Image Denoising Using Mean Curvature of Image Surface Tony Chan - - PowerPoint PPT Presentation
Image Denoising Using Mean Curvature of Image Surface Tony Chan (HKUST) Joint work with Wei ZHU (U. Alabama) & Xue-Cheng TAI (U. Bergen) In honor of Bob Plemmons 75 birthday CUHK Nov 18, 2013 1 How I Got to Know Bob Plemmons
1
2
3
4
5
Hike to Lei Yue Mun Nov 2011 Trip to Grass Island Aug 2010 6
7
Image denoising using mean curvature of image surface, SIIMS 2012.
Augmented Lagrangian method for a mean curvature based image denoising model, Inverse Probl Imag, In Press, 2013.
Image Segmentation Using Euler’s Elastica as the Regularization, J. Scientific Computing, 2013.
A fast algorithm for a mean curvature based image denoising model using augmented Lagrangian method, To appear in LNCS 2014, in “Efficient Algorithms for Global Optimisation Problems in Computer Vision”.
8
many other ones
Given image Desired clean image Noise
1
How to decompose the given noisy image using appropriate regularizers?
9
– Powerful & popular, excellent analytical properties – Preserve edges and sweep noise very efficiently – Cannot preserve corner & image contrast – Suffers from the staircase effect
λ , u) (f u λ E(u)
Ω 2 Ω
> − + ∇ =
goal true image goal boundary positive parameters
+ ∇ + − =
Ω 1 K \ Ω 2 2
(K) μH u λ u) (f K) E(u,
10
– Originally proposed for the disocclusion problem – Noise removal efficiently, no staircase effect – Need to solve a fourth-order PDE
– Excellent noise suppression, no staircase effect – Need to solve a fourth-order PDE
Ω Ω
− + ∇ ∇ ∇ ⋅ ∇ + =
2 2
) ( 2 1 ) ( u f u u u b a u E
Ω Ω
2 2 2 2 2
yy yx xy xx
11
Its zero level set corresponds to the image surface , whose mean curvature reads:
2 1
f(x)) (x,
f 2 x x x x x z) (x, z) (x, z) (x, z) (x,
H f 1 f 2 1 1 f, 1 f, 2 1 Φ Φ 2 1 = ∇ + ∇ ⋅ ∇ = − ∇ − ∇ ⋅ ∇ = ∇ ∇ ⋅ ∇
12
small oscillation part can be removed effectively.
− + ∇ + ∇ ⋅ ∇ = − + = Ω 2 u) (f 2 1 Ω | 2 | u | 1 u | 2 λ Ω 2 u) (f 2 1 Ω | u H | λ E(u)
u) (f )) u (H ' (Φ P) (I 2 u 1 1 λ t u − + ∇ − ∇ + ⋅ ∇ − = ∂ ∂ | x | Φ(x) , u 1 u u 1 u ν ) ν P( , ν ) ν I(
2 2
= ∇ + ∇ ∇ + ∇ ⋅ = =
u) (f u 2 λΔ t u 1, | u | − + − ≈ ∂ ∂ << ∇
The two operators are defined as
2 R 2 R : P I, →
13
If is an open set with boundary, and , then , the perimeter of set inside the domain (independent of h).
contrasts, as the regularizer doesn’t rely on the height of signal.
Let be an image defined on . Define , then there exists a constant , such that if , then the following holds:
Ω) P(E, | H |
Ω f
=
E Ω E
2
C
} shown. as profile
type same the takes g ), x g( y) u(x, : ) (R u {
2 2 2 2
y C S + = ∈ =
C > C < λ S} u : inf{E(u) E(f) ∈ =
R) B(0,
h f χ =
2R) (-2R, ) 2R , R 2
× = Ω
This property shows that the model attains a minimum at if is small enough, i.e. the model restores exactly and thus preserves contrast.
