
t˜l elm ylm (n(x)) = 28,· ω) 2010 tl yJune l0 (n(x)l,m
l=0 B(x) = X t˜l elm ylm (n(x)) L X X l,m ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t l,m 2 2 U H S x l=0 l,m 2 but n(x) w ω ∈ SM 2 n(x) w ω ∈ SM
L X
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
H2
X L X l2 X B(ω) = X t˜l elm ylm (ω) ˜ ˜ 2 ≈ kB − BkU = kTE − TEk U e y (ω) E(ω) = lm lm X B(ω) = l,m t˜l elm ylm (ω) X ˜l l=0 m=−l t B(ω) = l,m b =X lm ylm (ω) l,m L 2 X X SM = l,m blm ylm (ω) b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
˜l el0 m0 ylm (n Formulas 2 t = ˜ 2 δ S 0 δmm˜ 0 2 ˜ t e y (n(x)) 0 0 ll l lm lm kB{z − BkU = kTE l,m l ,m | } − TEkU0 ≈ 0 X l,m l ,m 24 l,m ˜ δ δ 0 0 ll mm = X tl elm ylm (n(x)) ∗ Christian Lessig X = l,m t˜l elm ylm (n(x)) ˜l elm ylm (n(x)) L X t = X l,m June 28, 2010 t˜l elm ylm (n(x)) B(x) = T (n(x) · ω) = t y (n(x) · ω) l,m l l0 X l,m ˜ l=0 B(x) = X tl elm ylm (n(x)) L X X l,m ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t 2 2 2 n(x)l,m w ω ∈ SM U H S x l=0 l,m
=
Simplification
2 but n(x) w ω ∈ SM
L X
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
2 n(x) w ω ∈ S M X H2 t˜l elm ylm (ω) B(ω) = X l,m ˜ L X l2 2 X t B(ω) =X e y (ω) ˜ ˜ l lm lm kB − Bk = kTE − TEk X U U ≈ elm ylm (ω) E(ω) = X (ω) lmy t˜l eylm B(ω)== l,mblm (ω) lm X ˜l l=0 m=−l t B(ω) = l,m l,m b =X lm ylm (ω) l,m L 2 X X SM = l,m blm ylm (ω) b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
˜l el0 m0 ylm (n Formulas 2 t = ˜ 2 δ S 0 δmm˜ 0 2 ˜ t e y (n(x)) 0 0 ll l lm lm kB{z − BkU = kTE l,m l ,m | } − TEkU0 ≈ 0 X l,m l ,m 25 l,m ˜ δ δ 0 0 ll mm = X tl elm ylm (n(x)) ∗ Christian Lessig X = l,m t˜l elm ylm (n(x)) ˜l elm ylm (n(x)) L X t = X l,m June 28, 2010 t˜l elm ylm (n(x)) B(x) = T (n(x) · ω) = t y (n(x) · ω) l,m l l0 X l,m ˜ l=0 B(x) = X tl elm ylm (n(x)) L X X l,m ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t 2 2 2 n(x)l,m w ω ∈ SM U H S x l=0 l,m
=
Simplification
2 but n(x) w ω ∈ SM
L X
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
2 n(x) w ω ∈ S M X H2 t˜l elm ylm (ω) B(ω) = X l,m ˜ L X l2 2 X t B(ω) =X e y (ω) ˜ ˜ l lm lm kB − Bk = kTE − TEk X U U ≈ elm ylm (ω) E(ω) = X (ω) lmy t˜l eylm B(ω)== l,mblm (ω) lm X ˜l l=0 m=−l t B(ω) = l,m l,m b =X lm ylm (ω) l,m L 2 X X SM = l,m blm ylm (ω) b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
4π |{z}
|
λi ≈1
C K() = + log 4π |{z} |
Approximation λi ≈1
{z
}
λi ∈[1,]
26
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0 λi ∈[1,]
Objective 2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U
[email protected]
Dynamic Graphics Project
University of Toronto
4π |{z}
i=1
|
λi ≈1
{z
} 27
λi ∈[1,]
1 2−
C 2+ o(log C) log(C) K() = B(ω) + −log ˜bi kϕ ˜
= B(∂U) B(ω) (ω)k i U | {z } 4π U |{z} | {z } i=K+1 λi ≈0
N X
Approximation λi ≈1
Objective
λi ∈[1,]
N X 2 kϕ (ω)k ˜ 2 ≈ ˜ i kB(ω) − B(ω)k = kTE − TEk U U i=K+1
We are interested in a basis
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L i
2
kϕi (ω)kU
[email protected]
,
i = K + 1...N
Dynamic Graphics Project
University of Toronto
4π |{z}
i=1 Formulas {z
|
λi ≈1
} 28
λi ∈[1,]
Lessig ∗ Christian N 1 2− X
C 2+ o(log C) log(C) K() = B(ω) + −log ˜bi kϕ ˜
= B(∂U) B(ω) (ω)k i U | {z } 4π U |{z} {z 2010 } i=K+1 {ϕi (ω)} , i |= 1 . .June . N 23, λi ≈0
Approximation λi ≈1
λi ∈[1,]
Objective
span (ϕi ) = i
N X 2 kϕ (ω)k ˜ 2 ≈ ˜ i kB(ω) − B(ω)k = kTE − TEk U U span (yL{ϕ)i=K+1 H≤L (ω)} , i = 1...N i=
l,m We are interested in a basis
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L span (ϕi ) = span (yi L ) = H≤L K X
i
l,m
˜bdimension ˜ effective B(ω) ≈ with B(ω) = , K N such that i ϕi 2
i=1kϕi (ω)kU
,
iK = K + 1...N X ˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = i=1
2
˜
=
B(ω) − B(ω) U [email protected]
N X
˜bi kϕi (ω)kN2 U X
Dynamic Graphics Project 2
˜
˜
2University of Toronto
span (ϕi ) = span (yL ) = H≤L i
l,m
Approximation ˜ B(ω) ≈ B(ω) =
K X
29
˜bi ϕi
,
K N
i=1
Approximation error N X
2 ˜bi kϕi (ω)k2 ˜
B(ω) − B(ω)
= U U i=K+1
N X
kϕi (ω)k i=K+1
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L i
[email protected]
2
kϕi (ω)kDynamic , Graphics i =Project K + 1...N U
University of Toronto
(yL ) = H≤L span (ϕi ) = spanX N
˜bi kϕi (ω)k2 ˜
B(ω) −i B(ω)
2 =l,m U U
30
i=K+1
Approximation ˜ B(ω) ≈ B(ω) N=
K X
X
˜bi ϕi
,
K N
i=1 i=K+1 Approximation error
kϕi (ω)k
N X
2
˜bi kϕi (ω)k2 ˜
=
B(ω) − B(ω) U U 2i=K+1 ϕi (ω) | i = 1 . . . (L + 1) , span (ϕi ) = H≤L i
which, for arbitrary input signals, is minimized if N X (ω)k , ikϕ = iK + 1...N
2
kϕi (ω)kU
i=K+1
is minimal.
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L i
∗ [email protected] [email protected]
2
kϕi (ω)kDynamic , Graphics i =Project K + 1...N U
University of Toronto
31
Spatio-Spectral Concentration Theory
[email protected]
Dynamic Graphics Project
University of Toronto
C K() = + log 4π |{z} | λi ≈1
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0
32
λi ∈[1,]
Spatio-Spectral Concentration Theory D gi = λi gi
Objective: Extremize concentration measure1 λ=
2 kgkU 2 kgkS 2
R 2 g dω U =R 2 dω g 2 S
g ∈ H≤L
,
2
C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1
1− C + log B(∂U) log(C) +o(log C) K() = 4π |{z} [email protected] Project | Dynamic Graphics{z } University of Toronto 1
Simons FJ, Dahlen FA, Wieczorek MA. Spatiospectral Concentration on a Sphere. SIAM Review. 2006;48(3):504-536; Simons FJ. Slepian Functions and Their Use in Signal Estimation and Spectral Analysis. In: Freeden W Handbook of Geomathematics.; 2010.
