Adaptive Reciprocal Cell based Sparse Representation for Satellite Image Restoration

Transcription

1 Vol.49 (SoftTech 04),.8-85 htt://dx.doi.org/0.457/astl Adative Recirocal Cell based Sarse Reresentation for Satellite Image Restoration Yanfei He and Shunfeng Wang Deartment of Math and Statistic, Nanjing University of Information Science & Technology, Nanjing 0044, China Abstract. Satellite images are unavoidably corruted by aliasing, blur and noise, leading to the restoration roblem, which is usually an ill-osed inverse roblem. To address the roblem, various regularization methods have been roosed in the ast decades. Among them, the sarse reresentation methods have drawn great attention. In this aer, we utilize adative recirocal cell to analyze the three degradation factors, in order to enhance the erformance of sarse reresentation for satellite image restoration. Exerimental results show that our method can roduce better restored results. Keywords: Image Restoration, Sarse Reresentation, Aliasing Introduction In earth observation, sometimes there are no other images about the scene of interest but a single satellite image, usually corruted by aliasing, blur and noise. Therefore, the image needs to be rocessed to better reflect its radiometric and geometric quality. The rocess is called satellite image restoration. Its goal is reconstruction or recovery of the degraded image using a rior knowledge of image degradation rocess [-3]. Regularly, the discrete degradation model can be reresented by ( h f ) n I + = Γ () where I is the observed (measured) image. f is the natural scene, defined on a continuous suort(a bounded set included in R ). n is the additive Gaussian white h f is the convolution roduct of f by the oint-sread function h, which noise. ( ) is normalized, ositive and symmetric with resect to the x and y axes. The Fourier transform of h is called the MTF. Actually, The effect of the imerfect otical system is characterized by MTF, which is similar to a low-ass filter leading to blurred aearance. is the samling comb comosed of delta-functions δ, Γ Γ = δ () ( Γ i, j ) ISSN: ASTL Coyright 04 SERSC

2 Vol.49 (SoftTech 04) where Γ is the samling grid. It reresents the geometry of the array of sensors, which are assumed to be distributed on a regular grid: where { e,e } is a basis of Γ, e = (, 0) T,and e ( 0, )T 4 { n e + n e : n n Ζ} Γ =, (3) R. For square samling grid δ, f f x dx = f = δ, samling on Γ 4 can =. since ( ) be exressed as simly multilying by Γ 4. The restoration of satellite image is an ill-osed inverse roblem in general[4-7]. In this work, we resent a new adative recirocal cell [8] based image restoration which uses the advanced sare and redundant reresentation technology[8. The adative recirocal cell is emloyed to analyze the degradation factors, such as blurring, noise and aliasing. Then, we introduce the analysis result into the sarse reresentation model. Proosed Method According to signal samling theory, the degradation model can be rewritten as a formulation of Fourier transform as follows: ~ I U ~ ~ ( X ) f ( n~ U k) T a V s = detv = E The traditional TV regularization model [9-] is effective in filtering the noise but tends to smooth the image, esecially the image texture. This is due to the iecewise smoothing constraint. In recent years, the sarse reresentation related methods have achieved romising restoration results [-5]. Because of the high dimensionality of image, sarse reresentation method focuses on small atches of natural images. So the whole image is usually divided into image atches. Each image atch is rocessed indeendently, and the final result image is roduced by stitching and averaging the atches. The sarse reresentation model assume that an image atch s k a (4) f can be arox- n K imately reresented via a vector over a dictionaryφ R (each column in φ is called an atom). Image atch can be aroximately reresented as: f 0 φ, s. t. T (5) 8 Coyright 04 SERSC

