Image Denoising through Multi-Scale Learnt Dictionaries
Jeremias Sulam, Boaz Ophir and Michael Elad
Computer Science Department, Technion
Contribution
Experiments
We present a patch-based denoising algorithm relying on a sparsity-inspired model (K-SVD) within multiscale analysis framework, that overcomes some of the disadvantages of the popular algorithms.
Our method is competitive with state of the art methods in terms of PSNR while giving superior results
with respect to visual quality.
Set Up
Background
Multi-scale K-SVD denoising
Problem Statement: recorver z ∈ Rn from
y =z+η
η ∼ N (0, σ 2 ).
• Noisy image Y
•
W
Yb
•
W
Ẑb :
• Images from the National Oceanic and Atmospheric Administration (US) landscape
image library
• Quality Measures: Peak Signal to Noise
Ratio (PSNR) and Structural Similarity
Index (SSIM)
Results
= (WA Y)b : its wavelet transform coefficients
2.5
denoised coefficients per band b
• S decomposition levels.
2
Sparse Model:
Original Image
K−SVD
M.S. K−SVD
Fused K−SVD
BM3D
Global MAP in the wavelet domain:
Noisy Image
z = Dx,
D ∈ Rn×m , (n < m), ||x||0 m
W
∀b, {xij,b , Db , Ẑb }
+
Sparse Coding:
X
µij,b ||xij,b ||0 +
ij
min ||x||0 subject to ||y −
x
2
Dx||2
= arg min
xij,b ,Db ,ZW
b
X
W
λ||Yb
−
W 2
Zb ||2
∆ P SN R
1.5
0.5
ij
K-SVD
2
≤ ,
Multi-Scale Dictionaries
PSNR = 31.0,
SSIM = 0.857
0
WS : wavelet synthesis matrix.
−0.5
10
σ
40
50
BM3D
0.1
K−SVD
M.S. K−SVD
Fused K−SVD
BM3D
Fused K-SVD
PSNR = 33.1,
SSIM = 0.940
Approximation
D
∆SSI M
0.08
Fusing Single and Multi-Scale
0.06
0.04
ZSingle Scale
• Better treatment of smooth areas
• Different size patches managed naturally
0.02
Diagonal
D
Vertical
30
0.12
subject to ||xi ||0 ≤ T, ∀i
WA : wavelet analysis matrix.
20
PSNR = 33.1,
SSIM = 0.923
3. Apply inverse wavelet transform on the
denoised coefficients Ẑ = WA ẐW
Dictionary Learning:
D,X
1. Apply wavelet transform YbW = (WA Y)b
2. Approximate the MAP estimator by K-SVD
on patches from the different bands b → ẐW
b
D̃ = WS D,
min ||WA Y −
1
2
||Db xij,b − Rij,b ZW
||
b
2
Numeral Solution
2
DX||F
σ = 30
0
10
ZMulti Scale
20
30
σ
40
50
• Incorporates global information through
different scales
x
Horizontal
α̂ = arg min ||α||0 subject to ||ỹ −
α
Fused K-SVD denoised patch:
{jsulam,boazo,elad}@cs.technion.ac.il
2
Aα||2
ẑf =
2
≤ c
√1
2
Dα̂
References
[1] M. Elad, M. Aharon. Image Denoising via Sparse and Redundant Representations over Learned Dictionaries. IEEE Trans.
Image Process., vol. 15, no. 12, 2006.
[2] B. Ophir, M. Lustig, M. Elad. Multi-Scale Dictionary Learning Using Wavelets. IEEE J. Sel. Top. Signal Process., vol. 5,
no. 5, 2011.