IGARSS_AMASM_woo_20110727.pdf

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Alternating minimization algorithm
for shifted speckle reduction variational model
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.. .

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Sangwoon Yun1 Hyenkyun Woo2

1 School of Computational Science

Korea Institute for Advanced Study
2 Institute
of Mathematical Sciences
Ewha Womans University

International Geoscience and Remote Sensing Symposium,
July 2011

. . . . . .

S. Yun, H. Woo (KIAS,Ewha Univ.) AMA for shifted speckle reduction model IGARSS 2011 1 / 28

Motivation

Figure: Proposed Algorithm (1Mpixel@8sec, 2.1GHz CPU Notebook)

We solve variational models with Total Variation (TV) prior
min f (u) + λ u
u∈U

The proposed method is the fastest one to solve total variation
(TV) based speckle reduction variational. models .
. . . .

Outline

.
. . Previous Speckle Reduction Methods
1
Convex Variational Models with Total Variation
Augmented Lagrangian based algorithms

.
. . Proposed Speckle Reduction Methods
2
Shifted Variational Models with Total Variation
Alternating Minimization Algorithm

.
. . Numerical Experiments
3

. . . . . .


Previous Speckle Reduction Methods

Outline

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1

.
2

.
3

. . . . . .


Previous Speckle Reduction Methods Convex Variational Models with Total Variation

Multiplicative Speckle Noise of L-look SAR image

b(SAR image) = u(unknown reﬂectivity) × η(speckle)

b is the L-look SAR intensity image
η is fully developed multiplicative speckle noise

PDF of η is the Gamma distribution

LL η L−1 −Lη 1
P(η) = e → E(η) = 1; Var(η) =
Γ(L) L

. . . . . .



Variational Model with Total Variation regularizer

Maximum a Posteriori (MAP) Criterion :

max P(u|b) ∝ P(b|u)P(u),
u∈U

where b
LL bL−1 e−L u
P(u) = e−λ
˜ u
P(b|u) = ,
Γ(L) u L
Variational model with Total Variation (Aubert & Aujol 08) :

min Fb (u) = −log[P(b|u)P(u)]
u∈U
b ˜
= log u + , 1 +λ u
u

Assume that u ∈ U = (0, C]n , where n is the size of image
. . . . . .



Algorithm for MAP based Variational Model
The original MAP based model :

b ˜
min Fb (u) = log u + , 1 +λ u
u∈U u

We can ﬁnd solution by Gradient Descent method

du ∂Fb (u)
=−
dτ ∂u
u k+1 − u k b − uk ˜ uk
= + λ div( )
δ (u k )2 | uk | + ε
Difﬁculties of this approach :
Fb (u) is not convex
δ is small and the algorithm is slow
TV is convex but non-differentiable : ∂u u is not unique
. . . . . .



Convex Variational Model with TV
The original MAP based model :
b ˜
min Fb (u) = log u + , 1 +λ u
u∈U u
(Shi & Osher 08) recommended log transformation.
EXPonential model (ul = log u)
∗
u ∗ = e ul ; ul∗ = arg min ul + be−ul , 1 + λ ul
ul ∈log U

(Steidl & Teuber 10) show that the solution of the I-divergence
model equals to that of the exponential model.
I-DIVergence model

u ∗ = arg min u − b log u, 1 + λ u
u∈U
. . . . . .


Previous Speckle Reduction Methods Augmented Lagrangian based algorithms

Augmented Lagrangian for Exponential Model
min {f (ul ) + g(d) = ul + be−ul , 1 + λ d | ul = d}
ul ,d

Augmented Lagrangian Function:
α
Lα (ul , d, p) := f (ul ) + g(d) + p, d − ul + d − ul 2
2.
2
(Bioucas-Dias & Figueiredo 10) proposed MIDAL1 :

 ulk +1 = arg min ul + be−ul , 1 + α ||d k − ul + pk ||2


 u ∈log U
2 2
l
d k +1 = arg min λ d + α ||d − ulk + pk ||2


2 2


d
pk +1 = pk + (d k +1 − ulk+1 ).

Use Newton iteration to obtain ulk+1 and solve subproblem to get
d k+1 at each iteration.
1
Multiplicative Image Denoising by Augmented Lagrangian . . . . . .



Augmented Lagrangian for I-divergence model

(Steidl & Teuber 10) recommend the following reformulation
(ADMM-III)
{ ( ) ( ) }
∗ u I
u = arg min u − b log u, 1 + λ z = d
u∈U z

The corresponding Augmented Lagrangian based algorithm
becomes
 ( k ) ( ) ( k ) 2
 k +1
 d p1 I u

 = arg min + d− ,

 k
p2 zk

 d 2

 k +1 { }
 u
 = arg min u − b log u, 1 + α p1 + d k +1 − u 2 ,
k
 2 2
u∈U
 { }
 z k +1 = arg min λ z + α p2 + d k +1 − z 2 ,

k

 2 2

 (
z

 ) ( k ) ( ) ( k +1 )


k
p1 +1 p1 I u

 k +1 = k + d k +1 − .
p2 p2 z k +1

. . . . . .



