Mouvement et Vidéo - Centre de Morphologie Mathématique

Htotal nspl
Htotal (λ/λ0, nreg)
Htotal nreg
λ/λ0 = 0.25
Htotal nreg
λ/λ0 = 0.125
Herr Hv
Htotal Hv
Htotal ∆v
φ00
xij φ00

R(ϑ, ρ) ρ = 0.96 ρ = 0.97 ρ = 0.98
P3,0
P3,1

R(ϑ, ρ) ρ = 0.96 ρ = 0.97 ρ = 0.98
P3,0
P3,1

R(ϑ, ρ) ρ = 0.96 ρ = 0.97 ρ = 0.98
P3,0
P3,1

∗ 0
r∗ 0
r∗ 0
r∗ 0
r∗ 0
r∗ 0
= 6.7
= 6.7
= 6.7
= 8.0
= 8.0
= 8.0
(αl)

∗ 0
r∗ 0
r∗ 0
r∗ 0
r∗ 0
r∗ 0
= 6.7
= 6.7
= 6.7
= 8.0
= 8.0
= 8.0
(αl)

x x
v v
M M
s s
Id
diag(λ1, λ2, . . . , λn) λ1, λ2, . . . , λn
t v v
t M M
vp M M
tr M M
F
W
Hf f
∇f f
˜z z
1E
E
card A A
δ
f f
f ⋆ g f g
w w
med(X) X
E X X
arrondi(t) t
arrondi(x) x
H(X) X
Hd(X) X
I(X; I) X Y

R
Z
N
ℓ2(Z)
L2(R)

R
Z
N
ℓ2(Z)
L2(R)

R
Z
N
ℓ2(Z)
L2(R)

R
Z
N
ℓ2(Z)
L2(R)

I(t, x)
t p
d
∂I
I(t, x(t)) = + v · ∇I,
dt ∂t
v(x) x(t)
∂I
∂t + v · ∇I = 0

I(t, x)
t p
d
∂I
I(t, x(t)) = + v · ∇I,
dt ∂t
v(x) x(t)
∂I
∂t + v · ∇I = 0

I(t, x)
t p
d
∂I
I(t, x(t)) = + v · ∇I,
dt ∂t
v(x) x(t)
∂I
∂t + v · ∇I = 0

I(t, x)
t p
d
∂I
I(t, x(t)) = + v · ∇I,
dt ∂t
v(x) x(t)
∂I
∂t + v · ∇I = 0

I(t, x)
t p
d
∂I
I(t, x(t)) = + v · ∇I,
dt ∂t
v(x) x(t)
∂I
∂t + v · ∇I = 0

f
f(t) =
R
f(u)δ(t − u)du.
f(t) =
R
f(ω)e iωt du.
f(t) f t
f(ω) f ω f(t)
f f(ω)
t ↦→ δ(t − u)
t ↦→ e iωt

f
L2 1
|f(t)| 2 dt = 1.
c(f) ∆(f)
c(f) = t|f(t)| 2 dt,
∆(f) =
(t − c(f)) |f(t)| 2 dt.
∆(()f)
f L2 1
∆(f)∆( f) ≥ 1
.
2
∆( f) f
Gt0,ω0,∆t(t) = Ae − (t−t0 )2
2δt2 e i∆0t
,
A Gt0,ω0,∆t
(t, ω) (c(f), c( f)) (∆(f), ∆( f))
f L2(R)

ω
t
ω
t0 ω0 ∆t
δt t0
ω0
gt0,ω0 (t) = g0(t − t0)e iω0t
t2
− g0(t) = A0e 2∆t2
ω0∆t c0
gt0,∆t(t) = 1
√ ∆t g0
t − t0
∆t
t2
− g0 = A0e 2∆t2 eic0t
∆ω
ω0
ω
t
ω
t
t

ψ
|
Cψ =
R
ψ(ω)| 2
dω < +∞.
ω
Wf f
f(t) = 1
Wf(t, s) =
Cψ
R 2
R
f(τ) 1
τ − t
√ ψ dτ.
s s
Wf(τ, σ) 1
√ σ ψ
t − τ
σ
dτ dσ
.
σ2
f L2(R) = 1
Cψ
R 2
|Wf(t, s)| 2 dt ds
.
s2 Wf
f
gt0,∆t(t) = 1
√ ∆t g
t − t0
∆t
∆t
∆t
∆t = b j
t0 = k∆t
2 −j
g0(2 j (t−2 −j k)) = g0(2 j t−k) j
k L2
(ψjk)j,k∈Z ψjk(t) = 2 j/2 ψ(2 j t−k)

ψ
m0 m1
k ↦→ m0[k] k ∈ Z
k ↦→ m1[k] k ∈ Z
ω ↦→ m0(ω) ω ↦→ m1(ω) 2π
φ ψ L2(R)
+∞
φ(ω) = m0(
k=1
ω
)
2k ψ(ω) = m1( ω
2 ) φ( ω
2
)
(ψjk)
L2(R) (Vj)j∈Z
j∈Z
Vj =
akφjk : ak ∈ R
k∈Z
Vj ⊂ Vj+1
Vj = {0}
Vj = L2(R).
j∈Z

Vj
(φjk)k∈Z
φjk
φ φjk
L2
ℓ2(Z) → L2(R)
(ak)k∈Z ↦→
k∈Z
akφ0k
φ
(φjk)k∈Z Vj+1
Vj
φ ∈ V1
φ(t) = 2
m0[k]φ(2t − k).
k∈Z
m0
L2(R)
φ φ 0
Vj
Vj+1 W0
V0 ⊕ W0 = V1.
W0 ψ
W0 =
t ↦→
dkψ(t − k) : dk ∈ Z
k∈Z
ψ V1
.
ψ(t) =
m1[k]φ(2t − k),
k∈Z
m1

j
Vj ⊕ Wj = Vj+1,
Vj ⊕ Wj ⊕ · · · ⊕ Wj ′ −1 = Wj ′ j < j′ .
j ′ +∞ j −∞
L2(R) = Vj ⊕
L2(R) =
+∞
j=−∞
+∞
j ′ =j
Wj
Wj ′ ∀j ∈ Z
Bj = {φjk : k ∈ Z} ∪ {ψj ′ k : j ′ ≥ j, k ∈ Z}
B = {ψjk : j ∈ Z, k ∈ Z}
f = 2
f[k/2 j ]
j/2
f[k/2 j ]φjk
k∈Z
f[k/2 j ] f(k/2 j ).
f Vj
L < j.
VL ⊕ WL ⊕ · · · ⊕ Wj−1
Vj ′ Vj ′ −1 ⊕ Wj ′ −1
Vj−1 ⊕ Wj−1
Vj−2 ⊕ Wj−2 ⊕ Wj−1
VL ⊕ WL ⊕ WL+1 ⊕ · · · ⊕ Wj−1.

Vj+1 → Vj ⊕ Wj.
ℓ2(Z) → ℓ2(Z) × ℓ2(Z)
(aj+1,k)k∈Z ↦→ [(ajk)k∈Z, (djk)k∈Z].
Aj Dj 2π
k ↦→ ajk k ↦→ djk
Aj(ω) =
k∈Z
Dj(ω) =
k∈Z
ajke −ikω
djke −ikω
Aj(2ω)
Dj(2ω)
=
m0(ω) m0(ω + π)
m1(ω) m1(ω + π)
Aj+1(ω)
Aj+1(ω + π)
T (ω) =
m0(ω) m0(ω + π)
m1(ω) m1(ω + π)
[0, 2π] [0, 2π]
˜ T (ω) = T (ω) −1 2π
˜m0 ˜ m1 ˜ T (ω)
˜T (ω) =
˜m0(ω) ˜m0(ω + π)
˜m1(ω) ˜m1(ω + π)
˜ φ ˜ ψ
˜φ(ω) =
+∞
k=1
˜ψ(ω) = ˜m1
˜m0
ω
2
ω
2 k
˜ φ
ω
2
˜ φ ˜ ψ j
L2(R)
f =
k∈Z
f, ˜
φjk φjk +
j ′ ≥j,k∈Z
f, ˜ ψj ′
k ψj ′ k

j ∈ Z f ∈ ℓ2(Z) j −∞
f =
j,k∈Z
f, ˜
ψjk ψjk
m0 m1 ˜m0 ˜m1
{φjk : k ∈ Z} ∪ {ψjk : k ∈ Z} ↔ {φj+1,k : k ∈ Z}
ajk = 2
l∈Z
djk = 2
l∈Z
˜m0[k]aj+1,2l−k
˜m1[k]aj+1,2l−k
aj+1,k = 1
2
m0[2l − k]ajl + m1[2l − k]djl
l∈Z
j = 0 j = −3
N
A × N
A
N log N
ψ t ↦→ 2j/2ψ(2jt− k)
j,k∈Z L2(R) φ = ˜ φ ψ = ˜ ψ
ω
m0 m1
|m0(ω)| 2 + |m0(ω + π)| 2 = 1 ∀ω
m0(ω)m1(ω) + m0(ω + π)m1(ω + π) = 0 ∀ω
|m1(ω)| 2 + |m1(ω + π)| 2 = 1 ∀ω

a−3,k
d−3,k
a0,k
1 ↑ 2
1 ↑ 2
˜m0
˜m1
m0
m1
2 ↓ 1
2 ↓ 1
+
d−2,k
˜m0
˜m1
2 ↓ 1
2 ↓ 1
˜m0
˜m1
2 ↓ 1
2 ↓ 1
a−3,k
d−1,k d−2,k d−3,k
1 ↑ 2
1 ↑ 2
m0
m1
+
d−1,k
1 ↑ 2
1 ↑ 2
m0
m1
+
a0,k

m0 m1
m0 m1
m1(ω) = e iω m0(ω + π)
L2
j → +∞
˜ ψ p
˜ψ(t)t k dt = 0 ∀k ∈ {0, . . . , p − 1}.
R
˜ ψ
p ω = 0 ψ
p
f p
p I 2 −j(p+1/2)

