Michal Kuneš

Transcription

1 A (Zernike) moment-based nonlocal-means algorithm for image denoising Michal Kuneš ZOI UTIA, ASCR, Friday seminar

2 Introduction - 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, 3-4 (November 009), DOI= /j.ipl

3 a) Noise σ = 0 (PSNR =.16) b) PM model (8.83) c) Bilateral f. (9.16) e) NL-means (31.09) e) Exemplar-based method (3.64) f) SIFT based m. (31.6) g) rotationally invariant block matching (31.75) h) Moment base NL-means (3.9) (blockmatching and 3D f. (33.05)) i) real noise component j) p) corresponding noise component of each method PSNR = 10log 55 ( ( )( ) 0 ( )) 10 i I NL u i u i / I [ db] 3

4 a) Noise σ = 0 (PSNR =.16) b) PM model (8.83) c) Bilateral f. (9.16) e) NL-means (31.09) e) Exemplar-based method (3.64) f) SIFT based m. (31.6) g) rotationally invariant block matching (31.75) h) Moment base NL-means (3.9) (blockmatching and 3D f. (33.05)) i) real noise component j) p) corresponding noise component of each method PSNR = 10log 55 ( ( )( ) 0 ( )) 10 i I NL u i u i / I [ db] 4

5 NL-means filter ω ( i, j) = NL( u)( i) ω( i, j) u( j) 1 0 ω( i, j) 1 ω ( i, j) = 1 j ( ) u( N ) u N i j I h = e C C ( i) ( i) u(j) intensity value w(i,j) weight, depends on the similarity between pixels i and j u(n i ) intensity gray level vector N i square neighborhood of fixed size and centered at a pixel i G ρ Gauss kernel with standard deviationρ. Gρ C(i) normalizing konstant h degree of filtering j Gρ = j e ( ) u( N ) u N i h j Gρ ( i ) ( j ) = ρ ( i ) ( j ) u N u N G u N u N 5

6 NL-means filter + Moments 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 redundancy S. Zimmer, S. Didas, J. Weickert, A rotationally invariant block matching strategy improving image denoising with non-local means, in: Proc. 008 Int. Workshop on Local and Non-Local Approximation in Image Processing, in: LNLA, vol. 008, > Zernike moments - global shape descriptors - particulary robust 6

7 Main points - 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) 7

8 Zernike polynomials / moments - mathematical simplicity and universality - set of orthogonal basis functions mapped over the unit circle Main properties: - orthogonality - rotation invariance - information compaction p + 1 Z V x y f x y dxdy = π x + y 1 * (, ) (, ) p q order repetition ( ) D = p, q ;0 p, q p, p q = even 8

9 ( ) Zernike polynomials / moments p + 1 Z V x y f x y dxdy = π x + y 1 * (, ) (, ) p + 1 * Z = V ( x, y) f ( x, y) ; x + y 1 π x y D = p, q ; 0 p, q p, p q = even ( ρ, θ ) = ( ρ ) iq V R e θ R ( ρ ) = p k = q p k = even p k p + k ( 1 )! k ρ p k k q k + q!!! p q order repetition V * p q complex conjugate of V R radial polynomial ρ length of vector from origin to pixel (x,y) θ angle of ρ from x axis 9

10 Zernike polynomials / moments Cartesian moments 3 ( ) Z3,3 = 8 3ρ cos 3θ 10

11 Zernike moments The Lena image with noise (σ = 0) shown in (a). Radius r = 3. (b) (g) are the images of Z 00, Z 11, Z 0, Z, Z 31, Z

12 Moment-based nonlocal filtering Normalization: Vector for each pixel Intensity values: Zˆ Z / Z p, q if Z p, q 0 and q < p = Z if Z p, q = 0 or q = p { } ( ) = ˆ ( ), ˆ ( ), ˆ ( ), ˆ ( ), ˆ ( ), ˆ ( ) v i Z i Z i Z i Z i Z i Z i u ( i) Zˆ z ; Zˆ z ; Zˆ z ; Zˆ z ; Zˆ z ; Zˆ z ; Similarity measurement: ( ) v( j) v i ω ( i, j) 1 ( ) v( j) v i h = e C ( i) C ( i) = j e ( ) v( j) v i h NL( u)( i) = ω( i, j) u( j) ( h = 95) j I 1

13 Weight (ω(i,j)) distribution used to estimate the central pixel Original NL-means proposed ω ( i, j) 1 = C i ( ) e ( ) u( N ) u N i h j Gρ ω ( i, j) = 1 C i ( ) e v i h ( ) v( j) 13

14 a) Noise σ = 0 (PSNR =.16) b) PM model (8.83) c) Bilateral f. (9.16) e) NL-means (31.09) e) Exemplar-based method (3.64) f) SIFT based m. (31.6) g) rotationally invariant block matching (31.75) h) Our (3.9) (block matching and 3D f. (33.05)) i) real noise component j) p) corresponding noise component of each method PSNR = 10log 55 ( ( )( ) 0 ( )) 10 i I NL u i u i / I [ db] 14

15 PSNR [db] 15

16 Questions? 16