C K() = + log 4π |{z} | λi ≈1
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0
33
λi ∈[1,]
Spatio-Spectral Concentration Theory D gi = λigi 1− C +Extremize log B(∂U) log(C) measure + o(log C)1 K() = Objective: concentration 4π R 2 2 U g dω kgk U C λ= 1 −= R , g ∈ H≤L 2 2 B(∂U) log(C) + o(log C) K() = + log kgk 2 g dω 2 S S | {z } 4π |{z} | {z } λi ≈0
λi ≈1Eigenvalue problem Solution: λi ∈[1,]
2
C = N A(U) = (L + 1) A(U) D gi = λi gi R 1g− C 2 2 dω B(∂U) log(C) + o(log C) kgk + K() = U logR U λ = 4π 2 = , g ∈ H≤L 2 |{z} g dω kgkS 2 S2 λi ≈1 Simons FJ, Dahlen FA, Wieczorek MA. Spatiospectral Simons FJ. Slepian Functions and Their Use in Concentration on a Sphere. SIAM Review. 2006;48(3):504-536; Signal Estimation and Spectral Analysis. In: Freeden W Handbook of Geomathematics.; 2010. 1− C + log B(∂U) log(C) +o(log C) K() = 2 4π= N A(U) = (L + 1) A(U) C |{z} [email protected] Project | Dynamic Graphics{z } University of Toronto 1
C 1− K() = B(∂U) log(C) + o(log C) + log June 27, 2010 | {z } 4π |{z} | {z } λi ≈0
λi ≈1
34
λi ∈[1,]
Spatio-Spectral Concentration Theory D gi = λigi 1kgk − 2 C 1 Uconcentration +Extremize log B(∂U) log(C) + o(log C) K() = Objective: measure λ = , g ∈ H ≤L 2 4π kgkS 2R 2 U g 2 dω kgk U C λ= 1 −= R , g ∈ H≤L 2 2 B(∂U) log(C) + o(log C) K() = + log kgk 2 g dω 2 S S | {z } T 4π D g g |{z} | {z } λi ≈0 λ = λ ≈1 T i Solution: Eigenvalue problem λi ∈[1,] g g 2 C = N A(U) = (L + 1) A(U) D gi = λi gi
where
D = {dlm,l0 m0 } RZ C 20 0 = 1 − 2y (ω) y 0 0 (ω) dω d g dω lm,l m lm B(∂U) lm kgk log(C) + o(log C) + K() = U logR U λ = 4π 2 = , g ∈ H≤L U 2 |{z} g dω kgkS 2 S2 λi ≈1 Simons FJ, Dahlen FA, Wieczorek MA. Spatiospectral Simons FJ. Slepian Functions and Their Use in Concentration on a Sphere. SIAM Review. 2006;48(3):504-536; Signal Estimation and Spectral Analysis. Z In: Freeden W Handbook of Geomathematics.; 2010. 1− C +D(ω, logω B(∂U) log(C) +o(log C) K() = ¯ ) g (¯ ω ) dω = 2 λi gi (ω) i 4π=U N A(U) = (L + 1) A(U) C |{z} [email protected] Project | Dynamic Graphics{z } University of Toronto 1
35
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
Dynamic Graphics Project
University of Toronto
36
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
Dynamic Graphics Project
University of Toronto
37
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
λ = 0.968958
Dynamic Graphics Project
University of Toronto
38
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
λ = 0.968958
Dynamic Graphics Project
λ = 0.814437
University of Toronto
39
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
λ = 0.968958
Dynamic Graphics Project
λ = 0.814437
λ = 0.814437
University of Toronto
40
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.814437
λ = 0.814437
λ = 0.000189436
[email protected]
Dynamic Graphics Project
University of Toronto
41
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.000189436
λ = 0.000189436
[email protected]
λ = 0.968958
Dynamic Graphics Project
λ = 0.814437
λ = 0.814437
University of Toronto
42
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.000189436
λ = 0.000189436
λ = 0.000125893
[email protected]
Dynamic Graphics Project
λ = 0.814437
λ = 0.814437
University of Toronto
43
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.814437
λ = 0.000189436
λ = 0.000189436
λ = 0.000125893
λ = 0.000100997
[email protected]
Dynamic Graphics Project
λ = 0.814437
University of Toronto
44
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.814437
λ = 0.814437
λ = 0.000189436
λ = 0.000189436
λ = 0.000125893
λ = 0.000100997
λ = 0.000100997
[email protected]
Dynamic Graphics Project
University of Toronto
45
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
46
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
47
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
48
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
49
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
50
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
51
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
52
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
53
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
54
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
55
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
56
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
57
Spatio-Spectral Concentration Theory 2
Spectrum (conjecture):
B(∂U) log(C) + o(log C)
1− C + log K() = 4π 1− C B(∂U) log(C) + o(log C) + log K() = | {z } 4π |{z} | {z } λi ≈0 λi ≈1
λi ∈[1,]
D gi = λi gi
λ= [email protected]
2 kgkU 2 kgkS 2
R 2 g dω U =R 2 dω g 2 S Dynamic Graphics Project
,
g ∈ H≤L University of Toronto
4π
C + log K() = 4π |{z} |
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0
58
λ ≈1 Spatio-Spectral Concentration Theory λ ∈[1,] i
2
i
D gi = λi gi
Spectrum (conjecture):
1R− 2 C 2 g dωB(∂U) log(C) + o(log C) + K() = kgklog U U R λ4π = , g ∈ H≤L 2 = 2 dω kgkS 2 S 2 g 1− C with B(∂U) log(C) + o(log C) + log K() = | {z } 4π |{z} | N A(U) = (L{z+ 1)2 A(U) } λi ≈0 C = λi ≈1 λi ∈[1,]
D gi = λi gi 1− C + log B(∂U) log(C) + o(log C) K() = R 4π |{z} 2 2 g dω kgk λi ≈1 U U λ = 2 = R 2 , g ∈ H≤L kgkS 2 1 −S2 g dω C + log Dynamic Graphics B(∂U) log(C) +o(log C) of Toronto K() = [email protected] Project University 4π
C 1− K() = + log B(∂U) log(C) + o(log C) 4πC 1 − B(∂U) log(C) + o(log C) + log K() = 4π |{z} 1 − C + log B(∂U) log(C) + o(log C) K() = λi ≈1 | {z } 4π |{z} | {z } C 1 − λi ≈0 λ ≈1 i K() = + log B(∂U) log(C) +o(log C) λi ∈[1,] 4π |{z} | {z } Spectrumλ(conjecture): i ≈1 D gi =λλ i gi i ∈[1,] C 1− B(∂U) log(C) + o(log C) K() = + log | {z } 4π R 2 |{z} 2 | {z } g dω kgk λi ≈0 U U λ ≈1 i R λ= g ∈ H≤L λi ∈[1,], 2 = 2 g dω kgkS 2 S2
59
Spatio-Spectral Concentration Theory
with
2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U 2 C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1 [email protected]
Project Dynamic Graphics
University of Toronto
C 1− K() = + log B(∂U) log(C) + o(log C) 4πC 1 − + log B(∂U) log(C) + o(log C) K() = 4π |{z} 1 − C + log B(∂U) log(C) + o(log C) K() = λi ≈1 | {z } 4π |{z} | {z } 1 − C λi ≈0 λ ≈1 i + log B(∂U) log(C) +o(log C) K() = λi ∈[1,] 4π |{z} | {z } Spectrumλ(conjecture): i ≈1 D gi =λλ i gi i ∈[1,] C 1− B(∂U) log(C) + o(log C) K() = + log | {z } 4π R 2 |{z} 2 | {z } g dω kgk λi ≈0 U U λ ≈1 i R λ= g ∈ H≤L λi ∈[1,], 2 = 2 g dω kgkS 2 S2
60
Spatio-Spectral Concentration Theory
with
2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U 2 C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1 [email protected]
Project Dynamic Graphics
University of Toronto