I M
|〈 ˜ ψjk, f〉| ≤ M2 −j(p+1/2) supp ˜ ψjk ⊂ I.
|〈 ˜ ψjk, f〉| ≤ M2 −j(p+1/2) ψ p
f r r < p
x = (xk)k∈Z
(x2k)k∈Z (x2k+1)k∈Z xe
xo
h
ℓ
xe
xo xo
P
h = xo − P (xe)
xe h xo
xo = d + P (xe)
P xo,k hk h
xo P
(xe, d) (xe, xo)

xe
h
ℓ = xe + U(h)
xe = ℓ + U(h)
(x)
(x2k)
(x2k+1)
P
−
+ (ℓk)
U
(hk)
(ℓk)
(hk)
−
U
P
+
(x2k)
(x2k+1)
T = GL1U1 · · · LnUn,
(Li)1≤i≤n (Ui)1≤i≤n
G
(xk)

xo,k
xe,k
hk = x2k+1 − x2k
ℓk = x2k + 1
2 hk
hk = x2k+1 − x2k
ℓk = 1
2 (x2k + x2k+1)
d
hk = x2k+1 − 1
2 (x2k + x 2(k+1))
ℓk = x2k + 1
4 (hk + hk+1)
hk = x2k+1 − 1
2 (x2k + x 2(k+1))
ℓk = 3
4 x2k + 1
4 (x2k−1 + x2k+1) − 1
8 (x2k−2 + x2k+2)

yk = x2k+1 − α(x2k + x 2(k+1)) α −1.58613
zk = x2k + β(x 2(k−1)+1 + x2k+1) β −0.05298
h ′ k = yk − γ(zk + zk+1) γ 0.88291
ℓ ′ k = zk + δ(yk−1 + yk) δ 0.44351
hk = ηh ′ k
ℓk = 1
η ℓ′ k
η 1.14960
(xk)
(x2k)
(x2k+1)
P U P U
−
+
−
+ (ℓk)
y z
(hk)

hk = x2k+1 − ⌊ 1
2 (x2k + x 2(k+1))⌋
ℓk = x2k + ⌊ 1
4 (hk + hk+1)⌋

x2k+1
x2k+1
P (x2k+1) = x2k + x2k
,
2
x2k+1
P (x2k+1) = x2k + x2k − x2(k−1) .
2
Gi
i ≤ j ⇒ Gi ⊂ Gj
Gi Gi+1 \ Gi
φ ψ
φℓℓ = φ ⊗ φ
ψℓh = φ ⊗ ψ
ψhℓ = ψ ⊗ φ
ψhh = ψ ⊗ ψ.
φℓℓ ψℓh ψhℓ ψhh

ma mb
(f ∗ ma) ∗ mb = (f ∗ mb) ∗ ma = f ∗ (ma ∗ mb).
d ψhℓ d ψℓh d ψhh
P U
xe = (xi,j)i,j∈Z,i+j≡0 mod 2
xo = (xi,j)i,j∈Z,i+j≡1 mod 2,
j −2
j − 1

j = 0 j = −1 j = −2 j = −3 j = −4
j
s
j = 0
j = −1
s
√ 2
s
2
j = −2
j = −3
s
2 √ 2
s
4
j = −4
j
2
hi,j = xi,j − med(xi−1,j, xi+1,j, xi,j−1, xi,j+1) i + j ≡ 1 mod 2
ℓi,j = xi,j + 1
2 med(hi−1,j, hi+1,j, hi,j−1, hi,j+1) i + j ≡ 0 mod 2
hi,j = xi,j − max(xi−1,j, xi+1,j, xi,j−1, xi,j+1) i + j ≡ 1 mod 2
ℓi,j = xi,j + max(hi−1,j, hi+1,j, hi,j−1, hi,j+1) i + j ≡ 0 mod 2

s t ⇐⇒ 0 ≤ s ≤ t t ≤ s ≤ 0 ∀s, t ∈ R,
K
K = med(0, min K, max K).
hi,j = xi,j − {xi−1,j, xi+1,j, xi,j−1, xi,j+1} i + j ≡ 1 mod 2
ℓi,j = xi,j + 1
2 {hi−1,j, hi+1,j, hi,j−1, hi,j+1} i + j ≡ 0 mod 2
h = xo − Padap(xe)
ℓ = xe + U(h)
Padap(xe)(n) = xe(k)Fn(n − k) Fn (xe)
n

(x)
(x2k)
(x2k+1)
Padap
−
+ (ℓk)
h xe
˜xe
xe
|Padap( ˜xe) − Padap(xe)| 0.
U
(hk)

(xk)
(x2k)
(x2k+1)
+ (ℓk)
U
Padap
−
(hk)

I(t, x)
t x = (x1, x2)
t x I
X1(t), X2(t), X3(t) (x1(t), x2(t))
v = dx(t)
dt
x(t)
I(t, x(t))

d
∂I
I(t, x(t)) = + v · ∇I
dt ∂t
d
I(t, x(t)) = 0
dt
∂I
+ v · ∇I = 0
∂t
∇I
?
?
?
?
?
?
?
t = 0 t = 1
∇I t = 0 t = 1

Image
échelle s
vδt
Image
Image
échelle s
? Image
?
vδt
|v| ≪ s |v| > s
s
v |v| s
εx0,t = I(t, x0 + v(x0)) − I(t + 1, x0)

G
W0
v0 = argmin
v∈G
W0
(I(t, x + v) − I(t + 1, x)) 2 d 2 x
I Ĩ
( Ĩ(t, x + v) − I(t + 1, x))2 d 2 x
v0 = argmin
v∈G W0
W1
W0
v0 = argmin
v∈G
W1
( Ĩ(t, x + v) − I(t + 1, x))2 d 2 x
W0 ⊂ W1

2 k
2 k−1
M(v) =
∂I
∂t + v · ∇I 2 d 2 x.
R(v) =
v 2 Hd 2 x
·H
v
v = argmin M(v) + λR(v).

λ
F
I(t, x) = I(0, x) ∗ δ(x − vt).
Î(ω, ξ) ∝ Î(0, ξ)δ(tvξ − ω).
t
x
(x, t) (ωx, τ)
x t
(vx, 1)
vx
1
τ
vx
ωx

