be the original image. Let be the Gaussian white noise with zero mean and variance σ. g ( x, denotes the motion blurred image with noise.

Transcription

1 3rd International Conference on Multiedia echnology(icm 3) Estiation Forula for Noise Variance of Motion Blurred Iages He Weiguo Abstract. he ey proble in restoration of otion blurred iages is how to get the point spread function and the noise inforation. It is coonly assued as the white Gaussian noise. he paper corrects istaes in literature [] and puts forward an accurate estiation forula for noise power of linear otion-blurred iages and provides the relative error forula. he relative error is very sall. Experients prove that the noise power can be coputed accurately by utilizing this ethod. eywords: Iage Restoration White Gaussian Noise Motion-blurred Iages Introduction he ey proble in otion-blurred iage restoration is how to get the point spread function (PSF) and the noise inforation. If the PSF and the power spectru of noise are given, a lot of ethods for restoring iages are discussed [-4]. he noise is coonly assued as the white Gaussian noise with zero ean. he variance of noise in sooth area is regarded as noise power. he coon ethod to estiate the variance of noise is to copute the variance of sooth area in the noisy-blurred iage []. Soeties, the sooth area does not exist and the result is rough in this way. Suppose the otion direction is horizontal, the length of otion blur can be estiated by utilizing these ethods in literature, such as Spectru, Cepstru, and Whitening and so on [5-8].he size of iage is assued as M N and σ denotes the variance of noise and the blurred length is a pixels. [] Proposed a ethod for estiating noise power that a superposition operator on the difference of the otion-blurred iage is used to produce a noise-doinated iage. he variance of the noise-doinated iage N is approxiately equal to σ. But there is a serious error in the literature. he 3 a estiation forula was not accurate and the relative error was overestiated. he paper will correct istaes and optiize reasoning and siulation. he correct forula will be proved in section two and the siulation will be done in section three. Guangdong University of Finance and Econoics, Guangzhou 53, China e-ail: jarbei@63.co 3. he authors - Published by Atlantis Press 958

2 he algorith on estiating the noise variance For convenience, assue that the direction of oveent is horizontal. In a very short period of tie, variable otion can usually be approxiated as unifor otion. he unifor otion is assued along x direction, the speed x ( t ) at,where a is constant and is the exposed interval. Let f ( x, be the original iage. Let n ( x, be the Gaussian white noise with zero ean and variance σ. g ( x, denotes the otion blurred iage with noise. g ( x, can be obtained as follows: g( x, a xa f [ x x ( t), y] dt + n( x, f ( τ, dτ + n( x, f [ x at, y] dt + n( x, at withτ x. Since otion is along x direction, y can be oitted. Eq. () can be rewritten a g( x) f ( τ ) dτ + n( x) () xa aing the derivative of g (x) with respect to x in Eq. () gives g ' ( x) f ( x) f ( x a) + n' ( x) (3) Given that the size of iage is M N. Let N x, a, a x z + a, where z [, a) (4) is the integer part of ( N / a ), is the integer part of ( x / a ). Inserting Eq. (4) into Eq.(3) gives f ( z + a) g'( z + a) + f ( z + ( ) a) z + a) (5) Let φ (z) f ( z a),we can rewrite Eq. (5) as f ( z + a) g'( z + + φ ( z) z + (6) () 959

3 Eq. (4) and Eq. (6) iply that f ( x) g'( x + φ ( x a) x (7) φ ( x a) f ( x) g'( x + x (8) When x varies fro to N, varies fro to -. We can derive Eq. (9) fro Eq. (8) φ( x) + + Hence φ( x a) f ( x + a) x + a x a + a Eq. (7) and Eq. () iply that g'( x a + a Let + f ( x) + g'( x + a f ( x + a a) x a + a f ( x + a a) g'( x x for x < a (9) g'( x a + a () () P ( x) g'( x + g'( x a + a () R ( x) x + x a + a (3) 96