Augmented Lagrangian for I-divergence model
We can express the algorithm in a more simpliﬁed form:
 k+1
 d
 = (I − ∆)−1 (u k − p1 − div (z k − p2 ))
k k

 [ ( √ )]

 k+1
 u
 = PU 1 p1 + d k +1 − α 1 + (p1 + d k +1 − α 1)2 + 4 α
k 1 k 1 b
 2
λ
 z k+1 = shrink (p2 + d k+1 , α )

k
 (
 ) ( k ) ( ) ( k+1 )

 k
p1 +1

 p1 I k +1 − u
 k +1 = k + d ,
p2 p2 z k+1

where
The discrete cosine transform (DCT) is used to inverse I − ∆,
where ∆ is the Laplacian operator.
PU (u) denotes the projection of u onto the set U
ai
shrink (ai , c) = max( ai 2 − c, 0) ai 2
. . . . . .


Proposed Speckle Reduction Methods

Outline

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1

.
2

.
3

. . . . . .


Property of multiplicative speckle noise model (b = uη)
Since E(η) = 1, averaging property is invariant w.r.t. T shifting:

E(b + T ) = E(u + T ), ∀T ∈ +.

In homogeneous region A, despeckled image u ∗ is given by

u ∗ = EA (b) = EA (b + T ) − T .

We can ﬁnd solution by the following process

1 Shift up : B =b+T
2 Despeckling : u ∗ = arg minu FB (u)
˜
3 Shift down : u ∗ = u ∗ − T
˜

The solution u ∗ roughly equals to the despeckled image without
shifting
u ∗ ≈ arg min Fb (u).
u
Reference : H. Woo and S. Yun, “Alternating minimization algorithm for speckle reduction with
shifting technique”, submitted to IEEE TIP, 2011. . . . . . .

Proposed Speckle Reduction Methods Shifted Variational Models with Total Variation

Shifted Exponential Variational Model
Let B = b + T and UT = U + T , where T ≥ 0 and U = [0, C]n

Shifted EXP model
∗
u ∗ = eul − T ; ul∗ = arg min f (ul ) + λ ul
ul ∈log UT

where
f (ul ) = ul + Be−ul , 1

The ﬁdelity term f (ul ) is strongly convex

min B T
2
ul f (ul ) ≥ I≥ I.
max UT C+T
T
The strongly convex modulus of f (ul ) is C+T .
. . . . . .


Proposed Speckle Reduction Methods Shifted Variational Models with Total Variation

Shifted I-divergence Variational Model
Let B = b + T and UT = U + T , where T ≥ 0 and U = [0, C]n

Shifted I-DIV model

u∗ = u∗ − T ;
˜ u ∗ = arg min f (u) + λ
˜ u
u∈UT

where
f (u) = u − B log u, 1

The ﬁdelity term f (u) becomes strongly convex

min B T
2
u f (u) ≥ 2
I≥ I.
(max UT ) (C + T )2
T
The strongly convex modulus of f (u) is (C+T )2
.
. . . . . .


Proposed Speckle Reduction Methods Alternating Minimization Algorithm

Alternating Minimization Algorithm (Tseng 1991)
min {f (u) + λ z | u = z}
u,z

Lagrangian & Augmented Lagrangian Function:

L(u, z, p) := f (u) + λ z + p, z − u
α
Lα (u, z, p) := f (u) + λ z + p, z − u + z− u 2.
2
2
Alternating Minimization Algorithm becomes (0 < α < σ/4)
 k +1
 u
 = arg min L(u, z k , pk )
 u∈log U
z k +1 = arg min Lα (u k +1 , z, pk )


 z
pk +1 = pk + α(z k +1 − u k+1 ),

where σ is a strongly convex modulus, i.e. 2 f (u)
u ≥ σI.
. . . . . .



AMA for Shifted Exponential Model

min {f (ul ) + g(z) = ul + Be−ul , 1 + λ z | ul = z}
ul ∈log UT ,z

(Proposed method) AMA for Shifted exponential Model (AMASM) :

 k+1
 ul
 = Plog UT [log( 1+div pk )],
B

k+1 = shrink ( u k − pk , λ ),
 z
 k+1 l α α
p = pk + α(z k +1 − ulk +1 ),
k +1
u ∗ = eul −T

where B = b + T 1, UT = U + T 1, and
Plog UT (ul ) denotes the projection of ul onto the set log UT
ai
. . . . . .