f λ f
E
hχ f =
14
Let be an image defined on . Define , then there exists a constant , such that if , then the following holds
Q} u : inf{E(u) E(f) ∈ =
R) (0, R) (0,
h f
×
= χ
R) (-R, ) R ,
( × = Ω
}
the along generatrix the rotating by
is y) u(x, z
surface the : u { Q = =
C > C < λ
For small enough regularization (e.g. low noise level), our model can preserve corners. Summary of our model:
15
minimization of the above functionals by Tai et al. (SIIMS 2010 & 2011)
subproblems
− + ∇ =
Ω Ω
u) (f u λ E(u)
2
Ω Ω
− + ∇ ∇ ∇ ⋅ ∇ + =
2 2
) ( 2 1 ) ( u f u u u b a u E
16
denoising through minimization of Euler’s Elastica
variables
17
Review of ALM for Euler’s Elastica Denoising (Tai,Hahn,Chung, SIIMS,2011)
Ω Ω
− + ∇ ∇ ∇ ⋅ ∇ + =
2 2
) ( 2 1 ) ( u f u u u b a u E
17
and we just need to minimize with one of them, the problem is convex
18
18
Features of the ALM in Tai et al SIIMS11:
solved by fast solvers like FFT.
very fast.
19
19
.
− + ∇ + ∇ ⋅ ∇ =
Ω 2 Ω 2
u) (f 2 1 | u | 1 u E(u) λ
n q u u n u p u f q
n q p u
⋅ ∇ = ∇ + ∇ = ∇ = − + ∫
Ω Ω
, 1 / , subject to ) ( 2 1 min
2 2 , , ,
λ
20
n q u u n u p ⋅ ∇ = ∇ ∇ = ∇ = , 1 , / 1 , , 1 ,
1 , , / 3 , 2 , 1 , 2 , 1 subject to 2 ) ( 2 1 , , , min u p p p n n n n n n q u f q n q p u ∇ = = = ⋅ ∇ = Ω − + Ω
λ
21
Original curve Result by our Model ( u ) Difference ( f-u ) Result by ROF Model ( u ) Difference ( f-u ) Noisy curve ( f )
Jumps preserved better Removed noise more uniform Staircase alleviated 22
Original curve Result by our Model ( u ) Difference ( f-u ) Difference ( f-u ) Noisy curve ( f ) Result by ROF Model ( u )
Staircase alleviated Removed noise more uniform 23
Original “Bars” Result by our Model ( u ) Difference ( f-u ) Difference ( f-u ) Noisy “Bars” ( f ) Result by ROF Model ( u )
Contrast preserved better 24
25
Original “Shapes” Result by our Model ( u ) Difference ( f-u ) Difference ( f-u ) Noisy “Shapes” ( f ) Result by ROF Model ( u )
As indicated in f-u, Contrast and corners Preserved better 25
Original “Barbara” Result by our Model ( u ) Difference ( f-u ) Difference ( f-u ) Noisy “Barbara” ( f ) Result by ROF Model ( u )
Large scale signal, such as face preserved better 26
27
Original “Barbara” Local patch By our model By ROF model
Staircase effect alleviated 27
28
Difference ( f-u ) Difference ( f-u ) Noisy “Peppers” ( f ) Original “Peppers” Result by our Model ( u ) Result by ROF Model ( u )
Large scale signal, such as surface of pepper, preserved better 28
29 29
Original “Peppers” Local patch By our model By ROF model
Staircase effect alleviated 29
30 30 30
noise-free image noisy image
By Euler’s elastica model By the LLT model By our model
Contrast and corners preserved better than other models A slice of the noise-free (B), noisy (R), and cleaned image (G) 30
31 31 31
Original image clean image
f u
3
10 . 3 × = λ
clean image u
4
10 5 . 2 × = λ
clean image u
4
10 . 4 × = λ
TV-L1 shares a similar property, but cannot preserve corners of objects
When the regularization parameter increases, objects of small scales will be removed first and then the
ultimately corners will be smeared. 31
inpainting
32
33