Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… ·...
Transcript of Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… ·...
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Robotics
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P F
{wfp = (xfp, yfp)T | f = 1, ..., F, p = 1, ..., P} .
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W =
⎡⎢⎢⎣w11 . . . w1P
wF1 . . . wFP
⎤⎥⎥⎦ .
W 2F × P
M S
W = MS .
W
2 × 3 Rf
2F × 3 P
WM
S
W
Rf 2× 3
M
S 2F × 3 3 × P
W
W = UDV T ,
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D U V T
U
3× 3 D V T
W ′ = U ′D′V ′T U ′D′V ′T
‖W −W ′‖2F s.t. rank(W ′) = 3 .
M = U ′√D′ S =√D′V ′T
S
Rf
G
MS = M(GG−1)S = (MG)(G−1S) = MS .
G MG
iTf GGT if = 1
jTf GGT jf = 1
iTf GGT jf = 0 .
Q = GGT
Q
MQ Q−1S
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P F
{wfp = (xfp, yfp)T | f = 1, ..., F, p = 1, ..., P}
W =
⎡⎢⎢⎣w11 . . . w1P
wF1 . . . wFP
⎤⎥⎥⎦ .
W 2F × P
M S W = MS
W
2F × 3F
R1...Rf 2× 3
3F × P
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P
WM
S
W 10P × 10s × 30fps × 2(x, y) = 6, 000
3 × 10
900 × 10
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K
B1...BK
cf = [cf1, cf2, ..., cfK ]
Sf =
K∑d=1
cfdBd ,
Sf 3×P f Bd 3×P
K cfd
B1, ..., BK c1, ..., cK
Wf = Rf (
K∑d=1
cfdBd) .
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K
W = M(C ⊗ I3)B = ΠB ,
I3 3×3 B Π = M(C⊗ I3)
W Π
R ∈ 2×3
WΠ
R ∈ 2×3
W K
K
K
Π ∈ 2F×3K B ∈ 3K×P
3K × 3K Q
Q Π
Ri
Q
Mf = [cf1Rf ...cfKRf ] Π
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Q
MS = MQQ−1S G = QQT
Qk Q
k Mk = MQk Mk, k = 1, ...,K
M Mk
kth cfk
M M = MQ
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MfGkMTf = cfkcfkRfR
Tf ,
Gk = QkQTk 3K × 3K
Qk Mf
ith
MfGkMTf = c2fkI2 ,
I2 2×2 Gk
cfk
M2f−1GkMT2f−1 − M2fGkM
T2f = 0
M2f−1GkMT2f = 0 .
Gk
QHkQT Hk
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Gk
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W = ΠGG−1B Π = M(C ⊗ I3)
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Π
Qk ∈ 3K×3K Qk = GkGTk Π2f−1:2fQkΠ
T2f−1:2f = c2fkI2
cfk Qk
Π2f−1QkΠT2f−1 = Π2fQkΠ
T2f
Π2f−1QkΠT2f = 0 .
Qk
2K2−K
A
W
vec(.) qk = vec(Qk)
vec(AXBT ) = (B ⊗A)vec(X)
[Π2f−1 ⊗ Π2f−1 − Π2f ⊗ Π2f
Π2f−1 ⊗ Π2f
]qk = Afqk = 0 ,
Π2f−1, Π2f 2f − 1th 2fth Π
(f = 1, ..., F )
Aqk = 0 ,
A = [AT1 , A
T2 , ..., A
TF ]
T qk (3K)(3K+1)/2
2F ≥ (3K)(3K +
1)/2 qk
2K2 −K
A
Qk
2K2 −K
A
{Avec(Qk) = 0} ∩ {Qk � 0} ∩ {rank(Qk) = 3} .
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Qk
trace(Qk) s.t.
Qk � 0 ,
Avec(Qk) = 0 .