AMA for Shifted I-divergence Model

min {f (u) + g(z) = u − B log u, 1 + λ z | u = z}
u∈UT ,z

(Proposed method)AMA for Shifted I-divergence Model (AMASM):


 u k+1 = PUT [ 1+div pk ],

B
k
uk p λ
 z k+1 = shrink ( k +1− α , α ),
k+1

p = pk + α(z − u k +1 ),
u ∗ = u k+1 − T

where B = b + T 1, UT = U + T 1, and
PUT (u) denotes the projection of u onto the set UT
ai
. . . . . .



AMASM performance vs. T shifting
[EXP] σ1 = min B
max UT ≥ T
C+T , [IDIV] σ2 = min B
(max UT )2
≥ T
(C+T )2

. . . . . .


Numerical Experiments

Outline

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1

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2

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3

. . . . . .


Twelve Test Images (256 × 256, ..., 1500 × 1500).
2.1GHz CPU Notebook (64bit Linux).
All algorithms(AMASM,MIDAL,ADMM-III,PPBit(Nonlocal)) are implemented in C.
Stopping Criterion u k +1 − u k 2 ≤ 3 × 10−4 u k 2 .
. . . . . .

Exponential model I-Divergence model Nonlocal
MIDAL[1] AMASM ADMM-III[2] AMASM PPBit[3]
L IMAGE PSNR/Time/Iter PSNR/Time/Iter PSNR/Time/Iter PSNR/Time/Iter PSNR/Time
Barbara 22.2/1.3/17 22.1/0.5/120 22.1/0.4/33 22.2/0.5/139 23.4/34.3
Boat 23.9/5.7/17 23.7/2.1/114 23.9/2.3/32 23.7/2.2/132 23.9/103.3
House 24.5/1.2/17 24.3/0.5/111 24.6/0.4/31 24.3/0.4/127 24.7/34.3
Lena 25.6/5.7/17 25.4/2.1/115 25.6/2.4/33 25.4/2.2/135 26.1/103.3
Remote1 21.1/4.5/18 21.0/1.9/124 21.1/1.9/32 21.1/2.0/150 21.0/90.6
3 Remote2 21.2/6.4/18 21.0/2.3/112 21.2/2.8/32 21.0/2.4/128 21.2/111.4
Remote3 20.8/1.6/18 20.7/0.6/118 20.7/0.5/32 20.8/0.6/138 20.6/42.9
Remote4 21.6/5.9/18 21.5/2.3/122 21.6/2.2/31 21.6/2.4/146 21.3/103.4
Remote5 20.7/5.8/18 20.5/2.4/128 20.7/2.5/34 20.6/2.6/161 21.0/102.3
Remote6 22.7/14.8/18 22.5/5.1/122 22.7/7.7/33 22.6/5.6/147 22.4/235.1
Remote7 23.8/26.5/18 23.7/8.5/122 23.8/14.2/33 23.8/9.7/147 23.7/360.5
Remote8 24.2/55.6/17 24.2/18.6/121 24.2/42.4/32 24.4/20.2/140 23.9/838.0
Average 22.7/11.2/18 22.6/3.9/119 22.7/6.6/32 22.6/4.2/141 22.8/180.0

[1] MIDAL : Bioucas-Dias and Figueiredo 2010 IEEE TIP
[2] ADMM-III: Steidl and Teuber 2010 JMIV
[3] PPBit : Deledalle, Denis, and Tupin. 2009 IEEE TIP

Remote7(1000x1000), Remote8(1500x1500) : Megapixel size
Proposed method (AMASM) for the exponential model is the
fastest method with reasonable image quality.
. . . . . .

Figure: Remote8 (1500x1500) Image
. . . . . .

Figure: Barbara (256x256) Image
. . . . . .

Figure: Real SAR Image (ENL = 3.5), c Sandia NL .
. . . . .


Conclusion

Alternating minimization algorithm for shifted exponential model is
the fastest one.

The proposed method is highly parallelizable and so can be
accelerated easily by using GPGPU (CUDA).

We can going to improve PSNR of the proposed method by using
nonlocal total variation regularization.

. . . . . .



Thank you for your attention!

H. Woo and S. Yun, “Alternating minimization algorithm for speckle reduction with shifting technique”, submitted
to IEEE TIP, 2011.
S. Yun and H. Woo, “A new multiplicative denoising variational model based on m-th root transformation”, submitted to
IEEE TIP, 2011.
H. Woo and S. Yun, “Proximal linearized alternating direction method for multiplicative denoising”, submitted to SIAM J.
Sci. Comp, 2011.
. . . . . .


IGARSS_AMASM_woo_20110727.pdf

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