M(v) =
x∈W (x0)
ρ(x) 2 ∇I · v + ∂I 2
∂t
W (x0)
ρ
N W (x0) t
t AR 2 Av = t AR 2 b
A = t ∇I(x1), . . . , ∇I(xN)
R = diag(ρ(x1), . . . , ρ(xN))
b = − t ( ∂I
∂t (x1), . . . , ∂I
∂t (xN))
t AR 2 A 2 × 2
t AR 2 A
v j Br r
2 j Z 2

vj−1
r
B r 2 2
v {j,j−1} = v j + v j−1
{v1 } + B r
2
2j−1Z2
(ψ n )1≤n≤N
x0
∇I · v + ∂I
∂t
∂I
∂x1
v1, ψ n x0
ψ n k (x − x0)d 2 x = 0 ∀n = 1 . . . N
〈f, g〉 =
f(x)g(x)d 2 x
ψ n x0 (x) = ψn (x − x0),
∂I
+ v2, ψ
∂x2
n
∂I
x0 +
∂t , ψn
x0 = 0 ∀n = 1 . . . N
I, ∂ψn x0
∂x1
v1(x0) +
I, ∂ψn x0
∂x2
v2(x0) = ∂
I, ψ
∂t
n
x0
∀n = 1 . . . N
|Av − b| A

vj−1
r
B r 2 2
v {j,j−1} = v j + v j−1
{v1 } + B r
2
2j−1Z2
(ψ n )1≤n≤N
x0
∇I · v + ∂I
∂t
∂I
∂x1
v1, ψ n x0
ψ n k (x − x0)d 2 x = 0 ∀n = 1 . . . N
〈f, g〉 =
f(x)g(x)d 2 x
ψ n x0 (x) = ψn (x − x0),
∂I
+ v2, ψ
∂x2
n
∂I
x0 +
∂t , ψn
x0 = 0 ∀n = 1 . . . N
I, ∂ψn x0
∂x1
v1(x0) +
I, ∂ψn x0
∂x2
v2(x0) = ∂
I, ψ
∂t
n
x0
∀n = 1 . . . N
|Av − b| A

vj−1
r
B r 2 2
v {j,j−1} = v j + v j−1
{v1 } + B r
2
2j−1Z2
(ψ n )1≤n≤N
x0
∇I · v + ∂I
∂t
∂I
∂x1
v1, ψ n x0
ψ n k (x − x0)d 2 x = 0 ∀n = 1 . . . N
〈f, g〉 =
f(x)g(x)d 2 x
ψ n x0 (x) = ψn (x − x0),
∂I
+ v2, ψ
∂x2
n
∂I
x0 +
∂t , ψn
x0 = 0 ∀n = 1 . . . N
I, ∂ψn x0
∂x1
v1(x0) +
I, ∂ψn x0
∂x2
v2(x0) = ∂
I, ψ
∂t
n
x0
∀n = 1 . . . N
|Av − b| A

vj−1
r
B r 2 2
v {j,j−1} = v j + v j−1
{v1 } + B r
2
2j−1Z2
(ψ n )1≤n≤N
x0
∇I · v + ∂I
∂t
∂I
∂x1
v1, ψ n x0
ψ n k (x − x0)d 2 x = 0 ∀n = 1 . . . N
〈f, g〉 =
f(x)g(x)d 2 x
ψ n x0 (x) = ψn (x − x0),
∂I
+ v2, ψ
∂x2
n
∂I
x0 +
∂t , ψn
x0 = 0 ∀n = 1 . . . N
I, ∂ψn x0
∂x1
v1(x0) +
I, ∂ψn x0
∂x2
v2(x0) = ∂
I, ψ
∂t
n
x0
∀n = 1 . . . N
|Av − b| A

N (F [k])0≤k

¯ J
[yn, yn+1[
n
[yn, yn+1[
¯ F [k] F [k] xn ∈ [yn, yn+1[
F [k] ∈ [yn, yn+1[ ⇒ ¯ F [k] = xn.
∆
T θ = T/∆
θ = 2
n
F [k]
F [k] ¯ F [k]
I(x, y)
Ī(x, y) mathcalI
(I, Ī) = 10 log 10
(x,y)∈I
(x,y)∈I I(x, y)2
(Ī(x, y) − I(x, y))2 .
Vmax
(I, Ī) = 10 log 10
(x,y)∈I
V 2 max card I
.
(Ī(x, y) − I(x, y))2

X
p X
H(X)
H(X) = −
p(x)log2p(x)
x∈X
p

ε Eε
Eε = {v |Ax − b| < ε}

ε Eε
Eε = {v |Ax − b| < ε}

ε Eε
Eε = {v |Ax − b| < ε}

E N
(ei)1≤i≤N
I = I1 × I2 × · · · × IN,
Ii R
(Gr )rmin≤r≤rmax
G r = 2 r Z ∩ I.
p Gr Ar p m × n
br p n vr p m Ar br vr Gr
(A r v r )p = A r pv r p.
ar
r
a = a r p.
p∈G r
A r pv r p = b r p.
A r = F(v r+1 , D)
b r = G(v r+1 , D)
D It It+1
v rmax+1 = 0.
v rmin
v r rmax rmin
p r
argmin v r p |A r pv r p − b r p| 2
t A r pA r pv r p − t A r pb r p = 0,

t A r pA r p
A r
A r (v r ) = |A r v r − b r | 2 .
argmin
vr A r (v r ) + λB r (v r )
Br
B r (v r
) = |∇v r i | 2 (v r i )p = v r p,i
1≤i≤m
λ
A
B
B r M r H r c r
(B r v r )p,i,j = ∂vr q,j
(p) B
∂qi
r ∇
M r = t A r A r
H r = t B r B r H r
c r = t A r b r
A r B r M r H r
(A r ) i′ 1 ...i′ N ,j′
i1...iN ,j = 0 ∃ k ik = i ′ k
(B r ) i′ 1 ...i′ N ,j′
i1...iN ,j,l = 0 j = j′
(M r ) i′ 1 ...i′ N ,j′
i1...iN ,j = 0 ∃ k ik = i ′ k
(H r ) i′ 1 ...i′ N ,j′
i1...iN ,j
= 0 j = j′

argmin
v r
|A r v r − b r | 2 + λ tv r H r v r ,
argmin
vr tvr r r r r t r r
((M + λH )v − 2c ) + b b .
λ
λ
λ
tr M r
λ0 = .
tr Hr λ0
(H r ) i′ 1 ...i′ N ,j
i1...iN ,j = (Hr ) (i′ 1 −i1)...(i ′ N −iN ),1
0...0,1 ,
λ0 =
=
tr M r
i1,...iN ,j (Hr ) i1...iN ,j
i1...iN ,j
tr M r
mN card Gr(H r ) 0...0,1
0...0,1
λ0
λ/λ0
∇(A + λB) = 2[(M r + λH r )v r − c r ]
(v r,0
p )
A

nreg
v r,i+1 = (1 − ɛ(M r + λH r ))v r,i + ɛc r
ɛ
0 < ɛ <
2
max vp(M r + λHr .
)
H r κ = 4N
0 < ɛ <
2
max vp(M r ) + λκ
ɛ
λ/λ0 nreg
M r p
∂vl,k
∂x = vl+1,k − vl,k.
H r
∂ 2 vl,k
∂x 2 = vl+1,k − 2vl,k + vl−1,k
l,k (M r l,k,xx + M r l,k,yy )
8N r xN r y
2
ɛ <
max vp M r + 8λ
λ0 =

Nx Ny
v r,i+1
l,k,x
v r,i+1
l,k,y
= vr,i
l,k,x − ɛ(M r l,k,xx vr,i
l,k,x + M r l,k,xy vr,i
l,k,y )
− ɛλ(v r,i
l−1,k,x − 2vr,i
l,k,x + vr,i
l+1,k,x )
+ ɛc r l,k,x
= vr,i
l,k,y − ɛ(M r l,k,yx vr,i
l,k,x + M r l,k,yy vr,i
l,k,y )
− ɛλ(v r,i
l,k−1,y − 2vr,i
l,k,y + vr,i
l,k+1,y )
+ ɛc r l,k,y
Hv
Herr
S
Hv Herr
Htotal
Htotal = 0.25Hv + Herr.
λ/λ0 nreg
v It It−1
v ˜v
∆v ˜v Q1(˜v)
Hv
Qv(˜v) ¯˜v
¯˜v ¯v
I pred
t
It−1 ¯v
Et I pred
t It
Et ˜ Et
∆err ˜ Et Qerr( ˜ Et)
Herr
Qerr( ˜ Et) ¯˜ Et

¯˜ Et Ēt
Īt I pred
t
Īt It
Entrées
It−1 ME v CD1 ¯v
It
−
Et
Īt
I pred
t
+
MC
CD2
Ēt
Ēt
Ev
Eerr
PSNR
Qi ∆i
× ×
ɛ
nspl = 3
nspl = 1
Sorties

46
45
44
43
42
A
B
A
B
0.10 0.15 0.20 0.25
A
B
A
40
39
38
37
36
A
B
A
0.2 0.3 0.4 0.5 0.6
B
A
= f(Htotal) λ/λ0 = 0.25 ∆v = 1 nreg = 1000
∆err nspl = 1
nspl = 3
A

λ/λ0
λ/λ0
λ/λ0
47
46
45
44
43
42
A
D
C
B
A
D
C
0.05 0.10 0.15 0.20
B
A
D
C
B
A
D
38
36
34
32
30
A
D
C
B
A
D
C
B
0.6 0.8 1.0 1.2 1.4 1.6 1.8
= f(Htotal) ∆v = 4 nspl = 1 ∆err
nreg = 100 λ/λ0 = 0.25
nreg = 100 λ/λ0 = 0.125 nreg = 1000 λ/λ0 = 0.25
nreg = 1000 λ/λ0 = 0.125
λ/λ0 = 0.25 λ/λ0 = 0.125
A
D
C
B
A