C 1− K() = + log B(∂U) log(C) + o(log C) 4πC 1 − + log B(∂U) log(C) + o(log C) K() = 4π |{z} 1 − C + log B(∂U) log(C) + o(log C) K() = λi ≈1 | {z } 4π |{z} | {z } 1 − C λi ≈0 λ ≈1 i + log B(∂U) log(C) +o(log C) K() = λi ∈[1,] 4π |{z} | {z } Spectrumλ(conjecture): i ≈1 D gi =λλ i gi i ∈[1,] C 1− B(∂U) log(C) + o(log C) K() = + log | {z } 4π R 2 |{z} 2 | {z } g dω kgk λi ≈0 U U λ ≈1 i R λ= g ∈ H≤L λi ∈[1,], 2 = 2 g dω kgkS 2 S2
61
Spatio-Spectral Concentration Theory
with
2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U 2 C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1 [email protected]
Project Dynamic Graphics
University of Toronto
62
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
63
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
64
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
65
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
66
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
67
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
68
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
69
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
70
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
71
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
72
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
73
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
Formulas
N X
2 ˜i (ω)} =
B(ω) − {ϕB(ω) , i = 1 ˜.b.i .kϕ Ni (ω)k2U U
74
i=K+1
Christian Lessig∗
Effective Dimension N = span (yL ) = H≤L span (ϕi )X i
l,m kϕ
June 23, 2010 (ω)k i
Error is minimizedi=K+1 if K X
˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = ϕi (ω) | i = 1 . . . (L +i=1 1)2 , span (ϕi ) = H≤L
i
i = 1...N is approximated with{ϕi (ω)}for ,which N X
2 2 2 ˜ ˜
B(ω) − B(ω) = kϕ (ω)k b kϕi (ω)kU ,U i = K + 1i . . .iN U i=K+1
span (ϕi ) = span (yL ) = H≤L N X i
l,m
kϕi (ω)k i=K+1 ∗ [email protected]
[email protected]
Dynamic Graphics Project
˜
K X
˜
University of Toronto
Formulas
N X
2 ˜i (ω)} =
B(ω) − {ϕB(ω) , i = 1 ˜.b.i .kϕ Ni (ω)k2U U
75
i=K+1
Christian Lessig∗
Effective Dimension N = span (yL ) = H≤L span (ϕi )X i
l,m kϕ
June 23, 2010 (ω)k i
Error is minimizedi=K+1 if K X
˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = ϕi (ω) | i = 1 . . . (L +i=1 1)2 , span (ϕi ) = H≤L
i
i = 1...N is approximated with{ϕi (ω)}for ,which N X
2 2 2 ˜ ˜
B(ω) − B(ω) = kϕ (ω)k b kϕi (ω)kU ,U i = K + 1i . . .iN U i=K+1
=> Optimal basis functions Slepian functions. ) = span (yL ) = H≤L span (ϕare i N X i
l,m
kϕi (ω)k i=K+1 ∗ [email protected]
[email protected]
Dynamic Graphics Project
˜
K X
˜
University of Toronto
Formulas
N X
2 ˜i (ω)} =
B(ω) − {ϕB(ω) , i = 1 ˜.b.i .kϕ Ni (ω)k2U U
76
i=K+1
Christian Lessig∗
Effective Dimension N = span (yL ) = H≤L span (ϕi )X i
l,m kϕ
June 23, 2010 (ω)k i
Error is minimizedi=K+1 if K X
˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = ϕi (ω) | i = 1 . . . (L +i=1 1)2 , span (ϕi ) = H≤L
i
i = 1...N is approximated with{ϕi (ω)}for ,which N X
2 2 2 2 ˜ ˜
B(ω) − B(ω) = kϕ (ω)k b kϕi (ω)kU ,U i = K + 1i . . .iN U i=K+1
=> Optimal basis functions Slepian functions. ) = span (yL ) = H≤L span (ϕare i
C 1− => Effective dimension is given by K(). = + log kϕi (ω)k 4π i=K+1 ∗ [email protected] K X 1− C [email protected] Dynamic Graphics Project University of Toronto + log K() ˜ = ˜ N X i
l,m
77
Effective Dimension
B(∂U) log(C) + o(log C
C 1− K() = + log 4π C 1− K() = + log B(∂U) log(C) + o(log C | {z 4π |{z} | {z } λi ≈0 λi ≈1
λi ∈[1,]
D gi = λi gi [email protected]
Dynamic Graphics Project
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78
Applications • Transport Matrix compression. — Optimal cluster size. — Analytic basis functions for neighborhoods.
[email protected]
Dynamic Graphics Project
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79
Applications • Transport Matrix compression. — Optimal cluster size. — Analytic basis functions for neighborhoods. • Representation of signals localized on the sphere. — Compact representation with most of the advantages as Spherical Harmonics.
[email protected]
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80
Applications • Transport Matrix compression. — Optimal cluster size. — Analytic basis functions for neighborhoods. • Representation of signals localized on the sphere. — Compact representation with most of the advantages as Spherical Harmonics. • Exploitation of coherence of light transport in sampling-based algorithms.
[email protected]
Dynamic Graphics Project
University of Toronto
81
2
Future Work
C 1− • Boundary function for K(). = + log B( 4π 1− C + log B( K() = 4π |{z} | {z λi ≈1
λi ∈[1
D gi = λi R
λ= [email protected]
Dynamic Graphics Project
2 2 g dω kgkU U R = 2 University of Torontog 2 dω kgk 2 S2
82
2
Future Work
C 1− • Boundary function for K(). = + log B( 4π • Effective dimension of light transport for non linear approximation schemes. C 1− + log B( K() = 4π |{z} | {z λi ≈1
λi ∈[1
D gi = λi R
λ= [email protected]
Dynamic Graphics Project
2 2 g dω kgkU U R = 2 University of Torontog 2 dω kgk 2 S2
83
2
Future Work
C 1− • Boundary function for K(). = + log B( 4π • Effective dimension of light transport for non linear approximation schemes. C 1− + log B( K() = 4π • Slepian functions for Riemannian manifolds. |{z} | {z λi ≈1
λi ∈[1
D gi = λi R
λ= [email protected]
Dynamic Graphics Project
2 2 g dω kgkU U R = 2 University of Torontog 2 dω kgk 2 S2
84
Conclusion • Characterization of effective dimension of light transport in a local neighborhood. — Improved estimate for dimensionality. — Closed form expression for basis functions.
[email protected]
Dynamic Graphics Project
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85
Conclusion • Characterization of effective dimension of light transport in a local neighborhood. — Improved estimate for dimensionality. — Closed form expression for basis functions. • Introduction of Slepian functions. — Efficient representation of localized signals. — Advantages of Spherical Harmonics.
[email protected]
Dynamic Graphics Project
University of Toronto
86
More information and source code: www.dgp.toronto.edu/people/lessig/effective-dimension/
[email protected]
Dynamic Graphics Project
University of Toronto
87
Effective Dimension Diffuse 0.5 Θ = 15° Θ = 20° Θ = 25° Θ = 30°
0.45
0.35 0.3 0.25 0.2 0.15
Eigenvalue magnitude
0.4
0.1 0.05 1
2
3
4
5
6
7
8
9
0
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
88
Effective Dimension Phong, s = 132 1 Θ = 5° Θ = 10° Θ = 15° Θ = 20°
0.9
0.7 0.6 0.5 0.4 0.3
Eigenvalue magnitude
0.8
0.2 0.1 0
10
1
10
2
0
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
89