Qk Gk ∈ 3K×3
Gk
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K
K
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K(LK
)L
2×3
S F × 3P S
S K K
B S = C B
C ∈ F×L K Π = M(C ⊗I3) Π
2× 3 M 2× 3
W L
K
L
3× 2
B,Π‖W T −BΠ‖2F s.t. ‖Πi‖0,3 ≤ K, i = 1 : N/2 ,
‖Πi‖0,3 3 × 2 Π
B
Π
S
2 × 3
3K × 3K
3× 3 3L× 3L
G
Π B
G
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643
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![Page 47: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/47.jpg)
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G(V, E)V
E
![Page 49: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/49.jpg)
α
α
![Page 50: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/50.jpg)
R3
R3
(s, t, u) = o + s + t + u ,
o ,
0<s<1, 0<t<1 0<u<1 i,j,k
i,j,k = o +i
l+
j
m+
k
n,
l m n
l+1,m+1 n+1 , ,
(s, t, u) =
l∑i=0
m∑j=0
n∑k=0
Bi,l(s)Bj,m(t)Bk,n(u) i,j,k,
Bθ,n(x)
![Page 51: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/51.jpg)
n
i,j,k i, j, k
n
Bθ,n(x) =
(n
θ
)xθ(1− x)n−θ,
(nθ
)i,j,k
= ,
∈ RN×3 ∈ R
N×M
∈ RM×3 N
M
t s
u
l,m, n=3( , , ) i,j,k
![Page 52: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/52.jpg)
l,m, n=3
S TS
T
S( S , S) T ( T , T )
ST S ′( ′, S)
![Page 53: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/53.jpg)
Δ
1
2‖ − ( A ⊗ 3)( +ΦΔ )‖22 +
γ
2‖ΦΔ ‖22.
∈ R3P
vec([ T ]A) [ T ]A ∈ R3×P
P A A⊗
∈ R3M vec( �) M
Δ ∈ R3M
Φ ∈ R3M×3M
x−Φ
L2
γ L2
Δ
x−, y−z− Φ
vec(·)
![Page 54: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/54.jpg)
sdist
sdist =1
| ′|∑∈ ′
f( ,ST ; θ) +1
| T |∑∈ T
f( ,S ′; θ),
f( ,S; θ) =
⎧⎨⎩1 dist( ,S) > θ
0
θ
sIoU =V ′ ∩ VTV ′ ∪ VT
,
V ′ VT
G sdist < θdist sIoU > θIoU
θdist θIoU
G
Ω Sc( c, c)
i ∈ Ω
S( , )
= αc c +∑i∈Ω
αiic, = c,
![Page 55: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/55.jpg)
α
![Page 56: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/56.jpg)
G∈ R
2×P P
S
L
∈ R2×3 ∈ R
2×1
s
Δ , ,s,
1
2‖ L −
(( L ⊗ s )( +ΦΔ ) +
)‖22
+γ
2‖ΦΔ ‖22, � = 2.
L
LΔ
Φ
L2 γ
![Page 57: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/57.jpg)
Δ , , ,
1
2‖ L −
(( L ⊗ )( +ΦΔ ) +
)‖22
+γ
2‖ΦΔ ‖22, � = s2 2, = ,
s
Lρ( , ,Δ , ,Λ) =1
2‖ L −
(( L ⊗ )( +ΦΔ ) +
)‖22
+γ
2‖ΦΔ ‖22 + 〈Λ, − 〉F +
ρ
2‖ − ‖2F ,
Λ ρ
〈., .〉F
k = Lρ( , k−1,Δ k−1, k−1,Λk−1),
� = s2 2;
k = Lρ(k, ,Δ k−1, k−1,Λk−1);
Δ k =Δ
Lρ(k, k,Δ , k−1,Λk−1);
k = Lρ(k, k,Δ k, ,Λk−1);
Λk = Λk−1 + ρ( k − k).
Lρ( , k−1,Δ k−1, k−1,Λk−1) =[(σ1 + σ2)/2
(σ1 + σ2)/2
]�,
− Λρ = [ σ1
σ2 ]� σ
− Λρ
Lρ(k, ,Δ k−1, k−1,Λk−1) =
(( L − ) � +Λ+ ρ
)( � + ρ )+,
![Page 58: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/58.jpg)
= unvec( +ΦΔ ) L ∈ R3×|L| |L|
ΔLρ(
k, k,Δ , k−1,Λk−1) =
(Φ�( L ⊗ )�( L ⊗ )Φ+ γΦ
)+(Φ�( L ⊗ )�
(L − ( L ⊗ ) +
)).