47
46
45
44
43
42
A
C
B
A
C
0.05 0.10 0.15 0.20
B
A
C
B
A
38
36
34
32
30
A
C
B
A
C
0.4 0.6 0.8 1.0 1.2 1.4 1.6
B
A
= f(Htotal) λ/λ0 = 0.25 nspl = 1 ∆v = 4
∆err nreg = 0
nreg = 100 nreg = 1000
38
36
34
32
A
C
B
A
C
B
0.4 0.6 0.8 1.0 1.2
A
C
B
A
38
36
34
32
30
A
C
B
A
C
B
0.6 0.8 1.0 1.2 1.4 1.6 1.8
= f(Htotal) λ/λ0 = 0.125 nspl = 1 ∆v = 4
∆err nreg = 0
nreg = 100 nreg = 1000
A
C
C
B
B
A
A

Herr Hv
Htotal
∆v = 2
1.0
0.8
0.6
0.4
A
C
B
C
A
B
A
0.05 0.10 0.15 0.20
C
B
A
0.58
0.56
0.54
0.52
0.50
0.48
0.46
A
C
B
C
A
0.05 0.10 0.15
B
Herr = f(Hv) λ/λ0 = 0.25 nspl = 3 ∆err = 8
∆v nreg = 0
nreg = 100 nreg = 1000
nreg λ/λ0
C
A

1.2
1.1
1.0
0.9
0.8
0.7
0.6
A
C
B
C
A
B
A
C
0.05 0.10 0.15 0.20
B
A
0.68
0.66
0.64
0.62
0.60
0.58
0.56
0.54
A
C
B
A
C
0.05 0.10 0.15
Htotal = f(Hv) λ/λ0 = 0.25 nspl = 3 ∆err = 8
∆v nreg = 0
nreg = 100 nreg = 1000
47
46
45
44
43
42
A
A
B
C
A
B
0.05 0.10 0.15 0.20 0.25
C
A
B
C
40
38
36
34
32
A
C
B
A
0.2 0.4 0.6 0.8 1.0
C
B
= f(Htotal) λ/λ0 = 0.25 nreg = 1000 nspl = 1
∆err ∆v = 4
∆v = 2 ∆v = 1
B
A
A
A

λ/λ0 = 0.125

λ/λ0 = 0.125

λ/λ0 = 0.125

φij xij xij
v(x) =
i,j
vij1
vij2
φij(x)
φij

φ00
xij φ00
[v] opt = argmin
[v]
It+1(x + v(x)) − It(x) A
x
·A ·1 ·2
[v] opt = argmin
[v]
⎛
x
⎝It+1
x +
vij1
φij(x)
vij2
i,j
⎞
− It(x) ⎠
2

v
x
χ 2 ([vijl])
χ 2
[v]
=
⎛
x
⎝It+1
x +
vij1
φij(x)
vij2
ij
⎞
− It(x) ⎠
2
.
χ 2 ([vijl])
χ 2 ([v]) = γ − ∇χ 2 · [v] + 1
2 [v] · Hχ2 · [v]
[v]
[v] = [v] + D −1 (−∇χ 2 ([v] )).
[v] = [v] − · ∇χ 2 ([v] ),
x ′ = x + v(x).
x ′
χ 2 (v) =
It+1(x ′ ) − It(x) .
x

vijl
∂χ 2
∂vijl
= 2
∂ 2 χ 2
∂vijl∂vi ′ j ′ l ′
= 2
x
x
(φijφi ′ j ′)(x)
∂2It+1 φij(x) ∂It+1
(x
∂xl
′
) It+1(x ′
) − It(x)
∂xl∂xl ′
(x ′ )
It+1(x ′
) − It(x)
∂It+1
+
∂xl
∂It+1
∂xl ′
(x ′
)
∂ 2 χ 2
∂vijl∂vi ′ j ′ l ′
= 2
x
αkk
(φijφi ′ j ′)(x)
∂It+1 ∂It+1
βk = ∂χ2
∂vijl
1
′ =
2
∂ 2 χ 2
∂vijl∂vi ′ j ′ l ′
∂xl
,
∂xl ′
(x ′ )
k k ′ ijl i ′ j ′ l ′
αkk ′δvk ′ = βk,
δvk = × βk
1/αkk βk
λ
k
δvk = 1
λαkk
βk λαkkδvk = βk.
αkk
αkk

[α ′ ] [α]
α ′ kk = (1 + λ)αkk
α ′ kk ′ = αkk ′ k = k′ .
′
α kk ′δvk ′ = βk
λ
λ [α ′ ]
λ [α] [α ′ ]
[v]
λ
δ [v] ← (Hχ 2 ([v]) + λId) −1 ∇χ 2 ([v])
χ 2 ([v] + δ [v]) < χ 2 ([v])
λ ← λ/10
[v] ← [v] + δ [v]
λ ← 10λ
χ 2 < ε1 δχ 2 < ε2
t + 1
mij

q
It+1(q) =
mijψi(qx)ψj(qy) =
i,j
∇It+1(q) =
i,j
i,j
mijΨi,j(q)
mij∇Ψi,j(q)
s
E(s) = 0.

s Sp s
Sp s = E(|s|).
AutoCorr(s)(x) = E(
s(t)s(t + x)dt)
AutoCorr(s)(x) = E(
Sp s =
s(ω)e ixω s(ω)dω)
= F −1 E(ω ↦→ |ω| 2 ) (x)
AutoCorr s
f
Sp s ⋆ f = E(|s f| 2 )
= | f| 2 E(|s| 2 )
= Sp f Sp s
f s f
T
δT (x) =
δ(x − kT ).
k∈Z
FδT = 1
T
δ 2π .
T
T
Sp sδT = E(|s ⋆ 1
T
= E(ω ↦→ | 1
T
≤
k∈Z
δ 2π |
T
2 )
k∈Z
s(ω + 2kπ
T )|)
E(ω ↦→ | 1 2kπ
s(ω +
T T )|)

[− π
T
π , T ]
2π
T
T
2T
R+
hT
[− π π
T , T ] s h
Sp(s ⋆ hT ) = Sp s Sp hT .
h2T hT
h2T (ω) = hT (2ω)
h2T
i
Sp(s ⋆ hT ⋆ h2T ) = Sp f Sp hT Sp h2T .
ki(ω) =
=
i
hT (2 j ω)
j=0
i
j=0
h2 j T (ω).
| ki(ω)| 2 ≪ 1 ω ∈ [π/2, π],
i
(1/4, 1/2, 1/4)
Sp(h)(ω) = cos 4 ω
2 .

cos4 ω
2 cos4 ω
[π/4, π/2]
− π
2
− π
2
0
b k0(ω)
π
2
b k0(ω)
0
ω
π
2 ω
− π
2
− π
2
0
b k1(ω)
b k1(ω)
G = (1/4, 1/2, 1/4)
d
u
It
J 0 t = G ⋆ It
K (J i t )1≤i≤K
J i+1
t
= d(G ⋆ J i t ).
i i − 1
0
π
2
ω
π
2 ω

φ(x) = 1 [−1,1](|1 − x1|)1 [−1,1](|1 − x2|),
φij
n × n
φ n ij(x) = φ(i − x1 x2
, j −
n n )
u
φ2 00
J 0 t ← G ⋆ It)
i 1 K
J i t ← d(u(G) ⋆ J i−1
t )
vK+1 ← 0
i K
w i ← φ 2 00 ⋆ u(vi+1 )
v i ← J i t J i t+1 w i
w ← v 0
vopt ← It It+1 w

4 × 4 8 × 8
4 × 4
8 × 8
4 × 4
8×8

40
38
36
34
42
41
40
39
38
37
36
A
A
D
C
A
D
D
C
B
A
C
A
B
D
A
C C
A
B
5 10 15
A
B
D
D
C C
A
A
D
B
A
A
A
B D
5 10 15
B
D
B
A
C
C
D
A
B
D
B
41
40
39
38
37
36
35
42
40
38
36
A
A
D
D
C
C
A
A
AB
D
C
A
A
D
B
C
A
A
B
D
5 10 15
A
B
D
C C
A
A
B
D
A
A
D
B
5 10 15
4 × 4 8 × 8 4 × 4
8 × 8
C
BA
C
D
B
A
D
C

26.5
26.0
25.5
25.0
33.0
32.5
32.0
31.5
31.0
30.5
30.0
A
A
D
C C
D
B
A A
D
B
5 10 15
A
C
B
D
5 10 15
B
A
B
D
C
C
A
A
B
D
B
A
D
A
C
B
C
31.0
30.5
30.0
29.5
29.0
28.5
33.5
33.0
32.5
32.0
31.5
31.0
30.5
A
A
D
B
C
A A A
D
B
5 10 15
AC
B
D D
5 10 15
4 × 4 8 × 8 4 × 4
8 × 8
A
B
C
C
A
D
D
B
B
A
A
C
C
D