Related Work • Coherence has been exploited in precomputed radiance transfer for some time.1 • Mahajan et al.2 studied the effective dimension in flatland and discussed extension to 3D. • Shading equation has been studied previously by Ramamoorthi and co-workers3 using assumptions similar to ours. Liu X, Sloan P, Shum H, Snyder J. All-Frequency Precomputed Radiance Transfer for Glossy Objects. In: Eurographics Symposium on Rendering 2004.; 2004.; Sloan P, Hall J, Hart J, Snyder J. Clustered Principal Components for Precomputed Radiance Transfer. In: SIGGRAPH '03: ACM SIGGRAPH 2003 Papers. New York, NY, USA: ACM Press; 2003:382-391. 2 Mahajan D, Shlizerman IK, Ramamoorthi R, Belhumeur P. A Theory of Locally Low Dimensional Light Transport. ACM Trans. Graph. 2007;26(3 (Proceedings of ACM SIGGRAPH 2007):1-9. 3 Ramamoorthi R, Hanrahan P. A Signal-Processing Framework for Inverse Rendering. International Conference on Computer Graphics and Interactive Techniques. 2001. Available at: http://portal.acm.org/citation.cfm?id=383271; Ramamoorthi R, Hanrahan P. A Signal-Processing Framework for Reflection. ACM Transactions on Graphics (TOG). 2004;23(4).; Ramamoorthi R, Koudelka M, Belhumeur P. A Fourier Theory for Cast Shadows. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2005;27(2). 1
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Dynamic Graphics Project
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2
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
3
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
4
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
5
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
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Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
7
Problem statement
[email protected]
Dynamic Graphics Project
University of Toronto
Christian
Lessig∗
8
June 28, 2010 Problem statement
U
H2
S2
x
Z T (n(x) · ω) E(ω) dω
B(x) = H2
˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U U
L X
tl yl0 (n(x) · ω)
T (n(x) · ω) = l=0 L X
T (n(x) · ω) = [email protected]
tl Pl (n(x) · ω) l=0 Dynamic Graphics Project
University of Toronto
kB − BkU = kTE − TEkU ≈
Christian
Lessig∗
9
L X
June 2010 T (n(x) · ω) = statement tl yl0 (n(x)28, · ω) Problem l=0
L X
T (n(x) · ω) =
tl Pl (n(x) · ω) l=0 U H2 S 2 x l L X X = t˜l Z ylm (n(x)) ylm (ω) B(x) l=0 =m=−lT (n(x) · ω) E(ω) dω H2
L X l X ˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U elm ylm (ω) U E(ω) = l=0 m=−l 2 L SM X tl yl0 (n(x) · ω) T (n(x) · ω) = l=0 L X ˜ dim (T) dim (T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
l=0 Dynamic Graphics Project
University of Toronto
Christian Lessig∗ kB − BkU = kTE − TEkU ≈
Christian
June 28, 2010 T (n(x) · ω) = Problem statement
Lessig∗
10
L X
28, 2010 tl yJune · ω) l0 (n(x) l=0
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) Z l=0 U H2 S 2 x l L B(x) = T (n(x) · ω) E(ω) dω X X H2 = ylm (n(x)) ylm (ω) t˜l Z B(x) l=0 = m=−lT (n(x) · ω) E(ω) dω 2 2 ˜ ˜ kB − Bk H2 U = kTE − TEkU ≈
U
H2
S2
L X l X ˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U elm ylm (ω) U E(ω) = L X
l=0 m=−l
tl yl0 (n(x) · ω)
T (n(x) · ω) = l=0 L X
T (n(x) · ω) =
tl Pl (n(x) · ω)
2 L SM X T (n(x) · ω) = tl yl0 (n(x) · ω)
l=0
L X ˜ dim (T) dim (T) tl Pl (n(x) · ω) T (n(x) · ω) = l L X X l=0 [email protected] Graphics Project University of Toronto = ylmDynamic (n(x)) ylm (ω) t˜l l=0
Christian Lessig∗ kB − BkU = kTE − TEkU ≈
Formulas 11
Christian L Christian Lessig∗ X 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) statement l=0 June 28, 2010
June 28, 2010
Problem
Lessig∗
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) Z l=0 U H2 S 2 x l L B(x) = T (n(x) · ω) E(ω) dω X X U H2 S 2 x H2 = ylm (n(x)) ylm (ω) t˜l Z Z · ω) E(ω) dω B(x) l=0 = m=−lT (n(x) 2 2 ˜ ˜ kB − Bk H2 U = kTE − TEkU ≈ T (n(x) · ω) E(ω) B(x) = L X l H2 X 2 ˜ 2 ≈ ˜ = kTE − TEk kB − Bk U elm ylm (ω) U E(ω) = 2 2 ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = 2 L SM X l=0 T (n(x) · ω) = tl yl0X (n(x) · ω) L L X T (n(x)l=0 · ω) = tl yl0 (n(x) · ω) T (n(x) · ω) = tl Pl (n(x) · ω) L X l=0 ˜ dim (T) dim (T)l=0 t l Pl X (n(x) · ω) T (n(x) · ω) = L l L X X [email protected] Graphics Project of Toronto· ω) tl Pl (n(x) T (n(x)l=0 · ω) = University = ylmDynamic (n(x)) ylm (ω) t˜l
U
H2
S2
Christian Lessig∗ kB − BkU = kTE − TEkU ≈ Formulas ∗ 12 Christian Lessig Christian Lessig∗ June 28, 2010 ∗ L Christian X June Lessig 28, 2010 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) Problem statement l=0 June 28, 2010
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 U H S x 2 2 Z l=0 U H S x l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x H2 = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) 2 2 ˜ ˜ kB − Bk H2 U = kTE − TEkU ≈ T (n(x) · ω) E(ω) B(x) = ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ = kTE − TEk kB − Bk U elm ylm (ω) U ≈ E(ω) = 2 2 ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L X 2 L X · ω) = l=0 M TS(n(x) tl yl0 (n(x) · ω T (n(x) · ω) = tl yl0X (n(x) · ω) L L X l=0 T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) = tl Pl (n(x) · ω) X L X l=0 ˜ T tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) t l Pl X (n(x) · ω) T (n(x) · ω) = L l L X X l=0 [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University = ylmDynamic (n(x)) ylm (ω) t˜l l
U
H2
S2
Formulas Christian Lessig∗ kB − BkU = kTE − TEkU ≈ Formulas ∗ 13 Christian Lessig ∗ ∗ Christian Lessig June 28, 2010 Christian Lessig ∗ L Christian X June Lessig 28, 2010 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) June 28, 2010 Problem statement l=0 June 28, 2010
L X x Shading equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 U H S x 2 2 Z l=0 U H S x U H2 S 2 x l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) Z H2 = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) B(x) = T (n(x) · ω) E(ω) dω 2 2 ˜ ˜ 2 = kTE − TEk ≈ kB − Bk H2 U U Objective: H T (n(x) · ω) E(ω) B(x) = ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ 2 2 = kTE − TEk kB − Bk ˜ ˜ U elm ylm (ω) U ≈ = kB − BkU = kTE − TEkU ≈E(ω) 2 2 ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L X 2 L X · ω) = l=0 L M TS(n(x) tl yl0 (n(x) · ω X tl yl0X (n(x) · ω) L L t y (n(x) · ω) T (n(x) · ω) = T (n(x) · ω) = X l=0 l l0 T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) =l=0 tl Pl (n(x) · ω) X L X l=0 ˜ T L tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) X t l Pl X (n(x) · ω) T (n(x) · ω) = L l L t PX T (n(x) · ω) = X l=0 l l (n(x) · ω) [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University =l=0 t˜l ylmDynamic (n(x)) ylm (ω) l