Lρ(k, k,Δ k, ,Λk−1) =
∑l∈L( l − l)
|L| .
G
S α Sc( c, c)
Ω
S( , )
= αc c+∑
i∈Ω αiic c
ic
unvec(·)
![Page 59: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/59.jpg)
Sc Sic i
i α
s, ,α
1
2
∑l∈L
‖ l −(s (αc[ c]l +
∑i∈Ω
αi[ic]l) +
)‖22
+ μ
N∑l=1
C(s (αc[ c]l +
∑i∈Ω
αi[ic]l) +
)+
γ
2
∑i∈Ω
α2i ,
l l l = 1, . . . , P
N Cμ γ
L2
∇αi
αi
∇αi =∑l∈L
(l −
(s (αc[ c]l +
∑i∈Ω
αi[ic]l) +
))�s [ i
c]l
+ μN∑l=1
∇C�s [ ic]l + γαi,
∇C
e[ξ]×
[·]× ∇ξ
ξ
∇ξ =∑l∈L
(l −
(s (αc[ c]l +
∑i∈Ω
αi[ic]l) +
))�
(s
∂[ξ]×ξj
(αc[ c]l +∑i∈Ω
αi[ic]l)
)
+ μ
N∑l=1
∇C�(s
∂[ξ]×ξj
(αc[ c]l +∑i∈Ω
αi[ic]l)
).
![Page 60: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/60.jpg)
∇
∇ =∑l∈L
(l −
(s (αc[ c]l +
∑i∈Ω
αi[ic]p) +
))+ μ
N∑l=1
∇C.
s α
eRP
epose e3D
![Page 61: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/61.jpg)
l,m, n = 3 4×4×4
![Page 62: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/62.jpg)
sdist
sIoU sdist < θdist
sIoU > θIoU θdist θIoU
S Tsdist T
S ′ 1e−4 θdist = 1e−3
S ′
S ′
TS ′
T S ′
VT V ′ 0.1694 θIoU = 0.25
θdist θIoU
1283
![Page 63: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/63.jpg)
eRP epose
e3D
eRP epose e3D eRP epose e3D eRP epose e3D eRP epose e3D
eRP epose e3D
![Page 64: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/64.jpg)
S T T VT
S′ S′ V ′
ST
VT TS ′ V ′
S ′
T sdist S ′
sIoU
![Page 65: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/65.jpg)
![Page 66: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/66.jpg)
FFD
FFD
![Page 67: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/67.jpg)
![Page 68: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/68.jpg)
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c Δ αG cG
Δα
c Δ
α G
![Page 70: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/70.jpg)
G
Ω Sc( c, c)
i ∈ Ω
S( , )
= αc c +∑i∈Ω
αiic, = c,
α
G
= Φ( +Δ ),
∈ RN×3 ∈ R
N×M
∈ RM×3 N M
Δ
Φ ∈ R3M×3M
Δ
α
![Page 71: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/71.jpg)
c Δ α
Δ α
Δ
α
{8, 16, 32} {5, 3, 3} {3, 3, 3}
{16, 8, 1} {3, 3, 5} {3, 3, 3}220× 220
2202 → 722 → 242 → 82 → 242 → 722 → 2202
∈ R2,048
![Page 72: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/72.jpg)
8 × 3 × 3
Δ α
f : → {Δ ,α}
κ ∈ RMN M N
α
κ ∈ R126
(32×3) α
α
Gα α
![Page 73: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/73.jpg)
256 × 192
70 30
sdist sdist
dist3D =1
| |∑∈
dist( ,S) + 1
| |∑∈
dist( , S),
S S
IoU =V ∩ VV ∪ V
,
