A (Zernike) moment-based nonlocal-means algorithm

for image denoising

Michal Kunešxkunes@utia.cas.cz

ZOI UTIA, ASCR, Friday seminar 13.03.2015

Introduction

2

- Uses nonlocal (NL) means filter

- Introduce the Zernike moments (rotation invariant)

- Zernike moments in small local windows of each pixel are computed

(local structure information)

- similarities are computed (insted of pixel intensity)

- it can gat much more pixels with higher similarity measure

Zexuan Ji, Qiang Chen, Quan-Sen Sun, and De-Shen Xia: A moment-based nonlocal-means algorithm for image denoising. Inf. Process. Lett. 109, 23-24

(November 2009), 1238-1244. DOI=10.1016/j.ipl.2009.09.007

http://dx.doi.org/10.1016/j.ipl.2009.09.007

3

a) Noise σ = 20 (PSNR = 22.16)b) PM model (28.83)c) Bilateral f. (29.16)e) NL-means (31.09)e) Exemplar-based method (32.64)f) SIFT based m. (31.26)g) rotationally invariant

block matching (31.75)h) Moment base NL-means (32.29)

(blockmatching and 3D f. (33.05))

i) real noise component

j) – p) corresponding noise component of each method

( )( ) ( )( )[ ]

2

10 2

0

25510log

/i I

PSNR dBNL u i u i I

∈

=−∑

4

a) Noise σ = 20 (PSNR = 22.16)b) PM model (28.83)c) Bilateral f. (29.16)e) NL-means (31.09)e) Exemplar-based method (32.64)f) SIFT based m. (31.26)g) rotationally invariant

block matching (31.75)h) Moment base NL-means (32.29)

(blockmatching and 3D f. (33.05))

i) real noise component

j) – p) corresponding noise component of each method

( )( ) ( )( )[ ]

2

10 2

0

25510log

/i I

PSNR dBNL u i u i I

∈

=−∑

NL-means filter

5

u(j) – intensity value

w(i,j) – weight, depends on the similarity between pixels i and j

u(Ni) – intensity gray level vector

Ni – square neighborhood of fixed size and centered at a pixel i

Gρ – Gauss kernel with standard deviation ρ.

C(i) – normalizing konstant

h – degree of filtering

( )( ) ( , ) ( )j I

NL u i i j u jω∈

=∑

0 ( , ) 1i jω≤ ≤ ( , ) 1j

i jω =∑

( )( )

( ) ( )2

21,

i jG

u N u N

hi j eC i

ρ

ω

−

−

= ( )

( ) ( )2

2

i jG

u N u N

h

j

C i e

ρ−

−

=∑

( ) ( ) ( ) ( )2 2

i j i jG

u N u N G u N u Nρ

ρ− = ∗ −

NL-means filter + Moments

6

NL-means:

- improves image quality

- high computational cost

- similarity of patches is only translation invariant

Zimmer et al. uses the Hu moments

+ common, simplest

- not efficient for image features representation

- certain degree of information redundancyS. Zimmer, S. Didas, J. Weickert, A rotationally invariant block matching strategy improving image denoising

with non-local means, in: Proc. 2008 Int. Workshop on Local and Non-Local Approximation in Image Processing, in: LNLA, vol. 2008, 2008.

-> Zernike moments

- global shape descriptors

- particulary robust

Main points

7

- compute Zernike moments within a small window around each pixel

- adds orientation invariants for pixels with similarity

- removes the Gauss kernel used in NL-means algorithm

- every moment has equal possibility to influence the brightness of the

central pixel

- Result: higher signal-to-noise ratio (on synthetic images)

Zernike polynomials / moments

8

- mathematical simplicity and universality

- set of orthogonal basis functions mapped over the unit circle

Main properties:

- orthogonality

- rotation invariance

- information compaction

p – order

q – repetition( ), ;0 , ,D p q p q p p q even= ≤ ≤ ∞ ≤ − =

( ) ( )2 2

*

1

1, ,pq pq

x y

pZ V x y f x y dxdy

π+ ≤

+= ∫∫

Zernike polynomials / moments

9

( ), ; 0 , ,D p q p q p p q even= ≤ ≤ ∞ ≤ − =

( ) ( )2 2

*

1

1, ,pq pq

x y

pZ V x y f x y dxdy

π+ ≤

+= ∫∫

p – order

q – repetition

Vp*q – complex conjugate of Vpq

Rpq – radial polynomial

ρ – length of vector from origin to

pixel (x,y)

θ – angle of ρ from x axis

( ) ( ), iq

pq pqV R eθρ θ ρ=

( )( ) 21 !

2

! ! !2 2 2

p k

pk

pq

k q

p k even

p k

Rp k k q k q

ρ ρ

−

=

− =

+−

=− − +∑

( ) ( )* 2 21, , ; 1pq pq

x y

pZ V x y f x y x y

π

+= + ≤∑∑

Zernike polynomials / moments

10http://astro.if.ufrgs.br/telesc/aberracao.htm

Cartesian moments

( )3

3,3 8 3 cos3Z ρ θ=

11

Zernike moments

The Lena image with noise (σ = 20) shown in (a). Radius r = 3.

(b)–(g) are the images of Z00, Z11, Z20, Z22, Z31, Z33.

Moment-based nonlocal filtering

12

Normalization:

Vector for each pixel

Intensity values:

Similarity measurement:

2, 2,

2,

/ 0ˆ

0

pq p q p q

pq

pq p q

Z Z if Z and q pZ

Z if Z or q p

− −

−

≠ <=

= =

( ) ( ) ( ) ( ) ( ) ( ) ( ){ }1 2 3 4 5 6ˆ ˆ ˆ ˆ ˆ ˆ, , , , ,v i Z i Z i Z i Z i Z i Z i=

1 00 2 11 3 20 4 22 5 31 6 33ˆ ˆ ˆ ˆ ˆ ˆ; ; ; ; ; ;Z z Z z Z z Z z Z z Z z≈ ≈ ≈ ≈ ≈ ≈

( )u i

( ) ( )2

v i v j−∑

( )( )

( ) ( )2

21,

v i v j

hi j eC i

ω

−−∑

= ( )( ) ( )

2

2

v i v j

h

j

C i e

−−∑

=∑

( )( ) ( , ) ( )j I

NL u i i j u jω∈

=∑ ( 95)h =

Weight (ω(i,j)) distributionused to estimate the central pixel

13

Original NL-means proposed

( )( )

( ) ( )2

21,

v i v j

hi j eC i

ω

−−∑

=

( )( )

( ) ( )2

21,

i jG

u N u N

hi j eC i

ρ

ω

−

−

=

14

a) Noise σ = 20 (PSNR = 22.16)b) PM model (28.83)c) Bilateral f. (29.16)e) NL-means (31.09)e) Exemplar-based method (32.64)f) SIFT based m. (31.26)g) rotationally invariant

block matching (31.75)h) Our (32.29)

(block matching and 3D f. (33.05))

i) real noise component

j) – p) corresponding noise component of each method

( )( ) ( )( )[ ]

2

10 2

0

25510log

/i I

PSNR dBNL u i u i I

∈

=−∑

PSNR [dB]

15

Questions?

16