4 Hence Q ( x) f ( x) + f ( x + a a) (4) P ( x) Q( x) + R (x) (5) σ ( R(, σ ( f ( denote respectively the variance of R (x), f (x). Inserting Eq. (4) into Eq. (3) gives R( z + a) z + ( j) a) + z + a z + + [ z) + ( ) z + a) + + n( z + ( ) a)] [ z + a) z + a) z + a) (6) + ( ( + )) z + ( + ) a) + + z + ( ) a) + z + ( ) a)] n '( z + ia) for i, z < a are independent rando variables with zero ean and variance σ. So R ( z + a) for, z < a are independent rando variables with zero ean and variance as: σ ( R( z + a)) [ ( ( + )) + + (7) + ]σ σ ( R( σ ( R( + a)) (8) here is an error in []. he error is σ ( R( + a)) [ + O () ] σ. 6 In fact, we can get the exactly result Eq. (9) fro Eq. (8). σ ( R( σ ( R( + a)) σ [( ) * + ( ) * +. + ( ( )) *( ) ] 3 96

5 σ [( * + * +. + *( ) + *( ) ) 3 (* + * +. + ( ) *( ) + ( ) *( ) )] σ 3 3 [( * ( ( ) ) ( ( ) 3 )] 3 Siplifying the above forula and getting Eq. 9. σ ( R( ( ) σ (9) 3 Inserting Eq. (4) into Eq. (4) gives Q ( z + a) [ f ( z + a) f ( z + a)] () σ ( Q( ( f ( σ () Eq. (5), Eq. (9) and Eq. () iply that σ ( P( σ ( R( + σ ( Q( ( ) σ + ( f ( 3 σ σ ( f ( σ [ + ( )] N σ [ + O( )] 3 σ 3 σ. 3 a Finally, we get the estiation forula Eq. (3) 3a σ σ ( P( N ( ) '( ) + (3) P x g x ja g'( x a + a P(x) is the result of superposition operation on the difference of the otion-blurred iage. We copute σ ( P( and get the noise variance σ fro equation (3). he relative error e r can be obtained as follows: σ ( f ( e r ( ) (4) σ 96

6 he relative error e r is O ( ) O ( ) in [] is wrong.. I f is large enough, er is nearly zero. hat e r is 3. Siulation In order to validate the algorith, we have carried out siulation experients and repeated the experients in []. We use the iages Lena (5x5), Caeraan (56x56), Peppers (384x5) and the gray value is fro to. Siulation iages about Lena are shown in Fig.. he results about noise variance estiation are shown in able. able proves that the estiation ethod is considerably accurate. able Variance of white Gaussian noise in iages iage Lena (5x5) Caeraan (56x56) Peppers (384x5) Blur length rue variance Estiation of variance % % % % % % % % % % % % Relative error 963

7 (a) Original iage (b) Blur length 9, noise variance. (c) Blur length 9, noise variance.5. (d) Blur length 7, noise variance.5 Fig. Original iage and noisy otion-blurred iages of different blur length and noise variance about Lena. 4. Conclusions he paper corrects istaes in [] and puts forward an accurate estiation forula for noise power of linear otion-blurred iages. he estiation forula relative error is very sall. Experients prove that the noise power can be coputed accurately by utilizing this ethod. Disadvantage of this approach is the need to nown blur length. 964

8 References. Wei-Guo He, Estiation of the Noise Variance of Unifor Linear Motion Blurred Iages, Inforatics in Control, Autoation and Robotics (CAR), nd International Asia Conference on (Volue:3 ), pp Rafael C.Gonzalez, Richard E.Woods, Digital iage processing, Beijing: Publishing house of electronics industry, 3. Mar R.Banha,Aggelos.atsaggelos, Digital iage restoration, IEEE Signal Processing Magazine, 997, 4(): Yitzha YitzhaY,ed., Coparison of direct blind deconvolution ethods for otion-blurred iages, applied optics, 999, 38(): Wei-Guo He, Shao-Fa Li, Gui-Wu Hu, Blur identification using an adaptive Adaline networ, in: IEEE International Conference on Machine Learning and Cybernetics, Guangzhou, 8- August M. E. Moghadda and M. Jazad, Blur identification in noisy iages using radon transfor and power spectru odeling, in Proceedings of the th IEEE International Worshop on Systes, Signals and Iage Processing (IWSSIP '5), pp , Chalida, Greece, Septeber Panchapaesan, A. Nishihara, Identification of Piecewise Linear Unifor Motion Blur, June 8 IEICE ransactions on Fundaentals of Electronics, Counications and Coputer Sciences, Volue E9-A Issue 6 Publisher: Oxford University Press 8. M. Ebrahii Moghadda, M. Jazad, Linear otion blur identification in noisy iages using fuzzy sets, EURASIP J. Adv. Signal Process, 7, doi:.55/7/