U
H2
S2
∗ kB − BkU = kTE − TEkU ≈ Formulas Christian Lessig L X Formulas ∗ 14 Christian Lessig T (n(x) · ω) = tl yl0 (n(x) · ω) ∗ ∗ Christian Lessig June 28, 2010 Christian Lessig l=0 ∗ L Christian Lessig X June 28, 2010 L X 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) June 28, 2010 Problem statement tl Pl (n(x) · ω) T (n(x) · ω) = l=0 June 28, 2010 l=0
L 2 2 X U H S x l L X X Shading˜equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 = ylm (n(x)) ylm (ω) tl U H S x 2 2 Z l=0 2 U H S x U Hm=−l S2 x l=0 l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x H2 = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) ZX L X l = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) B(x) = T (n(x) · ω) E(ω) dω e y (ω) E(ω) = 2 2 lm lm ˜ ˜ 2 = kTE − TEk ≈ kB − Bk H2 U U Objective: H m=−l T (n(x) · ω) E(ω) B(x) = l=0 ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ 2 2 = kTE − TEk kB − Bk ˜ ˜ U 2 elm ylm (ω) U ≈ = kB − BkU =SkTE − TEkU ≈E(ω) 2 2 M ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L with X 2 L S(n(x) X · ω) = l=0 L M T tl yl0 (n(x) · ω X ˜ T) dim (T) T (n(x) · ω) = tl yl0X (n(x) · ω) L L t T (n(x) · ω)dim = (X l=0 l yl0 (n(x) · ω) T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) =l=0 tl Pl (n(x) · ω) X L X l=0 ˜ T L tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) X t l Pl X (n(x) · ω) T (n(x) · ω) = L L t PX l T (n(x) · ω) = X (n(x) · ω) l=0 l l ˜ T˜ T [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University =l=0 tl ylmDynamic (n(x)) ylm (ω) l
∗ kB − BkU = kTE − TEkU ≈ Formulas Christian Lessig L X Formulas ∗ 15 Christian Lessig T (n(x) · ω) = tl yl0 (n(x) · ω) ∗ ∗ Christian Lessig June 28, 2010 Christian Lessig l=0 ∗ L Christian Lessig X June 28, 2010 L X 28, 2010 T (n(x) · ω) = tl yJune · ω) l0 (n(x) June 28, 2010 Problem statement tl Pl (n(x) · ω) T (n(x) · ω) = l=0 June 28, 2010 l=0
L 2 2 X U H S x l L X X Shading˜equation: T (n(x) · ω) = tl Pl (n(x) · ω) 2 2 = ylm (n(x)) ylm (ω) tl U H S x 2 2 Z l=0 2 U H S x U Hm=−l S2 x l=0 l L B(x) = T (n(x) · ω) E(ω) dω X X U ZH2 S 2 x H2 = ylm (n(x)) yTlm(n(x) (ω) · ω) E t˜l Z B(x) ZX L X l = Z ·H ω)2 E(ω) dω B(x) l=0 = m=−lT (n(x) B(x) = T (n(x) · ω) E(ω) dω e y (ω) E(ω) = 2 2 lm lm ˜ ˜ 2 = kTE − TEk ≈ kB − Bk H2 U U Objective: H m=−l T (n(x) · ω) E(ω) B(x) = l=0 ˜2 2 = kTE − TEk ˜ L X l H kB − Bk X U 2 2 ˜ ˜ 2 2 = kTE − TEk kB − Bk ˜ ˜ U 2 elm ylm (ω) U ≈ = kB − BkU =SkTE − TEkU ≈E(ω) 2 2 M ˜ ˜ = kTE − TEk kB − Bk L l=0 m=−l U U ≈ X tl yl0 (n(x) · ω) T (n(x) · ω) = L with X 2 L S(n(x) X · ω) = l=0 L M T tl yl0 (n(x) · ω X ˜ T) dim (T) T (n(x) · ω) = tl yl0X (n(x) · ω) L L t T (n(x) · ω)dim = (X l=0 l yl0 (n(x) · ω) T (n(x)l=0 · ω) = tlLyl0 (n(x) · ω) T (n(x) · ω) =l=0 tl Pl (n(x) · ω) X effective dimension L X l=0 ˜ T L tl Pl (n(x) · ω (n(x) ω)l=0 = dim (T) dim· (T) X t l Pl X (n(x) · ω) T (n(x) · ω) = L l L t PX T (n(x) · ω) = X (n(x) · ω) l=0 l l ˜ T˜ T [email protected] Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University =l=0 tl ylmDynamic (n(x)) ylm (ω) l
2
˜ 2 ≈ ˜ 2 = kTE − TEk kB − Bk U U
Formulas Christian Lessig∗ Lessig∗
16
Christian ∗ L Christian Lessig June 28, 2010 X 28, 2010 SimplificationT (n(x) · ω) = tl yJune l0 (n(x) · ω) June 28, 2010 l=0 Z
T (n(x) ·Tω) E(ω)· ω) dω = (n(x)
B(x) =
L X
tl Pl (n(x) · ω) U H2 S 2 x Z U H2 S 2 x l=0 2 2 l L Z T (n(x) · ω) E(ω) dω X B(x) = U H S x X Z S2 = ylm (n(x)) yTlm (ω) · ω) E t˜l B(x) = (n(x) Z X X Z ·H ω)2 E(ω) dω B(x) = m=−lT (n(x) l=0 ˜ tl ylm (n(x)) ylm (ω) = el0Hm20 yl0 m0 (ω) dω T (n(x) · ω) E(ω) B(x) = S 2 l,m l0 ,m0 ˜2 2 = kTE − TEk ˜ H kB − Bk L l Z U 2 2 X X ˜ ˜ XX kB − Bk U = kTE − TEkU ≈ y0 (ω) E(ω) = ylm (ω) yellm 0˜ t˜l el0 m0 ylm (n(x)) dω = 2lm (ω) m ˜ 2 ≈ = kTE − TEk kB − Bk U U S 2l=0 m=−l l,m l0 ,m0 Z L X XX L 2X ˜ S T (n(x) ω)0 (ω) = dω tl yl0 (n(x) · ω tl el0 m0 ylm (n(x)) ylm (ω) = M yl·0 m T (n(x) S· 2ω) = tl yl0X (n(x) · ω) L l=0 l,m l0 ,m0 | {z } l=0 T (n(x)δll·0 ω) tlLyl0 (n(x) · ω) δmm= 0 X L X X · ω)l=0 tl Pl (n(x) · ω T (n(x) = ˜ dim· ω) (T)= dim (T) = t˜l elm ylm (n(x))T (n(x) t l Pl X (n(x) · ω) L l=0 [email protected] l,m Dynamic Graphics Project of Toronto· ω) tlLPl (n(x) T (n(x)l=0 · ω) = University l H2
17
2
Simplification Z T (n(x) · ω) E(ω) dω
B(x) = H2
Z
T (n(x) · ω) E(ω) dω
B(x) = S2
Z
X
X t˜l ylm (n(x)) ylm (ω)
= S 2 l,m
el0 m0 yl0 m0 (ω) dω
Z
XX
l0 ,m0
t˜l el0 m0 ylm (n(x))
=
S2
l,m l0 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
=
ylm (ω) yl0 m0 (ω) dω
l,m l0 ,m0
ylm (ω) yl0 m0 (ω) dω S2 | {z } δll0 δmm0
X = [email protected] l,m
t˜l elm ylm (n(x)) Dynamic Graphics Project
University of Toronto
2 18
Z
Simplification T (n(x) · ω) E(ω) dω B(x) = H2
Z
T (n(x) · ω) E(ω) dω
B(x) = S2
Z
X
X t˜l ylm (n(x)) ylm (ω)
= S 2 l,m
el0 m0 yl0 m0 (ω) dω
Z
XX
l0 ,m0
t˜l el0 m0 ylm (n(x))
=
S2
l,m l0 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
=
ylm (ω) yl0 m0 (ω) dω
l,m l0 ,m0
ylm (ω) yl0 m0 (ω) dω 2 |S {z } δll0 δmm0
X t˜l elm ylm (n(x))
= l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
2 19
Z
Simplification T (n(x) · ω) E(ω) dω B(x) = H2
Z
T (n(x) · ω) E(ω) dω
B(x) = S2
Z
X
X t˜l ylm (n(x)) ylm (ω)
= S 2 l,m
el0 m0 yl0 m0 (ω) dω
Z
XX
l0 ,m0
t˜l el0 m0 ylm (n(x))
=
S2
l,m l0 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
=
ylm (ω) yl0 m0 (ω) dω
l,m l0 ,m0
ylm (ω) yl0 m0 (ω) dω 2 |S {z } δll0 δmm0
X t˜l elm ylm (n(x))
= l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
2 Z 20
T (n(x) · ω) E(ω) dω
B(x) = 2
ZH Z
T (n(x) · ω) E(ω) dω B(x) = Simplification B(x) = S T (n(x) · ω) E(ω) dω 2 H2
Z Z
X X = el0 m0 yl0 m0 (ω) dω t˜l ylm (n(x)) ylm (ω) B(x) = S 2 T (n(x) · ω) E(ω) dω l0 ,m0 S 2 l,m Z Z X XX 0 ylm (n(x)) = (n(x)) ylm (ω) ylme(ω) t˜t˜llyelm l0 m0 yl0 m0 (ω) dω l0 m S 2 ll,m 0 ,m0 l,m
Sl20 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
ylm (ω) yl0 m0 (ω) dω S2 l,m l0 ,m0 | {z } Z XX δll0 δmm0 t˜l el0 m0 ylm (n(x)) ylm (ω) yl0 m0 (ω) dω =X 2 = l,m tl˜0l,m elm ylm (n(x)) 0 |S {z } =
l,m
δll0 δmm0
X =
t˜l elm ylm (n(x)) X l,m t˜l elm ylm (n(x)) B(x) = l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
2 Z 21
T (n(x) · ω) E(ω) dω
B(x) = 2
ZH Z
T (n(x) · ω) E(ω) dω B(x) = Simplification B(x) = S T (n(x) · ω) E(ω) dω 2 H2
Z Z
X X = el0 m0 yl0 m0 (ω) dω t˜l ylm (n(x)) ylm (ω) B(x) = S 2 T (n(x) · ω) E(ω) dω l0 ,m0 S 2 l,m Z Z X XX 0 ylm (n(x)) = (n(x)) ylm (ω) ylme(ω) t˜t˜llyelm l0 m0 yl0 m0 (ω) dω l0 m S 2 ll,m 0 ,m0 l,m
Sl20 ,m0
Z
XX t˜l el0 m0 ylm (n(x))
ylm (ω) yl0 m0 (ω) dω S2 l,m l0 ,m0 | {z } Z XX δll0 δmm0 t˜l el0 m0 ylm (n(x)) ylm (ω) yl0 m0 (ω) dω =X 2 = l,m tl˜0l,m elm ylm (n(x)) 0 |S {z } =