30
29
28
27
26
34
33
32
31
30
29
28
A
A
D
D
C
A
C
A
C
B
D
A
C
A
C
B
A
D D
5 10 15
C C
D
A
B
A
D
C
A
D
A
C
A
BD
C C
5 10 15
B
A
D
B
A
33
32
31
30
29
28
34
32
30
28
A
A
D
D
C
C
CB
A
A
C
A
D
D
A
C
A
D
B
5 10 15
C C
D
A
BA
D
C
A
D
C
B
A
D
A
B
C C
5 10 15
4×4 8×8 4×4
8 × 8
D
A
B
C
A

37
36
35
34
40
39
38
37
36
35
34
A
A
D
D
C
B
C
A
B
D
A
C
5 10 15
B
A
B
A
D
A
D
B
D
5 10 15
C
A
B
C
D
A
C
B
A
D
B
C
A
D
39
38
37
36
35
34
40
39
38
37
36
35
A
A
D
D
C
B
C
B
A
A
D
B
5 10 15
A
B
A
D
C
C
A
B
D
A
B
D
5 10 15
4 × 4 8 × 8
4 × 4 8 × 8
C
D
A
B
A
C
D
B

29
28
27
26
25
24
30
28
26
24
A
D
A
C
D
C
B
C C C
D
A
B
A
A
D
5 10 15
B
A
B
C C
B
D
A
5 10 15
A
B
D
A
A
D
B
30
28
26
24
30
28
26
24
A
A
D
D
C
C
B
A
C
A
D A D A
B
5 10 15
B
A
B
A
A
B
C
C C
D
5 10 15
4 × 4 8 × 8 4 × 4
8 × 8
B
D
A
B

40
39
38
37
36
35
34
33
42
41
40
39
38
37
36
C
A
D
A
D
AD A
B
C
C
B
A
D
B
C
A
D
B
5 10 15
B
A
D
A
C
B
A
D
5 10 15
A
C
A
B
C
A
D
A
DB
C
B
C
A C
A
B
D
D
B
41
40
39
38
37
36
35
42
41
40
39
38
37
36
A
D
A
D
C
C
B A
D
C
A
B
D
A
A
C
B
D
5 10 15
B
A
D
A
C
B
A
D
5 10 15
C
A
B
A
A
D
C
A C
B
4 × 4 8 × 8 4 × 4
8 × 8
C
B
D
A
D
B
C

I2t
I2t+1
Dnr I2t
arrondi(v)
x ∈ Dnr ⇐⇒ ∀y ∈ I2t+1, y − arrondi(v(y)) = x

I2t
arrondi(v)
Dnc I2t+1
arrondi(v)
x ∈ Dnc ⇐⇒ ∃y ∈ I2t, ∃x ′ ∈ I2t+1, y = x−arrondi(v(x)) = x ′ −arrondi(v(x ′ )),
|I2t(y) − I2t+1(x ′ )| < |I2t(y) − I2t+1(x ′ )|.
Dnr Dnc
˜ J J
H(x) = 1
√ I2t+1(x) −
2
Ĩ2t(x
− v(x))
L(y) = √ 2I2t(y).
y I2t x I2t+1 y = x−v(x)
y
L(y) = √ 2I2t(y) + ˜ H(y + v(x))
I2t I2t+1
L H

I2t
arrondi(v)
Dnc I2t+1
arrondi(v)
x ∈ Dnc ⇐⇒ ∃y ∈ I2t, ∃x ′ ∈ I2t+1, y = x−arrondi(v(x)) = x ′ −arrondi(v(x ′ )),
|I2t(y) − I2t+1(x ′ )| < |I2t(y) − I2t+1(x ′ )|.
Dnr Dnc
˜ J J
H(x) = 1
√ I2t+1(x) −
2
Ĩ2t(x
− v(x))
L(y) = √ 2I2t(y).
y I2t x I2t+1 y = x−v(x)
y
L(y) = √ 2I2t(y) + ˜ H(y + v(x))
I2t I2t+1
L H

' () * $ &
!"# $%" &
&' ( # %
!" #$" %

) *+ , &" (
"#$% &'$ (
!

) *+ , &" (
"#$% &'$ (
!

) *+ , &" (
"#$% &'$ (
!

F [k] N
B = {gk}0≤k

B ∗ = {g ∗ k }0≤k

]yk, yk+1]
p(x) pk ∆k
∆k = yk+1 − yk,
∀x ∈]yk, yk+1] p(x) = pk
∆k
p(x)
xk
[yk, yk+1]
d = 1
pk∆k
12
∀k ∈ Z yk+1 − yk = ∆.
d = ∆
12 .
X
H(X) = −
+∞
−∞
p(x) log 2 p(x)dx.
X ¯ X
H( ¯ X) ≥ Hd(X) − 1
2 log 2(12d)
Q
¯ X RX
¯ X
RX = H( ¯ H) = Hd(X) − log 2(12d).
d(RX) = 1
12 22Hd(X) 2 −2RX .

FB[k]
∆k = 12d
N
0 ≤ k < N
d( ¯ R) = N Hd
¯ −2
22 2
12 ¯ R
,
¯ Hd

D
H
R
G
F
M
A
AM AM
W
∆d R/F
B
R = D/F
∆d = 0
H = Rprec
W = AM
W > M
D/F

M
AM
W
0
∆b
AM
M
W
0
∆b
AM
W
M
0
Saut de trame
∆b
AM W < AM
W > AM

W < AM
AM − W
W > AM
W/F
W ← max(W + H − D/F, 0)
skip ← 1
W > M
W ← max(W − D/F, 0)
skip ← skip + 1
skip.G/F − 1
W > AM
∆d ← W/F
∆d ← W − AM
B ← D/F − ∆d
R
QP
QPprec
W/F

Npix
Ri i
σi i
∆i i
Nmb
N = NpixNmb
C
K
Ri = Npix(K σ2 i
∆ 2 i
+ C).
¯ Xi
∆i X
σ2
∆ 2 i
< 1
2e
H( ¯ Xi) = e
ln2
σ2 i
∆2 i
.
K e
ln2
K C
i i − 1
αi
αi =
2 R
N (1 − σi) + σi
1
R < 0.5N

i i Nmb
ni i
ni = Nmb + 1 − i.
Li
Li = ri − NpixniC.
Si
Nmb
Si =
αiσi.
σi
Li > 0
∆ ∗ i =
j=i
NpixKiσi
Liαi
∆ ∗ i
C K
ρ R Rz
Rnz
Si,

(a[k], r[k])
( ¯ FB[k])1≤k≤Npix ↦→ ((a[k], r[k]))1≤k≤m
a[k] k
r[k]
m Npix
ρ = 1
N
Nmb mi
r[k],
i=0 k=0
ν = 1
Nmb
N
Qnz
Qnz = 1
N
i=0
mi
Nmb mi
⌈log2 |a[k]|⌉ + 1.
i=0 k=0
Qz
Qz = 1
nN
Nmb mi
⌈log2 |r[k]|⌉ + 1.
i=0 k=0
Qnz Qz ∆
ρ
Qnz ρ
Qnz = ϑ(1 − ρ).
ϑ
ϑ Qz ϑ ρ
Qz = Pρ(ϑ)
Pρ
ρ

R(ρ) = ξ1(ρ)Qnz(ρ) + ξ2(ρ)Qz(ρ) + ξ3(ρ).
ξ1 ξ2 ξ3
Pρ ξ1 ξ2 ξ3
ρ
R ρ R
R ρ
ρ
T
ρ ∆ ∗ θ = T/∆
∆ ∗ ↦→ ρ
ρ
0.9 < ρ < 0.95.
Qnz
ϑ
ϑ Qz
ρ R
ρ ↦→ R
R ↦→ ρ
ρ QP0
QP
ρ
ρ +
ρ −

κ = ρ−
,
ρ +
QP0 − 3 QP0 + 3 κ
κ
Rz
Rnz
ρ Rz
FB[k] s[k]
s[k] =
0 ¯ FB[k] = 0
1 ¯ FB[k] = 0
S ρ ν = 1 − ρ
H(S) = −ρ log 2 ρ + ν log 2 ν.
rz
rz = Rz
νN
S rz
Rz
N ≤ −ρ log 2 ρ + ν log 2 ν.
x ∈ [0, 1[ −x log 2 x ≤ (1 − x) log 2 e
rz = Rz
νN ≤ log 2 e − log 2 ν