V V
![Page 74: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/74.jpg)
MSE ∼ t Acc( ) Prec( ) Rec( ) ∼ t MSE ∼ t dist3D IoU
Acc Prec Rec t
GG
Δα
G
G
![Page 75: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/75.jpg)
α
α
43
Δ ∈ R32×3
α ∈ R30
κ ∈ R126
c κ
c
Δ
![Page 76: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/76.jpg)
= Φ( + Δ )
α
= αc c +∑
i∈Ω αiic
![Page 77: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/77.jpg)
![Page 78: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/78.jpg)
![Page 79: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/79.jpg)
![Page 80: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/80.jpg)
dist3D
![Page 81: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/81.jpg)
dist3D
IDX
FFD LC
IDXGT
![Page 82: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/82.jpg)
dist3D IoU dist3D IoU
IDX FFD LC
IDX + LCGT
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IDX + FFDGT + LCGT IDX + FFD + LCGT IDX
1e−3 1e−5
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dist3D IoU dist3D IoU dist3D IoU dist3D IoU
IDX FFD LCFFD LC
dist3D IoU dist3D IoU dist3D IoU dist3D IoU
IDXFFD LC IDX
IDXGT
dist3D IoU dist3D IoU dist3D IoU dist3D IoU dist3D IoU
2.62 −3 . −2.16 −5 . −3.59 −5 . −
. −2.58 −5 . −
1.34 −3 . −
IDX FFDLC
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FFDLC IDX
FFD LCGT
FFD LCGT
LC FFDGT
1e−3
Δ α
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Δ TΔ
γ
![Page 92: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/92.jpg)
c ∈ Rnv×3
∈ Znf×3, 0 ≤ Fij < nv i = [p, q, r]
p q r c = { , }c
c
λc
A B
λc(A,B) =∑a∈A b∈B
‖a− b‖2 +∑b∈B
a∈A‖b− a‖2.
λem
λem =φ:A→B
∑a∈A
‖a− φ(a)‖,
φ
IoU
![Page 93: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/93.jpg)
(A,B) =|A ∩B||A ∪B| .
,
l,m n l+1,m+1, n+1 , ,
(s, t, u) =l∑
i=0
m∑j=0
n∑k=0
Bil(s)Bjm(t)Bkn(u) ijk,
B·N (x)
N
ijk i, j, k
= ,
∈ RM×3 ∈ R
N×3 M
N ∈ RN×M
Δ (t)
T c(t) 0 ≤ t < T
(t) = (t) (t)
Δ˜ (qt)
![Page 94: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/94.jpg)
˜(qt) = (t)(
(t) +Δ˜ (qt)).
γ(qt)
λ0 =∑q, t
f(γ(qt)
)λc
(s(q), s(qt)
),
f
γ(qt) = (1− εγ) γ(qt)0 +
1
Tεγ ,
γ(qt)0
t εγ 0 < εγ � 1
t∗ =t
γ(qt),
c(q) = {˜(qt∗), (t∗)}.
Δ˜
![Page 95: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/95.jpg)
f
f(γ) = γ)
f
f(γ) = − (1− γ)
λe =∑t
γ(t)(γ(t)
),
γ(t) t
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λ0
λ′e = λ0 + κeλe.