l,m
δll0 δmm0
X =
t˜l elm ylm (n(x)) X l,m t˜l elm ylm (n(x)) B(x) = l,m
[email protected]
X
Dynamic Graphics Project
˜
University of Toronto
l,m
|
l0 ,m0
S2
˜l el0 m0 ylm (n Formulas = t 2 2 ˜ ˜ kB{z − BkU = kTE } − TEkU0 ≈ 0
X t˜l elm ylm (n(x))
= l,m
l,m l ,m
δll0 δmm0
SimplificationT (n(x) · ω) = X
22
∗ Christian Lessig X L X
= t˜l elm ylm (n(x)) 28,· ω) 2010 tl yJune l0 (n(x)l,m
l=0
L X X ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t l,m 2 2 U H S x l=0 l,m L X
2 n(x) w ω ∈ SM
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
H2
L X l2 2 X ˜ ˜ kB − Bk = kTE − TEk X U U ≈ elm ylm (ω) X t˜l elm ylm (ω) E(ω) = B(ω) = ˜l l=0 m=−l B(ω) = t l,m l,m X L 2X X SM blm ylm (ω) = b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
l,m
l0 ,m0
X = X t˜l elm ylm (n(x)) = l,m t˜l elm ylm (n(x)) l,m
|
S2
δll0 δ− ˜0 2 mm kB Bk {z U δll0 δmm0
Simplification T (n(x) · ω) = X
˜l el m Formulas t = 2 ˜ = kTE } − TEk ≈
L X
0
U l,m l0 ,m0
0
ylm (n
23
∗ Christian Lessig X
t˜l elm ylm (n(x)) = 28,· ω) 2010 tl yJune l0 (n(x)l,m
l=0 B(x) = X t˜l elm ylm (n(x)) L X X l,m ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t l,m 2 2 U H S x l=0 l,m 2 but n(x) w ω ∈ SM 2 n(x) w ω ∈ SM
L X
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
H2
X L X l2 X B(ω) = X t˜l elm ylm (ω) ˜ ˜ 2 ≈ kB − BkU = kTE − TEk U e y (ω) E(ω) = lm lm X B(ω) = l,m t˜l elm ylm (ω) X ˜l l=0 m=−l t B(ω) = l,m b =X lm ylm (ω) l,m L 2 X X SM = l,m blm ylm (ω) b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
˜l el0 m0 ylm (n Formulas 2 t = ˜ 2 δ S 0 δmm˜ 0 2 ˜ t e y (n(x)) 0 0 ll l lm lm kB{z − BkU = kTE l,m l ,m | } − TEkU0 ≈ 0 X l,m l ,m 24 l,m ˜ δ δ 0 0 ll mm = X tl elm ylm (n(x)) ∗ Christian Lessig X = l,m t˜l elm ylm (n(x)) ˜l elm ylm (n(x)) L X t = X l,m June 28, 2010 t˜l elm ylm (n(x)) B(x) = T (n(x) · ω) = t y (n(x) · ω) l,m l l0 X l,m ˜ l=0 B(x) = X tl elm ylm (n(x)) L X X l,m ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t 2 2 2 n(x)l,m w ω ∈ SM U H S x l=0 l,m
=
Simplification
2 but n(x) w ω ∈ SM
L X
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
2 n(x) w ω ∈ S M X H2 t˜l elm ylm (ω) B(ω) = X l,m ˜ L X l2 2 X t B(ω) =X e y (ω) ˜ ˜ l lm lm kB − Bk = kTE − TEk X U U ≈ elm ylm (ω) E(ω) = X (ω) lmy t˜l eylm B(ω)== l,mblm (ω) lm X ˜l l=0 m=−l t B(ω) = l,m l,m b =X lm ylm (ω) l,m L 2 X X SM = l,m blm ylm (ω) b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
˜l el0 m0 ylm (n Formulas 2 t = ˜ 2 δ S 0 δmm˜ 0 2 ˜ t e y (n(x)) 0 0 ll l lm lm kB{z − BkU = kTE l,m l ,m | } − TEkU0 ≈ 0 X l,m l ,m 25 l,m ˜ δ δ 0 0 ll mm = X tl elm ylm (n(x)) ∗ Christian Lessig X = l,m t˜l elm ylm (n(x)) ˜l elm ylm (n(x)) L X t = X l,m June 28, 2010 t˜l elm ylm (n(x)) B(x) = T (n(x) · ω) = t y (n(x) · ω) l,m l l0 X l,m ˜ l=0 B(x) = X tl elm ylm (n(x)) L X X l,m ˜ tl elm ylm (n(x)) B(x) = ˜l e tl Pl (n(x) · ω) B(x) = T (n(x) · ω) = t 2 2 2 n(x)l,m w ω ∈ SM U H S x l=0 l,m
=
Simplification
2 but n(x) w ω ∈ SM
L X
l X Z = ylm (n(x)) ylm (ω) t˜l dω w ω B(x) = m=−lT (n(x) · ω) E(ω) l=0 n(x)
2 n(x) w ω ∈ S M X H2 t˜l elm ylm (ω) B(ω) = X l,m ˜ L X l2 2 X t B(ω) =X e y (ω) ˜ ˜ l lm lm kB − Bk = kTE − TEk X U U ≈ elm ylm (ω) E(ω) = X (ω) lmy t˜l eylm B(ω)== l,mblm (ω) lm X ˜l l=0 m=−l t B(ω) = l,m l,m b =X lm ylm (ω) l,m L 2 X X SM = l,m blm ylm (ω) b T (n(x) · ω) = tl yl0 (n(x) · ω) = l,m l,m
l=0 L X ˜ dim (T) dim ( T) T (n(x) · ω) = tl Pl (n(x) · ω) [email protected]
Dynamic Graphics Project
l=0
University of Toronto
4π |{z}
|
λi ≈1
C K() = + log 4π |{z} |
Approximation λi ≈1
{z
}
λi ∈[1,]
26
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0 λi ∈[1,]
Objective 2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U
[email protected]
Dynamic Graphics Project
University of Toronto
4π |{z}
i=1
|
λi ≈1
{z
} 27
λi ∈[1,]
1 2−
C 2+ o(log C) log(C) K() = B(ω) + −log ˜bi kϕ ˜
= B(∂U) B(ω) (ω)k i U | {z } 4π U |{z} | {z } i=K+1 λi ≈0
N X
Approximation λi ≈1
Objective
λi ∈[1,]
N X 2 kϕ (ω)k ˜ 2 ≈ ˜ i kB(ω) − B(ω)k = kTE − TEk U U i=K+1
We are interested in a basis
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L i
2
kϕi (ω)kU
[email protected]
,
i = K + 1...N
Dynamic Graphics Project
University of Toronto
4π |{z}
i=1 Formulas {z
|
λi ≈1
} 28
λi ∈[1,]
Lessig ∗ Christian N 1 2− X
C 2+ o(log C) log(C) K() = B(ω) + −log ˜bi kϕ ˜
= B(∂U) B(ω) (ω)k i U | {z } 4π U |{z} {z 2010 } i=K+1 {ϕi (ω)} , i |= 1 . .June . N 23, λi ≈0
Approximation λi ≈1
λi ∈[1,]
Objective
span (ϕi ) = i
N X 2 kϕ (ω)k ˜ 2 ≈ ˜ i kB(ω) − B(ω)k = kTE − TEk U U span (yL{ϕ)i=K+1 H≤L (ω)} , i = 1...N i=
l,m We are interested in a basis
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L span (ϕi ) = span (yi L ) = H≤L K X
i
l,m
˜bdimension ˜ effective B(ω) ≈ with B(ω) = , K N such that i ϕi 2
i=1kϕi (ω)kU
,
iK = K + 1...N X ˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = i=1
2
˜
=
B(ω) − B(ω) U [email protected]
N X
˜bi kϕi (ω)kN2 U X
Dynamic Graphics Project 2
˜
˜
2University of Toronto
span (ϕi ) = span (yL ) = H≤L i
l,m
Approximation ˜ B(ω) ≈ B(ω) =
K X
29
˜bi ϕi
,
K N
i=1
Approximation error N X
2 ˜bi kϕi (ω)k2 ˜
B(ω) − B(ω)
= U U i=K+1
N X
kϕi (ω)k i=K+1
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L i
[email protected]
2
kϕi (ω)kDynamic , Graphics i =Project K + 1...N U
University of Toronto
(yL ) = H≤L span (ϕi ) = spanX N
˜bi kϕi (ω)k2 ˜
B(ω) −i B(ω)
2 =l,m U U
30
i=K+1
Approximation ˜ B(ω) ≈ B(ω) N=
K X
X
˜bi ϕi
,
K N
i=1 i=K+1 Approximation error
kϕi (ω)k
N X
2
˜bi kϕi (ω)k2 ˜
=
B(ω) − B(ω) U U 2i=K+1 ϕi (ω) | i = 1 . . . (L + 1) , span (ϕi ) = H≤L i
which, for arbitrary input signals, is minimized if N X (ω)k , ikϕ = iK + 1...N
2
kϕi (ω)kU
i=K+1
is minimal.
ϕi (ω) | i = 1 . . . (L + 1)2 , span (ϕi ) = H≤L i
∗ [email protected] [email protected]
2
kϕi (ω)kDynamic , Graphics i =Project K + 1...N U
University of Toronto
31
Spatio-Spectral Concentration Theory
[email protected]
Dynamic Graphics Project
University of Toronto
C K() = + log 4π |{z} | λi ≈1
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0
32
λi ∈[1,]
Spatio-Spectral Concentration Theory D gi = λi gi
Objective: Extremize concentration measure1 λ=
2 kgkU 2 kgkS 2
R 2 g dω U =R 2 dω g 2 S
g ∈ H≤L
,
2
C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1
1− C + log B(∂U) log(C) +o(log C) K() = 4π |{z} [email protected] Project | Dynamic Graphics{z } University of Toronto 1
Simons FJ, Dahlen FA, Wieczorek MA. Spatiospectral Concentration on a Sphere. SIAM Review. 2006;48(3):504-536; Simons FJ. Slepian Functions and Their Use in Signal Estimation and Spectral Analysis. In: Freeden W Handbook of Geomathematics.; 2010.