ρ
rz
rz
rz
T
∆ θ = T/∆
νN
rnz
rnz = Rnz
νN
νN ≫ 1 νN
pT (x) = 1
ν p(x)1 {|x|>T }(x).
XT pT
rnz = HdXT − log 2 ∆
∼
f ∼ g ⇐⇒ g = O(f) f = O(g)
νN > 1
ɛ ν < ɛ ɛ
F r B [k] FB[k]
k < k ′ ⇒ F r B[k] ≥ F r B[k ′ ]

|F r B[k]| ∼ k −γ(k) ,
γ k
N
rnz γ(ν) θ γ(ν)
rnz
γ γ > 0.5
γ = 1
1/2 < γ ≤ 1 γ
T
νN ∼ T −γ .
T ∆
νN ∼ ∆ −γ .
Rnz = Ra + Rs Ra
Rs
0 ≤ Rs ≤ νN.
pj
(j + 1/2)∆ + T
lj = − log 2 pj
Ha = −νN
+∞
j=0
pj log 2 pj.
nj = νNpj j |F r B [k]|
|F r B
[k]| ∈ [(j + θ)∆, (j + 1 + θ)∆[
1
1
− −
nj ∼ ((j + θ)∆) γ − ((j + θ + 1)∆) γ .
pj = nj
νN
1
1
−
∼ (j + θ)− γ − (j + θ + 1) γ .
+∞
j=0 pj log 2 pj νN
Ha ∼ νN

s
s
s = 1/2 Ra νN
Rnz = Ra + Rs ∼ νN,
rnz
Rz
Rnz
rz rnz
r ∗ = rz + rnz
R
N = Rz + Rnz
N
= rzν + rnzν
= r ∗ (1 − ρ)
ρ
γ = 1/2
R
N
1
∼ K ,
∆2
r ∗
r ∗
r ∗

Qnz Qz

∆ = W − AD/F W
AM
ρ ↦→ R

∗
r ∗
r ∗ r ∗
r ∗ t − 1 t − 2
t − 3 r ∗ 1 r ∗ 2 r ∗ 3 r ∗ t
˜r ∗ = 0.6r ∗ 1 + 0.3r ∗ 2 + 0.1r ∗ 3.
r ∗ = r ∗ 0
QP
r ∗ QP r ∗ QP r ∗ QP
r ∗
L
N = 25344 Nmb = 99
3
2 Nc1 + Nmbc2
Nmbc3
Nmbc4
NCtmn
QP ↦→ ρ ϑ
ϑ
Qnz

Nmbh1
3
2 Nh2
Lh3
ρ − R 32h4
ϑ Lh5
R − QP QP0 h6
Nmbh7
Nmbh8
NCh1 + LCh2
θ
ρ ↦→ R
θ
Nmbh1
3
2 Nh2
Lh3
ρ − QP 32h4
ρ − R QP0 a6
Nmba7
Nmba8
a6 < h6 a7 < h7 a8 < h8 NCa1 +LCa2 Ca1 < Ch1
Ca2 < Ch2

ρ

QP
r ∗
r ∗ 0

∗
r ∗

∗
r ∗

0.7
0.6
0.5
0.4
0.3
0.2
A
C
B
C
B
A
CB
3.0 3.5 4.0 4.5 5.0 5.5 6.0
R(ϑ, ρ) ρ = 0.96 ρ = 0.97 ρ = 0.98
R
ϑ
A

∗ 0 = 6.7
r∗ 0 = 6.7
3.5 ϑ 5.4 ρ ↦→ R
ρ
R ρ
ϑ

∗ 0 = 6.7
r∗ 0 = 8.0
r∗ 0 = 8.0

∗ 0 = 8.0
r ∗
r ∗ 0
r ∗
r ∗
r ∗
r ∗
r ∗
r ∗
r ∗ 0
r ∗
r ∗

∗
r∗
r ∗ 0
= 8.0

4000
3000
2000
1000
0
0 50 100 150 200 250
4000
3000
2000
1000
0
4000
3000
2000
1000
0 50 100 150 200 250
r ∗ = 6.7
0
0 50 100 150 200 250
r ∗ = 8
r ∗ QP r ∗ QP r ∗
QP r ∗ QP

3000
2500
2000
1500
1000
500
0
0 50 100 150 200 250
2500
2000
1500
1000
500
0
2000
1500
1000
500
0 50 100 150 200 250
r ∗ = 6.7
0
0 50 100 150 200 250
r ∗ = 8
r ∗ QP r ∗ QP r ∗
QP r ∗ QP

S(v) = |err| + λ(QP )C(v),
λ C(v)

C(v)
v
¯v

C(v)
v
¯v

H(X)
H(X) = −
p(x) log p(x)
x∈X
X X p
p

H(X, Y ) (X, Y )
p(x, y)
H(X, Y ) = −
p(x, y) log p(x, y)
x∈X y∈Y
H(Y |X)
H(X, Y ) = −
p(x, y) log p(y|x)
x∈X y∈Y
H(X|Y ) = H(Y |X) X = ⌊Y/5⌋
Y X
p q
D(p q) =
x∈X
p(x) log p(x)
q(x)
q
X p
I(X; Y )
I(X; Y ) = I(Y ; X) =
p(x, y)
p(x, y) log
p(x)p(y)
x∈X y∈Y
I(X; Y ) = H(X) − H(X|Y ) = H(Y ) − H(Y |X) = H(X) + H(Y ) −
H(X, Y )

n (V i
1
, V i
2 )0

20
15
10
5
0
−5
−10
−10 −5 0 5 10 15 20
10
0
−10
−20
−15 −10 −5 0 5 10

vx vy

140
120
100
80
60
40
20
0
0 50 100
150
∂vx
∂x
140
120
100
80
60
40
20
0
0 50 100
150
∂vx
∂y
140
120
100
80
60
40
20
0
0 50 100
150
∂vy
∂x
140
120
100
80
60
40
20
0
0 50 100
150
∂vy
∂y
∇v

(x k 1,n )0≤n≤N (x k 2,n )0≤n≤N
(i, j) ∈ {(1, 2), (2, 1)}
˜x k i,n,α = |x k i,2n+1 − Pα((x k i,2n))|,
Pα
αl = argmin ˜xi,l,α.
α
2l+1 (x k j,n )0≤n≤N Pαl
Uαl
Pαl
(x k i,n )0≤n≤N (x k j,n )0≤n≤N
˜xj,l,β = |x k+1
j,l
− Pβ((x k+1
j,l ))|,
Pβ β
γl = 1 + 2 argmin ˜xj,l,β.
β
2l + 1 Pγl
Uγl

P lin
−1 P lin
0 P lin
1
P lin
−1 ((x2k)) = ⌊− 1
2 (x 2(k−1)) + 3
2 (x2k)⌋
P lin
0 ((x2k)) = ⌊ 1
2 (x2k) + 1
2 (x 2(k+1))⌋
P lin
1 ((x2k)) = ⌊ 3
2 (x 2(k+1) − 1
2 (x 2(k+1))⌋
P lin
0
P lin
−1 P lin
1
(w t s) s∈{−1,0,1},t∈{d,b}
w d −1 = w d 1 = 2 w d 0 = 1
w b −1 = w b 1 = 1 w b 0 = 1
2
αl = argmin w
α∈{−1,0,1}
d α|x k i,2l+1 − Pα((x k i,2n))| + w b α
αl l
(xj,n) w0
αl
αl
U−1 U0
U1
U lin
−1 ((x2k+1)) = ⌊− 1
4 (x2k−3) + 3
4 (x2k−1)⌋
U lin
0 ((x2k+1)) = ⌊ 1
4 (x2k−1) + 1
4 (x2k+1)⌋
U lin
1 ((x2k+1)) = ⌊ 3
4 (x 2k+1) − 1
4 (x2k+3)⌋
αl ≤ 0 x k i,2l
x k i,2l+1 2l + 1
2l − 3 αl−1 ≥ 0
x k i,2l x k i,2l−1 2l − 1
2l + 3

αl ≤ 0 αl > 0
αl−1 ≥ 0 U lin
0
αl−1 < 0 U lin
1
U lin
−1
U lin
0
(αl)
αl > 0 αl−1 < 0
2l + 1 2l − 1
(x l i,n )
(x1)
Pα
n ˜xi,α,n
ρi
ρi =
α ′ =α (˜xi,α ′ ,n + εα ′)
(A − 1)
α ′(˜xi,α ′ ,n + εα ′),
A
εα