κe = e−b/b0κe0,
κe0 b b0
λr =∑q,t
γ(qt)|Δ˜ (qt)|2,
| · |2
λ′r = λ0 + κrλr,
κr κr0
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εe
α = 0.25
192 × 192
1 × 1
512
4 l = m = n = 3
16, 384
1, 024
192×256×3
30◦ 45◦
![Page 98: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/98.jpg)
192× 256× 36× 8× 256
1× 1 6× 8× 643, 072512
(t) 192 + 1
3 × 43 = 192 Δ˜ (qt)
γ(qt)
10−3 β1 = 0.9 β2 = 0.999 ε = 10−8
100, 000 b0 = 10, 000
εe = 0.02
323
εγ f(γ) κe0 κr0γ
− (1− γ)γγ
![Page 99: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/99.jpg)
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z ≥ 0 3.2
323
5
13
1000× (λc λem 1− )
λc λem
5
13
λc λem
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λIoU = 0.33
5
13
1000 × λc
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Template0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Norm
alizedfrequency
b
w
e
r
Template0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Norm
alizedfrequency
b
w
e
r
10−1 100
X
0.0
0.2
0.4
0.6
0.8
1.0
λc<X
b
w
e
r
10−2 10−1 100
X
0.0
0.2
0.4
0.6
0.8
1.0
λc<X
b
w
e
r
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10−1
100
X
0.0
0.2
0.4
0.6
0.8
1.0
λc<X
T=1
T=2
T=4
T=8
T=16
T=30
0.0 0.2 0.4 0.6 0.8 1.0
X
0.0
0.2
0.4
0.6
0.8
1.0
IoU>X
T=1
T=2
T=4
T=8
T=16
T=30
T > 1T = 1
T = 1
T ∈ {2, 4, 8, 16}N
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±
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Γ(t)
φ(x, t) : Ω×R �→ R
![Page 110: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/110.jpg)
φ
Γ(t) = {x : φ(x, t) = 0} ,
φ(x, t)
Γ
φ
Γ
φ
∂Γ
∂t= vn,
Γ v
n
Γ
∂Γ
∂t= −∂E(Γ)
∂Γ= vn.
φ
Γ
![Page 111: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/111.jpg)
∀tφ(Γ(t), t) = 0.
∂φ(Γ(t), t)
∂t= 0,
∂φ(Γ(t), t)
∂t=
∂φ
∂Γ
∂Γ
∂t+
∂φ
∂t= ∇φ
∂Γ
∂t+
∂φ
∂t,
∇φ = ∂φ∂Γ
∇φ‖∇φ‖
∂φ
∂t= −∇(φ)
∂Γ
∂t= −∇(φ)vn = −∇(φ)v(−1)
∇φ
‖∇φ‖ = v‖∇φ‖.
Γ v
φ v‖∇φ‖
φ
∂φ
∂t= v‖∇φ‖.
φ v
![Page 112: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/112.jpg)
Γ
Γ S ⊂R3
xi ∈ Sni xi Γ S
ni
ni ∼ ±ni S m
X = {xi}mi=1 N = {ni}mi=1 dX (x)
X
d(x,X ) =y∈X
‖x− y‖2.
EX (Γ) =
(∫Γd(s,X )pds
)1/p
, 1 ≤ p ≤ ∞,
Γ ds
Lp Γ
XN
Γ
SLp Γ
N
EN (Γ) =
(∫Γ(1− |N(s) · nΓ(s)|)p ds
)1/p
, 1 ≤ p ≤ ∞,
![Page 113: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/113.jpg)
N(s) = ni xi s
Γ
nΓ(s) =∇φ(s)
‖∇φ(s)‖ ,
EN (Γ)
EN (Γ) =
(∫Γ
(1−
∣∣∣∣N(s) · ∇φ(s)
‖∇φ(s)‖
∣∣∣∣)p
ds
)1/p
.
Γ
Earea =
∫Γds,
Evol =
∫Γds,
Γ ds
Γ
![Page 114: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/114.jpg)
f(x) = ±y∈Γ
‖x− y‖,
f(x) > 0 x Γ
|∇f | = 1
Γ
Esdf (φ) =
∫(‖∇φ(x)‖ − 1)2dx.
n Ij
Sj = {X j ,N j}θ
I φ(I; θ)
θ
![Page 115: Representation and Reconstruction of 3D Shapes in Computer ... Pontes_Thesis.… · 1dmodpdlq@qhþl@lc1dbþlpqorbqhþl þe #2g@mdphl"þkmrqdo5hphþl c?qmve 2b2kq/2hsqmi2b "x1x jxa+x](https://reader033.fdocuments.net/reader033/viewer/2022060608/605fe06398443459de08ad0d/html5/thumbnails/115.jpg)
L(θ) =∑j∈D
EX j (Γ(Ij ; θ)) + α1
∑j∈D
EN j (Γ(Ij ; θ))
+ α2
∑j∈D
Esdf (φ(Ij ; θ)) + α3
∑j∈D
Earea(Γ(Ij ; θ))
+ α4
∑j∈D
Evol(Γ(Ij ; θ)).