C K() = + log 4π |{z} | λi ≈1
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0
33
λi ∈[1,]
Spatio-Spectral Concentration Theory D gi = λigi 1− C +Extremize log B(∂U) log(C) measure + o(log C)1 K() = Objective: concentration 4π R 2 2 U g dω kgk U C λ= 1 −= R , g ∈ H≤L 2 2 B(∂U) log(C) + o(log C) K() = + log kgk 2 g dω 2 S S | {z } 4π |{z} | {z } λi ≈0
λi ≈1Eigenvalue problem Solution: λi ∈[1,]
2
C = N A(U) = (L + 1) A(U) D gi = λi gi R 1g− C 2 2 dω B(∂U) log(C) + o(log C) kgk + K() = U logR U λ = 4π 2 = , g ∈ H≤L 2 |{z} g dω kgkS 2 S2 λi ≈1 Simons FJ, Dahlen FA, Wieczorek MA. Spatiospectral Simons FJ. Slepian Functions and Their Use in Concentration on a Sphere. SIAM Review. 2006;48(3):504-536; Signal Estimation and Spectral Analysis. In: Freeden W Handbook of Geomathematics.; 2010. 1− C + log B(∂U) log(C) +o(log C) K() = 2 4π= N A(U) = (L + 1) A(U) C |{z} [email protected] Project | Dynamic Graphics{z } University of Toronto 1
C 1− K() = B(∂U) log(C) + o(log C) + log June 27, 2010 | {z } 4π |{z} | {z } λi ≈0
λi ≈1
34
λi ∈[1,]
Spatio-Spectral Concentration Theory D gi = λigi 1kgk − 2 C 1 Uconcentration +Extremize log B(∂U) log(C) + o(log C) K() = Objective: measure λ = , g ∈ H ≤L 2 4π kgkS 2R 2 U g 2 dω kgk U C λ= 1 −= R , g ∈ H≤L 2 2 B(∂U) log(C) + o(log C) K() = + log kgk 2 g dω 2 S S | {z } T 4π D g g |{z} | {z } λi ≈0 λ = λ ≈1 T i Solution: Eigenvalue problem λi ∈[1,] g g 2 C = N A(U) = (L + 1) A(U) D gi = λi gi
where
D = {dlm,l0 m0 } RZ C 20 0 = 1 − 2y (ω) y 0 0 (ω) dω d g dω lm,l m lm B(∂U) lm kgk log(C) + o(log C) + K() = U logR U λ = 4π 2 = , g ∈ H≤L U 2 |{z} g dω kgkS 2 S2 λi ≈1 Simons FJ, Dahlen FA, Wieczorek MA. Spatiospectral Simons FJ. Slepian Functions and Their Use in Concentration on a Sphere. SIAM Review. 2006;48(3):504-536; Signal Estimation and Spectral Analysis. Z In: Freeden W Handbook of Geomathematics.; 2010. 1− C +D(ω, logω B(∂U) log(C) +o(log C) K() = ¯ ) g (¯ ω ) dω = 2 λi gi (ω) i 4π=U N A(U) = (L + 1) A(U) C |{z} [email protected] Project | Dynamic Graphics{z } University of Toronto 1
35
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
Dynamic Graphics Project
University of Toronto
36
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
Dynamic Graphics Project
University of Toronto
37
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
λ = 0.968958
Dynamic Graphics Project
University of Toronto
38
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
λ = 0.968958
Dynamic Graphics Project
λ = 0.814437
University of Toronto
39
Spatio-Spectral Concentration Theory λ = 0.998093
[email protected]
λ = 0.968958
λ = 0.968958
Dynamic Graphics Project
λ = 0.814437
λ = 0.814437
University of Toronto
40
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.814437
λ = 0.814437
λ = 0.000189436
[email protected]
Dynamic Graphics Project
University of Toronto
41
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.000189436
λ = 0.000189436
[email protected]
λ = 0.968958
Dynamic Graphics Project
λ = 0.814437
λ = 0.814437
University of Toronto
42
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.000189436
λ = 0.000189436
λ = 0.000125893
[email protected]
Dynamic Graphics Project
λ = 0.814437
λ = 0.814437
University of Toronto
43
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.814437
λ = 0.000189436
λ = 0.000189436
λ = 0.000125893
λ = 0.000100997
[email protected]
Dynamic Graphics Project
λ = 0.814437
University of Toronto
44
Spatio-Spectral Concentration Theory λ = 0.998093
λ = 0.968958
λ = 0.968958
λ = 0.814437
λ = 0.814437
λ = 0.000189436
λ = 0.000189436
λ = 0.000125893
λ = 0.000100997
λ = 0.000100997
[email protected]
Dynamic Graphics Project
University of Toronto
45
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
46
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
47
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
48
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
49
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
50
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
51
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
52
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
53
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
54
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
55
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
56
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
57
Spatio-Spectral Concentration Theory 2
Spectrum (conjecture):
B(∂U) log(C) + o(log C)
1− C + log K() = 4π 1− C B(∂U) log(C) + o(log C) + log K() = | {z } 4π |{z} | {z } λi ≈0 λi ≈1
λi ∈[1,]
D gi = λi gi
λ= [email protected]
2 kgkU 2 kgkS 2
R 2 g dω U =R 2 dω g 2 S Dynamic Graphics Project
,
g ∈ H≤L University of Toronto
4π
C + log K() = 4π |{z} |
1− B(∂U) log(C) + o(log C) | {z } {z } λi ≈0
58
λ ≈1 Spatio-Spectral Concentration Theory λ ∈[1,] i
2
i
D gi = λi gi
Spectrum (conjecture):
1R− 2 C 2 g dωB(∂U) log(C) + o(log C) + K() = kgklog U U R λ4π = , g ∈ H≤L 2 = 2 dω kgkS 2 S 2 g 1− C with B(∂U) log(C) + o(log C) + log K() = | {z } 4π |{z} | N A(U) = (L{z+ 1)2 A(U) } λi ≈0 C = λi ≈1 λi ∈[1,]
D gi = λi gi 1− C + log B(∂U) log(C) + o(log C) K() = R 4π |{z} 2 2 g dω kgk λi ≈1 U U λ = 2 = R 2 , g ∈ H≤L kgkS 2 1 −S2 g dω C + log Dynamic Graphics B(∂U) log(C) +o(log C) of Toronto K() = [email protected] Project University 4π
C 1− K() = + log B(∂U) log(C) + o(log C) 4πC 1 − B(∂U) log(C) + o(log C) + log K() = 4π |{z} 1 − C + log B(∂U) log(C) + o(log C) K() = λi ≈1 | {z } 4π |{z} | {z } C 1 − λi ≈0 λ ≈1 i K() = + log B(∂U) log(C) +o(log C) λi ∈[1,] 4π |{z} | {z } Spectrumλ(conjecture): i ≈1 D gi =λλ i gi i ∈[1,] C 1− B(∂U) log(C) + o(log C) K() = + log | {z } 4π R 2 |{z} 2 | {z } g dω kgk λi ≈0 U U λ ≈1 i R λ= g ∈ H≤L λi ∈[1,], 2 = 2 g dω kgkS 2 S2
59
Spatio-Spectral Concentration Theory
with
2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U 2 C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1 [email protected]
Project Dynamic Graphics
University of Toronto
C 1− K() = + log B(∂U) log(C) + o(log C) 4πC 1 − + log B(∂U) log(C) + o(log C) K() = 4π |{z} 1 − C + log B(∂U) log(C) + o(log C) K() = λi ≈1 | {z } 4π |{z} | {z } 1 − C λi ≈0 λ ≈1 i + log B(∂U) log(C) +o(log C) K() = λi ∈[1,] 4π |{z} | {z } Spectrumλ(conjecture): i ≈1 D gi =λλ i gi i ∈[1,] C 1− B(∂U) log(C) + o(log C) K() = + log | {z } 4π R 2 |{z} 2 | {z } g dω kgk λi ≈0 U U λ ≈1 i R λ= g ∈ H≤L λi ∈[1,], 2 = 2 g dω kgkS 2 S2
60
Spatio-Spectral Concentration Theory
with
2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U 2 C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1 [email protected]
Project Dynamic Graphics
University of Toronto
C 1− K() = + log B(∂U) log(C) + o(log C) 4πC 1 − + log B(∂U) log(C) + o(log C) K() = 4π |{z} 1 − C + log B(∂U) log(C) + o(log C) K() = λi ≈1 | {z } 4π |{z} | {z } 1 − C λi ≈0 λ ≈1 i + log B(∂U) log(C) +o(log C) K() = λi ∈[1,] 4π |{z} | {z } Spectrumλ(conjecture): i ≈1 D gi =λλ i gi i ∈[1,] C 1− B(∂U) log(C) + o(log C) K() = + log | {z } 4π R 2 |{z} 2 | {z } g dω kgk λi ≈0 U U λ ≈1 i R λ= g ∈ H≤L λi ∈[1,], 2 = 2 g dω kgkS 2 S2
61
Spatio-Spectral Concentration Theory
with
2 ˜ 2 ≈ ˜ kB(ω) − B(ω)k = kTE − TEk U U 2 C = N A(U) = (L + 1) A(U)
1− C + log B(∂U) log(C) + o(log C) K() = 4π |{z} λi ≈1 [email protected]
Project Dynamic Graphics
University of Toronto
62
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
63