εα
εα

s
Hs ns
R
R =
nsHs.
s
vx
120
100
80
60
40
20
0
0 20 40 60 80 100 120
vx
120
100
80
60
40
20
0
0 20 40 60 80 100 120
vy

vy
vy
vx
vx
vy
vy
vx
vx

vy
vy
vx
vx
vy
vy
vx
vx

vy
vy
vx
vx
vy
vy
vx
vx

1.0
0.5
0.0
-0.5
-1.0
0 2 4 6 8 10
1.0
0.5
0.0
-0.5
-1.0
0 2 4 6 8 10

x2k+1 x2k x2k+1
P 2p − 1 2p
2k + 1
x2k−2p+2, . . . , x2k, x2k+2, . . . , x2k+2p
x2k+1 = P (2k + 1).
Mn (n, xn)
Mn (n, xn)
hk = x2k+1 − x2k+1
2k 2(k + 1) [xk ∧ x 2(k+1), x2k ∨ x 2(k+1)]

x−4 x−2 x0 x2 x4 x6
ˆx3
n
Pn
Pn,0
⎧
⎪⎨
Pn,0(x) =
⎪⎩
x2k ∧ x 2(k+1) Pn(x) ≤ x2k ∧ x 2(k+1)
x2k ∨ x 2(k+1) Pn(x) ≥ x2k ∨ x 2(k+1)
Pn(x)
[2k, 2(k + 1)]
Pn,0
Pn,0(x) = ((x2k ∧ x 2(k+1)) ∨ Pn(x)) ∧ (x2k ∨ x 2(k+1))
= med(x2k, Pn(x), x 2(k+1))
Pn,0
x−4 x−2 x0 x2 x4 x6
ˆx1
P3,0
∧ ∨ Pn,0
Pn,0

x2k, x2k+2, . . . , x2l
x2k, x2k+1, x2k+2, . . . , x2l
x1 (1, x1) ∆1 ∆2 ∆1
(M−2M2) M0 ∆2 (M0, M2)
M4 M2
M0 ∆ ′ 1 ∆ ′ 2
Pk,lm
Pk;l,m(2k) = x2k
Pk;l,m(2m) − Pk;l,m(2l) = x2m − x2l
2k + 1
Pk;k−1,k+1(x) Pn,1(x) =
∨ Pk;k,k+1(x) ∧ Pk+1;k,k+2(x) ∨ Pk+1;k,k+1(x)
∧ Pn(x)∨
Pk;k−1,k+1(x) ∧ Pk;k,k+1(x) ∨ Pk+1;k,k+2(x) ∧ Pk+1;k,k+1(x)
= med(Pk;k−1,k+1(x), med(Pk+1;k,k+2(x), Pn(x), Pk+1;k,k+1(x)), Pk+1;k,k+1(x))

x−2
x−2
M−2
(∆1)
M0
x0
(∆2)
ˆx1
M2
x2
M4
M−2
M0
x0
ˆx1
(∆ ′ 2)
M2
x2
(∆ ′ 1)
M4
P3,1
x4
x4

k
f
I ⊂ R R x ∈ I
f
∀ε > 0, ∃x ′ , x ′′ ∈ I, |x − x ′ | < ε, |x − x ′′ | < ε f(x ′ ) < 0, f(x ′′ ) > 0
f I ⊂ R R
f I ⊂ R R k
k − 1 i ∈ {0, . . . , k − 1} f (i)
f
f(x) =
0 x = 0
x 9 sin 1
x
k k
P3,1
C 4
O(2 −4j ) j
k k
k O(2 −jk )

k
pI I = [a, b]
⎧
⎪⎨ a x < a
pI(x) = x a ≤ x ≤ b
⎪⎩
b
I1 = [a1, b1]
I2 = [a2, b2] I1 ∩ I2 = ∅ I1 ∩ I2
I1 ∩ I2 = ∅ =⇒ pI1 ◦ pI2 = pI2 ◦ pI1 = pI1∩I2 .
I1 ∩ I2 = ∅ I1
I2
I1 ∩ I2 = ∅ =⇒ pI1
◦ pI2 = pI2 ◦ pI1
I1 I2
Rk k
k
intmed(I1, I2)
intmed(I1, I2) = [R1(a1, b1, a2, b2), R2(a1, b1, a2, b2)].
I1 I2
intmed(I1, I2)
intmed(I1, I2) ∩ I1 = ∅
intmed(I1, I2) ∩ I2 = ∅

k
intmed(([aj, bj])1≤j≤k) = [Rk(a1, . . . , ak, b1, . . . , bk), Rk+1(a1, . . . , ak, b1, . . . , bk)].
]−∞, b] [a, +∞[
Ik;d(x) Pk;k−d,k+d(x) Pk;k,k+d(x)
Pn,1(x) 2k + 1
Pn,1(x) = p intmed(Ik,1(x),Ik+1,−1(x))(Pn(x))
N
N
N
N

4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
250
200
150
100
50
0
0.0 0.2 0.4 0.6 0.8 1.0
N
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
250
200
150
100
50
0
0.0 0.2 0.4 0.6 0.8 1.0
N

6
5
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
100
50
0
-50
-100
-150
-200
0.0 0.2 0.4 0.6 0.8 1.0
N
6
5
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
100
50
0
-50
-100
-150
-200
0.0 0.2 0.4 0.6 0.8 1.0
N

4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
250
200
150
100
50
0
0.0 0.2 0.4 0.6 0.8 1.0
N
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
250
200
150
100
50
0
0.0 0.2 0.4 0.6 0.8 1.0
N

6
5
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
100
50
0
-50
-100
-150
0.0 0.2 0.4 0.6 0.8 1.0
N
6
5
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
100
50
0
-50
-100
-150
0.0 0.2 0.4 0.6 0.8 1.0
N

PSNR
50
45
40
35
30
25
20
15
1.0 2. 5. 10.0 20. 50. 100.0 200.
nombre de coefficients
N

EE EO
OE OO
EE
OE
EO
OO
P
P
U
U
P
x
y
⊗ I2 I2 ⊗
I U
0 I
P
U
U
I 0
P I
EE
OE
EO
OO
P
P
P
P
U
P/2
U
U
h2k+1 = x2k+1 − ˆx2k+1/2
U
LL
HL
LH
HH
LL
HL
LH
HH

EE
OE
EO
OO
P/2
P/2
P
h2k+1 = x2k+1 − ˆx2k+1
EE
OE
EO
OO
PHL
PLH
P
PHH UHH
P HL
n,0 (x) = med(x2k,2l, x, x2k+2,2l)
P LH
n,0 (x) = med(x2k,2l, x, x2k,2l+2)
P HH
n,0 (x) = med(x2k,2l, x2k,2l+2, x, x2k+2,2l, x2k+2,2l+2)
P/2
P/2
ULH
UHL
LL
HL
LH
HH
LL
HL
LH
HH

Pk;l,m(2k) = x2k
Pk;l,m(2m) − Pk;l,m(2l) = x2m − x2l
Ik,∆(x) Pk;k−∆,k+∆(x) Pk;k,k+∆(x)
∆ HH
1 = (1, 1) ∆ HH
2 = (−1, 1)
∆ LH = (0, 1) ∆ HL = (1, 0)
2k + ∆ HH
1
P HL
n,1 (x) = p intmed(Ik,∆ HL(x),I k+∆ HL ,−∆ HL(x))(x)
P LH
n,1 (x) = p intmed(Ik,∆ LH (x),I k+∆ LH ,−∆ LH (x))(x)
P HH
n,1 (x) = pintmed(Ik,∆HH (x),I
1
k+∆HH 1
,−∆HH (x),I
1
k,∆HH (x),I
2
k+∆HH 2
,−∆HH (x))(x)
2

Pk;l,m(2k) = x2k
Pk;l,m(2m) − Pk;l,m(2l) = x2m − x2l
Ik,∆(x) Pk;k−∆,k+∆(x) Pk;k,k+∆(x)
∆ HH
1 = (1, 1) ∆ HH
2 = (−1, 1)
∆ LH = (0, 1) ∆ HL = (1, 0)
2k + ∆ HH
1
P HL
n,1 (x) = p intmed(Ik,∆ HL(x),I k+∆ HL ,−∆ HL(x))(x)
P LH
n,1 (x) = p intmed(Ik,∆ LH (x),I k+∆ LH ,−∆ LH (x))(x)
P HH
n,1 (x) = pintmed(Ik,∆HH (x),I
1
k+∆HH 1
,−∆HH (x),I
1
k,∆HH (x),I
2
k+∆HH 2
,−∆HH (x))(x)
2

Pk;l,m(2k) = x2k
Pk;l,m(2m) − Pk;l,m(2l) = x2m − x2l
Ik,∆(x) Pk;k−∆,k+∆(x) Pk;k,k+∆(x)
∆ HH
1 = (1, 1) ∆ HH
2 = (−1, 1)
∆ LH = (0, 1) ∆ HL = (1, 0)
2k + ∆ HH
1
P HL
n,1 (x) = p intmed(Ik,∆ HL(x),I k+∆ HL ,−∆ HL(x))(x)
P LH
n,1 (x) = p intmed(Ik,∆ LH (x),I k+∆ LH ,−∆ LH (x))(x)
P HH
n,1 (x) = pintmed(Ik,∆HH (x),I
1
k+∆HH 1
,−∆HH (x),I
1
k,∆HH (x),I
2
k+∆HH 2
,−∆HH (x))(x)
2