Γ φ
I Γ(I; θ) = {x : φ(I; θ) = 0} D = {1, ..., n} α1 − α4
δ H
∑j∈D
EX j (Γ(Ij ; θ)) =
=∑j∈D
(∫R3
δ(φ(x, Ij ; θ))d(x,X j)pdx
)1/p
,
∑j∈D
EN j (Γ(Ij ; θ)) =∑j∈D
(∫R3
δ(φ(x, Ij ; θ))
(1−
∣∣∣N j(x) · ∇φ(x; Ij , θ)
‖∇φ(x, Ij ; θ)‖
∣∣∣)pdx
)1/p
,
∑j∈D
Esdf (φ(Ij ; θ)) =
∑j∈D
∫R3
(‖∇φ(x, Ij ; θ)‖ − 1)2dx,
∑j∈D
Earea(Γ(θ, Ij)) =
∑j∈D
∫R3
δ(φ(x, Ij ; θ)) dx,
∑j∈D
Evol(Γ(θ, Ij)) =
∑j∈D
∫R3
H(φ(x, Ij ; θ)) dx.
Ω
C1 C2
δ H
δε(x) =
{12ε
(1 + (πxε )
), |x| ≤ ε,
0, |x| > ε,
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Hε(x) =
⎧⎪⎨⎪⎩
12
(1 + x
ε +1π (πxε )
), |x| ≤ ε,
1, x > ε,
0, x > −ε,
H ′ε(x) = δε(x)
Lε φ
φj(x) = φ(x, Ij ; θ)) dj(x)p = d(x,X j)p
Lε(θ) =∑j∈D
(∑x∈Ω
δε(φj(x))dj(x)p
)1/p
+ α1
∑j∈D
(∑x∈Ω
δε(φj(x))
(1−
∣∣∣N j(x) · ∇φj(x)
‖∇φj(x)‖
∣∣∣)p)1/p
+ α2
∑j∈D
∑x∈Ω
(‖∇φj(x)‖ − 1)2 + α3
∑j∈D
∑x∈Ω
δε(φj(x))
+ α4
∑j∈D
∑x∈Ω
Hε(φj(x)).
Lε φ Ω
∂Lε
∂φ=
∑j∈D
1
p
(∑x∈Ω
δε(φj(x))dj(x)p
) 1−ppδ′ε(φ
j(x))dj(x)p
+α1
p
∑j∈D
(∑x∈Ω
δε(φj(x))
(1−
∣∣∣N j(x) · ∇φj(x)
‖∇φj(x)‖
∣∣∣)p) 1−p
p
(δ′ε(φ
j(x))(1−
∣∣∣N j(x) · ∇φj(x)
‖∇φj(x)‖
∣∣∣)p+
δε(φj(x))
∂
∂φ
(1−
∣∣∣N j(x) · ∇φj(x)
‖∇φj(x)‖
∣∣∣)p)
+ α2
∑j∈D
∑x∈Ω
(‖∇φj(x)‖ − 1) ∇ ·(
∇φj(x)
||∇φj(x)||
)
+ α3
∑j∈D
δ′ε(φj(x)) + α4
∑j∈D
δε(φj(x)).
Lε∂Lε
∂φφ
Γ φ
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500 2000
20
80/20
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p = 2 ε = 0.15 α1 = 0.8 α2 = 1
α3 = α4 = 0.1
E(p) = − 1
N
N∑n=1
[pn pn + (1− pn) (1− pn)],
p N
p
10−6
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A B
(A,B) =|A ∩B||A ∪B| .
0 1
P1 P2
d (P1,P2) =1
|P1|∑x∈P1
y∈P2
||x− y||+ 1
|P2|∑y∈P2
x∈P1
||y − x||.
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203 303
φ Δ φ Δ
203 303
Δ
203 303
φ Δ φ Δ
203
303 Δ
Δ
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O(NM) N M
203 303
2563
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203 203 303 303
203 203
303 303
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203 203 303 303
203 203
303 303
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203 203 303 303
203 203
303 303
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203 203 303 303
203 203
303 303
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203 203 303 303
203 203
303 303
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