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
64
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
65
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
66
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
67
Spatio-Spectral Concentration Theory L = 20 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
68
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
69
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
70
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
71
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
72
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
73
Spatio-Spectral Concentration Theory L = 10 1 Θ = 50 Θ = 40 Θ = 30 Θ = 20 Θ = 10 Θ=5
0.9
Eigenvalue Magnitude
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10
1
10
2
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
Formulas
N X
2 ˜i (ω)} =
B(ω) − {ϕB(ω) , i = 1 ˜.b.i .kϕ Ni (ω)k2U U
74
i=K+1
Christian Lessig∗
Effective Dimension N = span (yL ) = H≤L span (ϕi )X i
l,m kϕ
June 23, 2010 (ω)k i
Error is minimizedi=K+1 if K X
˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = ϕi (ω) | i = 1 . . . (L +i=1 1)2 , span (ϕi ) = H≤L
i
i = 1...N is approximated with{ϕi (ω)}for ,which N X
2 2 2 ˜ ˜
B(ω) − B(ω) = kϕ (ω)k b kϕi (ω)kU ,U i = K + 1i . . .iN U i=K+1
span (ϕi ) = span (yL ) = H≤L N X i
l,m
kϕi (ω)k i=K+1 ∗ [email protected]
[email protected]
Dynamic Graphics Project
˜
K X
˜
University of Toronto
Formulas
N X
2 ˜i (ω)} =
B(ω) − {ϕB(ω) , i = 1 ˜.b.i .kϕ Ni (ω)k2U U
75
i=K+1
Christian Lessig∗
Effective Dimension N = span (yL ) = H≤L span (ϕi )X i
l,m kϕ
June 23, 2010 (ω)k i
Error is minimizedi=K+1 if K X
˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = ϕi (ω) | i = 1 . . . (L +i=1 1)2 , span (ϕi ) = H≤L
i
i = 1...N is approximated with{ϕi (ω)}for ,which N X
2 2 2 ˜ ˜
B(ω) − B(ω) = kϕ (ω)k b kϕi (ω)kU ,U i = K + 1i . . .iN U i=K+1
=> Optimal basis functions Slepian functions. ) = span (yL ) = H≤L span (ϕare i N X i
l,m
kϕi (ω)k i=K+1 ∗ [email protected]
[email protected]
Dynamic Graphics Project
˜
K X
˜
University of Toronto
Formulas
N X
2 ˜i (ω)} =
B(ω) − {ϕB(ω) , i = 1 ˜.b.i .kϕ Ni (ω)k2U U
76
i=K+1
Christian Lessig∗
Effective Dimension N = span (yL ) = H≤L span (ϕi )X i
l,m kϕ
June 23, 2010 (ω)k i
Error is minimizedi=K+1 if K X
˜bi ϕi , K N ˜ B(ω) ≈ B(ω) = ϕi (ω) | i = 1 . . . (L +i=1 1)2 , span (ϕi ) = H≤L
i
i = 1...N is approximated with{ϕi (ω)}for ,which N X
2 2 2 2 ˜ ˜
B(ω) − B(ω) = kϕ (ω)k b kϕi (ω)kU ,U i = K + 1i . . .iN U i=K+1
=> Optimal basis functions Slepian functions. ) = span (yL ) = H≤L span (ϕare i
C 1− => Effective dimension is given by K(). = + log kϕi (ω)k 4π i=K+1 ∗ [email protected] K X 1− C [email protected] Dynamic Graphics Project University of Toronto + log K() ˜ = ˜ N X i
l,m
77
Effective Dimension
B(∂U) log(C) + o(log C
C 1− K() = + log 4π C 1− K() = + log B(∂U) log(C) + o(log C | {z 4π |{z} | {z } λi ≈0 λi ≈1
λi ∈[1,]
D gi = λi gi [email protected]
Dynamic Graphics Project
University of Toronto
78
Applications • Transport Matrix compression. — Optimal cluster size. — Analytic basis functions for neighborhoods.
[email protected]
Dynamic Graphics Project
University of Toronto
79
Applications • Transport Matrix compression. — Optimal cluster size. — Analytic basis functions for neighborhoods. • Representation of signals localized on the sphere. — Compact representation with most of the advantages as Spherical Harmonics.
[email protected]
Dynamic Graphics Project
University of Toronto
80
Applications • Transport Matrix compression. — Optimal cluster size. — Analytic basis functions for neighborhoods. • Representation of signals localized on the sphere. — Compact representation with most of the advantages as Spherical Harmonics. • Exploitation of coherence of light transport in sampling-based algorithms.
[email protected]
Dynamic Graphics Project
University of Toronto
81
2
Future Work
C 1− • Boundary function for K(). = + log B( 4π 1− C + log B( K() = 4π |{z} | {z λi ≈1
λi ∈[1
D gi = λi R
λ= [email protected]
Dynamic Graphics Project
2 2 g dω kgkU U R = 2 University of Torontog 2 dω kgk 2 S2
82
2
Future Work
C 1− • Boundary function for K(). = + log B( 4π • Effective dimension of light transport for non linear approximation schemes. C 1− + log B( K() = 4π |{z} | {z λi ≈1
λi ∈[1
D gi = λi R
λ= [email protected]
Dynamic Graphics Project
2 2 g dω kgkU U R = 2 University of Torontog 2 dω kgk 2 S2
83
2
Future Work
C 1− • Boundary function for K(). = + log B( 4π • Effective dimension of light transport for non linear approximation schemes. C 1− + log B( K() = 4π • Slepian functions for Riemannian manifolds. |{z} | {z λi ≈1
λi ∈[1
D gi = λi R
λ= [email protected]
Dynamic Graphics Project
2 2 g dω kgkU U R = 2 University of Torontog 2 dω kgk 2 S2
84
Conclusion • Characterization of effective dimension of light transport in a local neighborhood. — Improved estimate for dimensionality. — Closed form expression for basis functions.
[email protected]
Dynamic Graphics Project
University of Toronto
85
Conclusion • Characterization of effective dimension of light transport in a local neighborhood. — Improved estimate for dimensionality. — Closed form expression for basis functions. • Introduction of Slepian functions. — Efficient representation of localized signals. — Advantages of Spherical Harmonics.
[email protected]
Dynamic Graphics Project
University of Toronto
86
More information and source code: www.dgp.toronto.edu/people/lessig/effective-dimension/
[email protected]
Dynamic Graphics Project
University of Toronto
87
Effective Dimension Diffuse 0.5 Θ = 15° Θ = 20° Θ = 25° Θ = 30°
0.45
0.35 0.3 0.25 0.2 0.15
Eigenvalue magnitude
0.4
0.1 0.05 1
2
3
4
5
6
7
8
9
0
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
88
Effective Dimension Phong, s = 132 1 Θ = 5° Θ = 10° Θ = 15° Θ = 20°
0.9
0.7 0.6 0.5 0.4 0.3
Eigenvalue magnitude
0.8
0.2 0.1 0
10
1
10
2
0
10
Eigenvalue Index (Log) [email protected]
Dynamic Graphics Project
University of Toronto
89
Related Work • Coherence has been exploited in precomputed radiance transfer for some time.1 • Mahajan et al.2 studied the effective dimension in flatland and discussed extension to 3D. • Shading equation has been studied previously by Ramamoorthi and co-workers3 using assumptions similar to ours. Liu X, Sloan P, Shum H, Snyder J. All-Frequency Precomputed Radiance Transfer for Glossy Objects. In: Eurographics Symposium on Rendering 2004.; 2004.; Sloan P, Hall J, Hart J, Snyder J. Clustered Principal Components for Precomputed Radiance Transfer. In: SIGGRAPH '03: ACM SIGGRAPH 2003 Papers. New York, NY, USA: ACM Press; 2003:382-391. 2 Mahajan D, Shlizerman IK, Ramamoorthi R, Belhumeur P. A Theory of Locally Low Dimensional Light Transport. ACM Trans. Graph. 2007;26(3 (Proceedings of ACM SIGGRAPH 2007):1-9. 3 Ramamoorthi R, Hanrahan P. A Signal-Processing Framework for Inverse Rendering. International Conference on Computer Graphics and Interactive Techniques. 2001. Available at: http://portal.acm.org/citation.cfm?id=383271; Ramamoorthi R, Hanrahan P. A Signal-Processing Framework for Reflection. ACM Transactions on Graphics (TOG). 2004;23(4).; Ramamoorthi R, Koudelka M, Belhumeur P. A Fourier Theory for Cast Shadows. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2005;27(2). 1
[email protected]
Dynamic Graphics Project
University of Toronto