Pk;l,m(2k) = x2k
Pk;l,m(2m) − Pk;l,m(2l) = x2m − x2l
Ik,∆(x) Pk;k−∆,k+∆(x) Pk;k,k+∆(x)
∆ HH
1 = (1, 1) ∆ HH
2 = (−1, 1)
∆ LH = (0, 1) ∆ HL = (1, 0)
2k + ∆ HH
1
P HL
n,1 (x) = p intmed(Ik,∆ HL(x),I k+∆ HL ,−∆ HL(x))(x)
P LH
n,1 (x) = p intmed(Ik,∆ LH (x),I k+∆ LH ,−∆ LH (x))(x)
P HH
n,1 (x) = pintmed(Ik,∆HH (x),I
1
k+∆HH 1
,−∆HH (x),I
1
k,∆HH (x),I
2
k+∆HH 2
,−∆HH (x))(x)
2

Pk;l,m(2k) = x2k
Pk;l,m(2m) − Pk;l,m(2l) = x2m − x2l
Ik,∆(x) Pk;k−∆,k+∆(x) Pk;k,k+∆(x)
∆ HH
1 = (1, 1) ∆ HH
2 = (−1, 1)
∆ LH = (0, 1) ∆ HL = (1, 0)
2k + ∆ HH
1
P HL
n,1 (x) = p intmed(Ik,∆ HL(x),I k+∆ HL ,−∆ HL(x))(x)
P LH
n,1 (x) = p intmed(Ik,∆ LH (x),I k+∆ LH ,−∆ LH (x))(x)
P HH
n,1 (x) = pintmed(Ik,∆HH (x),I
1
k+∆HH 1
,−∆HH (x),I
1
k,∆HH (x),I
2
k+∆HH 2
,−∆HH (x))(x)
2

S
N : x ↦→ x + S
N : x ↦→ x + SD(x)
D

v +
v −
M + M −
M + (t, x, y) = (t + 1, x + v + x (t, x, y), y + v + y (t, x, y))
M − (t, x, y) = (t − 1, x + v − x (t, x, y), y + v − y (t, x, y))
M p ⎧
⎪⎨ Id p = 0
= (M
⎪⎩
+ ) p p > 0
(M − ) −p p < 0
T
T (a,b)(t, x, y) = (t, x + a, y + b).
M0
S x
V (x) = {x} + S(x)
V (x) = {(t + δt, x + δx, y + δy)|(δt, δx, δy) ∈ S(x)}
VM(x) = {M δt (T (δx,δy)(t, x, y))|(δt, δx, δy) ∈ S(x)}

WM(x) = {T (δx,δy)(M δt (t, x, y))|(δt, δx, δy) ∈ S(x)},
t
S − T
x
t
T − S
x

t
ST
x
t
T S
v(t, x, y) = (+∞, +∞).
VM,I(t, x, y) = VM(t, x, t) ∩ I.
x

B
A
t
A
B
A
B
k Rk
Rk(f)(x) = f(yx,k)
x

{yx,i} x f(yx,i) ≤ f(yx,j) i < j
k
med(f)(x) = med
y∈V (x) f(y)
α ∈ [0, 1]
k k
k = ⌊α card V (x)⌋.

9 8
7 4 3
7 3 2
α = 0.5
S1 S2
HOM(X) = {x|S1(x) ∈ X S2(x) ∩ X = ∅}
S1(x) ∩ S2(x) = ∅ ∀x ∈ I,

S2,M S ′ 2,M
S ′ 2,M(x) = S2,M(x) \ S1,M(x).
S1,M S2,M S ′ 1,M S ′ 2,M
S1,M

t
t
(S1, S2)
(S1,M , S ′ 2,M )
t
x
x
t
t
(S ′ 1,M , S ′ 2,M )
?
(S1,M , S2,M )
(S ′ 1,M , S2,M )
x
x
x

ε(X) = {x ∈ X|V (x) ⊂ X}
ε(f)(x) = min f(y)
y∈V (x)
ˇ V (x) V (x) x
δ(Y ) =
y∈Y
ˇV (y)
δ(f)(y) = max
x∈ ˇ f(x)
V (y)
Y ⊆ ε(X) ⇐⇒ δ(Y ) ⊆ X
δ(Y ) = {y|V (y) ∩ Y = ∅}
δ(f)(y) = max
x|y∈V (x) f(x)

x ∈ I
I ← +∞
x ′ ∈ V (x)
I(x) ← min(I(x), I(x ′ ))
x ∈ I
I ← −∞
x ∈ I
x ′ ∈ V (x)
I(x ′ ) ← max(I(x ′ ), I(x))

x ∈ I
I ← +∞
x ′ ∈ V (x)
I(x) ← min(I(x), I(x ′ ))
x ∈ I
I ← −∞
x ∈ I
x ′ ∈ V (x)
I(x ′ ) ← max(I(x ′ ), I(x))

352 × 288
4
3

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

1/8
I
Ĩ
Ĩ Ĩ
f : I → J ˜ f :
Ĩ → J

ϕ
ϕ
ϕ ψ
ϕ ◦ ψ = ψ ◦ ϕ
min max

h(i-1)
h(i-2)
h(i)
h(i+1)
h(i+2)

y I x Ĩ y ∈ V (x)
Ĩ
Ĩ
mathcalI ˜
x
g ≤ ε(f) ⇐⇒ δ(g) ≤ f
S = {(0, 0), (−1, 0), (1, 0), ( 1
2
, 1)}
, −1)}
y x y ∈ V (x) ˇ S = {(0, 0), (1, 0), (−1, 0), (− 1
2
S
g = ε(f)
δ(ε(f)) ≤ f.
I Ĩ
ε(f)(x) = min ˜f(y)
y∈V (x)
δ(g)(x) = max
x∈V (y) ˜g(y).
Ĩ
g ≤ ε( ˜ f) δ(˜g) ≤ f.

(0,0)
S
(0,0)
ˇ S
· · · · · · · ·
· 3 3 6 3 3 3 ·
· 4.5 6.5 8 5 2 2 ·
· 3 3 4 4 3.5 4.5 ·
ε(g)
1 3 5 8 11 3 5 8
3 3 6 7 9 3 4 7
8 7 9 11 13 5 2 7
9 6 3 4 9 8 10 5
g
· · · · · · · ·
· · 6 7.25 6.5 3.5 · ·
· · 8 8 8 8 · ·
· · · · · · · ·
δ1(ε(g))
δ ˇ S ε S
δ1(ε(g)) g
·
IV
IV =
V (x).
x∈I
g ≤ ε(f) ⇔ ∀x ∈ I, g(x) ≤ min ˜f(y)
y∈V (x)
⇔ δ(g) ≤ f
g h
∀x ∈ I, ∀y ∈ V (x), g(x) ≤ ˜ h(y),
h
h ≥ δ(g).
˜ h
h
h
δ(g)

h1
h2 min(h1, h2)
h ≥ δ(g)
∧ε(hi) = ε(∧hi),
I
Ĩ Ĩ
n
1
n
n
1
n
n
Ĩ
n
n n 2

x2
C1 : x1 ≥ 10 δ(g)
h1(x1)
h2(x1)
C
h2(x2)
h1(x2)
h1 h2 min(h1, h2)
C2 : x2 ≥ 20
x1
x2
C1 : x1 ≥ 10
A
C2 : x2 ≥ 20
C3 : 1
3 x1 + 2
3 x2 ≥ 20
C
C
h(xi) ≥
C1 C2
λixi ≥ A
C1 C2 C3
h A
h ≥ A ⇒ ε(h) ≥ g
x1

n (xi)1≤i≤n n > 1
m m < n M +
Z
∀i ∈ [1, n] yi,Z = M +
Z (xi)
M +
R yi,R xi
yi,R
(y j
i,Z )j∈N
(y0 i,Z )
n (ji,opt)1≤i≤n
i1 = i2 ⇒ y ji 1 ,opt
i1,Z = yji 2 ,opt
i2,Z ∀(i1, i2)

y ji,opt
i,Z
= argmin
1≤i≤n
|y ji
i,Z − yi,R| 2 .
n (ji,opt)1≤i≤n
n

y ji,opt
i,Z
= argmin
1≤i≤n
|y ji
i,Z − yi,R| 2 .
n (ji,opt)1≤i≤n
n

k

k

N

N

ρ
ρ

ρ
ρ

ρ
ρ