An improved non-blind image deblurring methods based on FoEs Qidan Zhu, Lei Sun College of Automation, Harbin Engineering University, Harbin, 5000, China ABSTRACT Traditional non-blind image deblurring algorithms always use maximum a posterior(map). MAP estimates involving natural image priors can reduce the ripples effectively in contrast to maximum likelihood(ml). However, they have been found lacking in terms of restoration performance. Based on this issue, we utilize MAP with KL penalty to replace traditional MAP. We develop an image reconstruction algorithm that minimizes the KL divergence between the reference distribution and the prior distribution. The approximate KL penalty can restrain over-smooth caused by MAP. We use three groups of images and Harris corner detection to prove our method. The experimental results show that our algorithm of non-blind image restoration can effectively reduce the ringing effect and exhibit the state-of-the-art deblurring results. Keywords: image deblurring, MRF, FoEs, KL divergence, MAP.. TRODUCTO Deblurring technology is to restore the sharp explanation from the blurring image. n this paper, we consider the nonblind image restoration method which is assuming that the PSF is known. Even the PSF is known, image restoration is an arduous task. Conventional non-blind algorithms involve maximum likelihood(ml) and maximum a posterior(map). The ML approaches are the simplest in the Bayesian inference and ignore the prior information. The deblurring result is often not unsatisfactory. Because the MAP approaches take into account the role of a prior information, they are applied more generally []. Recently, the prior model is studied extensively [-4] and it is used in the fields of super-resolution reconstruction [5,6], denoising [7], inpainting [8] and deblurring [9-]. Most techniques rely on natural image prior based on Markov random fields(mrf). Although MAP approaches take into account the prior information, they also have shortcomings. The MAP approaches always exhibit over-smooth. n this paper, we utilize MAP with KL penalty to solve the over-smooth issue caused by MAP estimation. And the FoEs as the prior is used in this paper which exhibits state-of -the-art results in denoising and inpainting. n the last, we use three groups of images to demonstrate our method. The remaining of this paper is organized as follows: mage deblurring used ML is in section, Generic image model is in section 3, MAP with KL penalty is in section 4, experimental Results is in section 5, conclusion is in section 6.. MAGE DEBLURRG USED ML The blurring image O is assumed to have been generated by convolution of a blur kernel H with latent image plus noise. Where is convolution, is additive noise. The ML term is: Assume that the noise is Gaussian distribution. O= H + () % = arg max PO ( ) () Fifth nternational Conference on Machine Vision (CMV 0): Computer Vision, mage Analysis and Processing, edited by Yulin Wang, Liansheng Tan, Jianhong Zhou, Proc. of SPE Vol. 8783, 87830G 03 SPE CCC code: 077-786X/3/$8 doi: 0.7/.034 Proc. of SPE Vol. 8783 87830G- Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/0/03 Terms of Use: http://spiedl.org/terms
( O H ) PO ( ) = exp πσ σ (3) The gradient ascent approach is always used in the Bayesian inference. So the gradient of ln-prior is: ln PO ( ) = H ( O H ) (4) σ With the iteration η,we can write the deblurring algorithm as: Where H ( xy, ) = H( x, y). = + η ( ) (5) σ n+ n H O H We deblur the image b to get the image c. But there are some ripples in the image c. t demonstrates that the prior information is very important. a b c Figure. ML deblurring results 3. GEREC MAGE MODEL mage pixels change very slowly except edge region. The image has a strong correlation between neighboring pixels. Markov random field is a very popular model in image processing. t can exhibit the image smoothness. As the equivalence of MRF and Gibbs distribution, the MRF is widely used. The Gibbs distribution equivalence term of MRF is: Where Z is the partition function. U( X ) and V ( X ) is the potential function. PX ( ) = exp( UX ( )) = ( Vc ( X)) (6) Z Z c C c Recently, the FoEs proposed by Stefan Roth and Michael J. Black is a new term of MRF to explain the image spatial structure information [3]. T PX ( ) = φi ( J i X( k) ; αi) (7) Z( Θ) k i= Where J is linear filter, α is expert parameter. φ i is expert function,we always choose student-t function as expert function. Proc. of SPE Vol. 8783 87830G- Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/0/03 Terms of Use: http://spiedl.org/terms