3 Vol.49 (SoftTech 04) where is a seudo norm that counts the number of nonzero items in vector 0. (5) indicates that the sarse coding of f can be calculated by solving the l 0 minimization roblem. As the l 0 minimization roblem is an NP roblem, it is often resolved by l roblem which is convex. The related formulation is as follow: { φ + β } ~ = arg min f (6) where constant β is the regularization arameter, and the second term is sarse coding which is the sarse aroximation rocess of f. In the view of image restora- tion, based on the (), to recover f from I, f can be sarsely reresented by solving the roblem: { ( φ ) + β } ~ = arg min f h (7) Γ It is exected that ~ could be close enough to. But due to the degradation factors, esecially the aliasing, the restoration task is very challenging. Few sarse reresentation methods fully considered the aliasing, or rather analyzed the aliasing during their rocessing. Here, we emloy the aliasing analysis tool, i.e. adative recirocal cell, to enhance the erformance of sarse reresentation for image recovery. We can rewrite the (7) as follow: where ( ) { ( f ) F( h φ ) + β } ~ = arg min F (8) F stands for Fourier Transform. In(8), the data-fitting term is defined on the suort region of adative recirocal cell so that the model can effectively describe the aliasing. Ω arc 3 Exerimental Results We alied restoration method to both simulated corruted images. In the simulated image restoration, the Gaussian function with standard deviation is used. Then, Additive Gaussian noise with level is added to the blurred image. After this, we the rocessed image is down samling. The basic arameter setting of our roosed method is as follows: the atch size is 9 9 and K = 0. In the exeriment, the roosed method is comared with the TV model. As show in Fig.. Fig.(a) is original image,(b) is degraded image, (c) is the image which is record by adative Recirocal Cell, (d) is the restoration result of TV model and (e) is our method. We can see that the result of our method is clear and natural. Coyright 04 SERSC 83

4 Vol.49 (SoftTech 04) (a) Original Image (b) Corruted Image (c) Adative Recirocal Cell (d) TV model (e) Proosed method Fig.. Simulated Image Restoration 84 Coyright 04 SERSC

5 Vol.49 (SoftTech 04) 4 Conclusions During the acquiring rocess, traditional satellite images are unavoidably corruted by blur, noise and aliasing. It is well known that the restoration of satellite image is an ill-osed inverse roblem. Accordingly, this ticklish roblem requires regularization to avoid unstable solutions. Recently sarse reresentation has been roved to be effective in image restoration we roosed an adative recirocal cell based image restoration that emloys the advanced sare reresentation to imrove the erformance of restoration. Exeriment results show that our method can roduce good quality results. References. A. Jalobeabu, L. Blanc-Feraud, J. Zerubia, satellite image deblurring using comlex wavelet ackets. International Journal of Comuter Vision, 5, 3(003). R. Neelamani, Choi Hyeokho, R.Baraniuk, ForVaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems, IEEE Transactions on Signal Processing, 5, (004) 3. A. Buades, B. Coll, J. M. Morel, A review of image denosing algorithms, with a new one. Multiscale Modeling & Simulation, 4, (005) 4. K. Dabov, A. Foi, V. Katkovnik, K. Egizarian, Image denoising by sarse 3-D transformdomain collaborative filtering. IEEE Transaction on Image Processing, 6, 8(007) 5. T. Peleg, Y. Eldar, M. Elad, Exloiting statistical deendencies in sarse reresentations for signal recovery, IEEE Transactions on Signal Processing, 60, 5(0) 6. A. A. Zohair, M. Dzulkifli, S. M. R. Mohd, S. L. Ghazali, Restoring degraded astronomy images using a combination of denoising and deblurring techniques, International Journal of Signal and Image Processing, 5, (0) 7. J. Yang, J. Wright, T. Huang, and Y. Ma, Image suer-resolution via sarse reresentation, IEEE Transactions Image Processing, 9, (00) 8. A. Andres, D.Sylvain, R.Bernard, Measuring and imroving image resolution by the adatation of the recirocal cell, Journal of Mathematical Imaging and Vision,,3 (004) 9. A.Chambolle, An algorithm for total variation minimization and alications. Journal of Mathematical imaging and vision. 0, - (004) 0. A.Chambolle, Image Recovery via Total Variational Minimization and Related Problems,Numerische Mathematik,76, (997). A. Andres, C. Vicent, H. Gloria, R. Bernard, Restoration and zoom of irregularly samled blurred and noisy images by accurate total variation minimization with local constraints. Multiscale Modeling & Simulation, 5, (006). M. Elad, M. Aharon, and A.M. Bruckstein. K-SVD: An algorithm for designining of overcomlete dictionaries for sarse reresentation. IEEE Transactions on Signal Processing, 54, (006) 3. J. Mairal, M. Elad, G. Sairo, Sarse reresentation for color image restoration. IEEE Transactions on image rocessing, 7, (008) 4. W. S. Dong, L. Zhang, G. G. Shi, X. Li, Nonlocally centralized sarse reresentation for image restoration, IEEE Transactions on Image Processing,,4 (03) 5. M.Elad, I. Yavneh, A lurality of sarse reresentations is better than the sarsest one alone, IEEE Trans.Image Process, 55, 0 (009) Coyright 04 SERSC 85