The basic idea We believe that the method-noise contains residual image content

that can be represented as a linear combination of the very same

atoms used in the initial denoising stage.

Sparsity-Based K-SVD Denoising

counts the non-zero elements.

is a matrix that extracts the (i,j)-patch from the image.

The K-SVD trains a dictionary D using the corrupted image itself

and represents each noisy patch (local approach) by a sparse

composition over D using the Orthonormal Matching Pursuit (OMP).

The denoised image is obtained by averaging the cleaned patches (the global need).

Removing Additive noise (called “denoising”) using Sparsity-

Based techniques have been shown to be very effective.

We aim to improve the K-SVD [1,2] denoising performance by a

second-layer processing stage that leverages on the initial

denoised result.

{yaniv,elad}@{tx,cs}.technion.ac.il

The Denoising Problem Given the noisy image , where is the clean image and

is Gaussian noise, our goal is to recover from .

This can be viewed as the need to separate between and .

A common patch-based denoising scheme:

This method (and variants of it) are very popular and lead to state-

of-the-art results in various applications.

Stopping Criterion

Apply “Pearson’s correlation test”

between the active regions of the

k-iteration denoised result and

the method-noise image.

The correlation closer to zero imply

less dependence between

these two images.

The iteration that obtains the

minimum absolute value of

the correlation is the best one.

Improvements We found that allowing a slight modification to the initial supports may

lead to further denoising improvement.

This can be achieved by sparse-coding additional (e.g. 1) atoms using

the given supports as initial solution for the OMP.

Denoising Performance

The proposed approach boost the K-SVD consistently.

The stopping criterion estimation-error is almost negligible.

References [1] M. Elad, “Sparse and redundant representations: From theory to applications in signal and image processing,”

Springer, 2010.

[2] M. Elad and M. Aharon, “Image denoising via sparse and redundant representations over learned

dictionaries,” IEEE Transactions on Image Processing, vol. 15, no. 12, pp. 3736–3745, Dec. 2006.

IMPROVING K-SVD DENOISING BY POST-PROCESSING

ITS “METHOD-NOISE”

Local vs. Global The problem: Rather than modeling the image, we model small

patches of it while disregarding their inter-relations.

The “method-noise” image contains “stolen” image information:

We could do much better if we “globalize” the models.

Boosting K-SVD Denoising Algorithm Init Stage:

- Apply K-SVD denoising and save its outputs:

the denoised image .

- Set k=0 and .

- Compute H, a mask obtained by

detecting the active-regions in .

Main Iteration:

(i) Project each one of the method-noise

patches onto the subspaces that

determined by Supp and D.

(ii) Reconstruct the residual content image by averaging the projected patches.

(iii) Increment k, and return to (i) unless stopping criterion is met.

Output: , where denote term-by-term matrix multiplication.

Yaniv Romano The Electrical Engineering Department

Michael Elad The Computer Science Department

Motivation and Goals

Remove Additive

Noise ?

Update the Model

Initial Model

Denoising each patch

Noisy Image

Denoised Image

Noisy image Denoised image Method Noise

(i) Divide the

image into

overlapping

patches

(ii) Initialize the

dictionary D

Compute ij

per patch

using OMP

Compute D one

column at a time

using SVD 0

ij 0α

22

ij 2

α =Min α

s.t. x- α εR D

2

ij 2,Aij

Min x- αD

R D

Compute ij per patch using OMP

Reconstruct the image

by averaging the

denoised overlaping

patches

0

ijR

α1 α2

+ +

2 1

?

Clean Image

K-SVD Denoise

Noisy Image

Fixed-Support K-SVD

Denoise

Fixed-Support K-SVD

Denoise

Fixed-Support K-SVD

Denoise

Activity mask detector

y

0x

0 *

H

*x

Supp ,D ijα

Supp ,D and ijα

0x

0x

00 y x

* 0 *x x H

k 0 kx x H* 0 *x x H minimum

|correlation|

x , y xk kCorrelation H H

δ

Barbara Couple Fingerprint House Boats Average Boost Est. Err.

15/24.61 32.49 32.61 31.52 31.61 30.04 30.16 34.37 34.37 31.80 31.89 32.04 32.13 0.08 0.02

20/22.11 30.88 31.12 30.08 30.22 28.45 28.66 33.22 33.21 30.43 30.57 30.61 30.76 0.15 0.01

25/20.18 29.57 29.87 28.90 29.12 27.24 27.53 32.19 32.20 29.34 29.52 29.45 29.65 0.20 0.01

50/14.16 25.40 25.87 25.27 25.69 23.21 23.88 28.02 28.23 25.92 26.17 25.56 25.97 0.41 0.03

75/10.63 22.92 23.13 23.56 23.79 19.99 21.65 25.06 25.42 24.01 24.20 23.11 23.64 0.53 0.06

100/8.14 21.85 21.89 22.61 22.68 18.31 20.02 23.64 23.77 22.85 22.92 21.85 22.26 0.41 0.04

Boosting results Regular K-SVD results

PSNR

Regular K-SVD

PSNR = 29.57 dB

Method-Noise of

regular K-SVD

Boosting K-SVD

PSNR = 29.87 dB

Noisy Image

PSNR = 20.18 dB

Extracted Information

from it’s method-noise

Activity mask Method-Noise of

Boosting K-SVD

Noisy Image

PSNR = 20.18 dB

Regular K-SVD

PSNR = 28.9 dB

Boosting K-SVD

PSNR = 29.12 dB

Method-Noise images

ijα

y x v x

v x y

vx

Technion – Israel Institute of Technology

SPARS 2013 Signal Processing with Adaptive

Sparse Structured Representations

This research was supported by

the European Research Council

under EU’s 7th Framework Program,

ERC Grant agreement no. 320649,

and by the Intel Collaborative

Research Institute for

Computational Intelligence.