The gradient of the ln-prior is: φ T T ( J X ) ( ) ( ); α = + J X i i k i i αi (8) where ψ ( y) = yln φ ( y; α ), J i is the mirror of J i. i i i With the iterationη, we can write the gradient ascent algorithm as: ln P ( ) = J i ψ i(j i ) (9) i= J (J ) ( ) (0) n+ n = + η i ψi i + H O H i= σ We deblur the image a to get image b. Although the image b does not have ripples, according to the specific details of the image c, it exhibits over-smooth. a b c Figure. Contrast different methods. 4. MAP WTH KL PEALTY To solve the over-smooth issue, we utilize MAP with KL penalty to replace traditional MAP. We develop an image reconstruction algorithm that minimizes the KL divergence between the reference distribution and the prior distribution. The distance penalty plays the role of a global image that steers the solution away from piecewise smooth image. Let qp ( x ) be a distribution of image % and qd ( x ) be a prior distribution. We use the KL divergence to measure between the two distributions. qp () x KL{ qp() x qd()= x } qp()ln x dx () x qd () x The qp ( x ) is parameterized using a FoEs models. Given image sample we can estimate the parameters α,j by contrasitive divergence. ow we can add the KL penalty to the MAP estimator: O H % = arg min + λ P( ) + λkl( qp qd) () σ Where the λ, λ are the weight parameters. Because the KL divergence calculation is intractable, we use the approximation of () and estimate KL divergence as: Proc. of SPE Vol. 8783 87830G-3 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/0/03 Terms of Use: http://spiedl.org/terms
qp () x KL( qp qd) qa( ) = ln( ) i i qd () x (3) n this algorithm, the KL penalty favors the distribution phenomenon in the image restoration. q to q, so it can efficiently restrain over-smooth P D 5. EXPERMETAL RESULTS This paper utilizes three groups of images to demonstrate our method. The PSF in the first group is (size 0 0, σ 0)Gauss kernel. The PSF in the second group is(size 5 5, σ 0) Gauss kernel. The PSF in the third group is motion kernel. We use Lucy-richardson and MAP-FoEs methods to contrast our methods. The results are summarized in Tab. using the common peak signal-to-noise ratio (PSR) and it demonstrates the proposed method is better than other methods. _ L ïf a blurring image b LR c MAP-FoEs d our methods Figure 3. Contrast different methods. Table PSR LR MAP-FoEs Our method photographer 6.048 6.637 9.47 books 9.9438 9.4055 3.0676 Proc. of SPE Vol. 8783 87830G-4 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/0/03 Terms of Use: http://spiedl.org/terms
cup 3.96 33.937 35.359 We also use Harris corner detection in the first group images to demonstrate the advantage of our methods. The numbers of a b c d Harris corners are 8,6,96,30. The numbers also prove our methods are practical. a blurring image b LR c MAP-FoEs d our methods Figure 4. Harris corners detection contrast. 6. COCLUSO On the basis of the study FoEs natural image statistical models, an improved non-blind image restoration algorithm is proposed. But there are some issues. The parameters of expert models are once trained, they are no longer changed. n the future we need to develop better natural image model to solve the above problems. ACOKOLEDGEMET This work is supported by ational atural Science Foundation of China under Grant (o. 675089), the Fundamental Research Funds for the Central Universities(HEUCF043, HEUCF044, HEUCFZ0), The Heilongjiang Province Postdoctoral Sustentation Fund(LBH-Q35). REFERECES [] D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, n PS,-9,(009). [] S.C. Zhu and D. Mumford, Learning generic prior models for visual computation, in Proc. CVPR, 463 469,(997). [3] S. Roth and M. J. Black, Fields of experts: A framework for learning image priors, in Proc. CVPR,, 860 867,(005). [4] S. Lyu and E.P. Simoncelli, Statistical modeling of images with fields of gaussian scale mixtures, in Proc. PS, 945 95,(006). [5] M.F. Tappen, B.C. Russell, and W. T. Freeman, Exploiting the sparse derivative prior for super-resolution and image demosaicing, in Proc.EEE Workshop Statist. Comput. Theories Vis., 4,(003). [6] J. Sun, J. Sun, Z.B. Xu, and H.Y. Shum, mage super-resolution using gradient profile prior, in Proc. EEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit., 8,(008). [7] S. Roth and M. J. Black, Steerable random fields, in Proc. CCV, 8,(007). [8] A. Levin, A. Zomet, and Y. Weiss, Learning how to inpaint from global image statistics, in Proc. CCV,, 305 3,(003). [9] R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, Removing camera shake from a single photograph, ACM Trans.Graph., 5, 787 794, (006). Proc. of SPE Vol. 8783 87830G-5 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/0/03 Terms of Use: http://spiedl.org/terms
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