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Image Denoising with K-SVD. Priyam Chatterjee EE 264 – Image Processing & Reconstruction Instructor : Prof. Peyman Milanfar Spring 2007. Sparseland Model. Defined as a set {D,X,Y} such that. t. Y. D. X. Figure courtesy Michael Elad. Sparse Coding.

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image denoising with k svd

Image Denoising with K-SVD

Priyam Chatterjee

EE 264 – Image Processing & Reconstruction

Instructor : Prof. Peyman Milanfar

Spring 2007

sparseland model
Sparseland Model

Defined as a set {D,X,Y} such that

t

Y

D

X

Figure courtesy Michael Elad

sparse coding
Sparse Coding
  • Given a D and yi, how to find xi
  • Constraint : xiis sufficiently sparse
  • Finding exact solution difficult
  • Approximate solution good enough ?
orthogonal matching pursuit
Orthogonal Matching Pursuit

Select dkwith max

projection on residue

xk = arg min ||y-Dkxk||

D, y

x

Check terminating

condition

Update residue

r = y - Dkxk

omp features
OMP : features
  • Greedy algorithm
  • Can find approximate solution
  • Close solution if T is small enough
  • Simplistic in nature
dictionary selection
Dictionary Selection
  • What D to use ?
  • A fixed overcomplete set of basis :
      • Steerable wavelet
      • Contourlet
      • DCT Basis
      • ….
  • Data Adaptive Dictionary – learn from data
k svd algorithm
Select atoms from input

Atoms can be patches from the image

Patches are overlapping

K-SVD Algorithm

Initialize Dictionary

Sparse Coding

(OMP)

Update Dictionary

One atom at a time

k svd algorithm8
Use OMP or any other fast method

Output gives sparse code for all signals

Minimize error in representation

K-SVD Algorithm

Initialize Dictionary

Sparse Coding

(OMP)

Update Dictionary

One atom at a time

k svd algorithm9
Replace unused atom with minimally represented signal

Identify signals that use k-th atom (non zero entries in rows of X)

K-SVD Algorithm

Initialize Dictionary

Sparse Coding

(OMP)

Update Dictionary

One atom at a time

k svd algorithm10
Deselect k-th atom from dictionary

Find coding error matrix of these signals

Minimize this error matrix with rank-1 approx from SVD

K-SVD Algorithm

Initialize Dictionary

Sparse Coding

(OMP)

Update Dictionary

One atom at a time

k svd algorithm11
[U,S,V] = svd(Ek)

Replace coeff of atom dk in X with entries of s1v1

dk = u1/||u1||2

K-SVD Algorithm

Initialize Dictionary

Sparse Coding

(OMP)

Update Dictionary

One atom at a time

denoising framework
Denoising framework
  • A cost function for : Y = Z + n
  • Solve for

Prior term

denoising framework13
Denoising Framework
  • Break problem into smaller problems
  • Aim at minimization at the patch level

Select i-th patch of Z

accounted for implicitly by OMP

denoising framework14
Denoising Framework
  • Solution :
  • Denoising by normalized weighted averaging

Initialize Dictionary

Sparse Coding

(OMP)

Update Dictionary

One atom at a time

Averaging of patches

proof of the pudding low noise
Proof of the pudding – low noise

PSNR 28.12 dB

PSNR 34.16 dB

Denoising under presence of AWGN of std. dev 10

high noise case std dev 50
High noise case – std dev 50

PSNR 24.93 dB

PSNR 14.75 dB

outside the math
Outside the math :
  • Similar atoms in dictionary should be replaced with signals that are least represented
  • Atoms which are least used should be replaced by signals that are least represented