Chen, K

Williams, BM

University of Liverpool Repository

 Bryan M. Williams & Ke Chen • University of Liverpool bryan@liv.ac.ukIn image deblurring, our aim is to remove the blur and noise degradation from a given corrupted im-age in order to find the hidden true image.  Filtering Approach Given the discrete form  z = h u + η of the forward problem, we attempt to recover the image by in-verting the blur function h:                      h = V diag ( si ) WΤ                      h-1 z = u + ∑ si-1 ( viΤ η ) wi  In order to avoid division by small singular values, we employ a regularisation filter function, thereby solving               u = ∑ ωα ( si2 ) si-1 ( viΤ z ) wi  ,   e.g. ωα ( s2 ) = s2 / ( s2 + α ).   Variational Approach In order to better cope with noise, we take a variational approach to recovering the image. We aim to minimise the below energy functional, composed of data fitting and smoothing, with respect to the image u. Using this, we can recover a good approximation of the true image.         F ( u ) = ʃΩ  ( [ h ◦ u ] ( x ) - z ( x ) ) 2  + α Ψ ( | u ( x ) |2 )  dx               (2)                                                  Fitting term                                    Regularisation term  where x = ( x1 , x2 )  and α is a small, non-negative parameter which measures the trade-off between data fitting and smoothness. We solve the following PDE resulting from the minimisation of (2).                   h†  ◦ ( h ◦ u - z ) - α · ( Ψ ( | u|2 ) u) = 0                                                                             where  h†  is the adjoint  h† ( x ) = h ( -x ). [1]  B. M. Williams, K. Chen, Y. L. Zheng & S. P. Harding. An enhanced model for semi-blind deblurring with parametric kernel identification and constrained image reconstruction. Submitted. 2014. [2]  A. Levin. Blind motion deblurring using image statistics. Advances in Neural Information Processing Systems (NIPS). 2007. [3]  R. Liu and J. Jia. Image partial blur detection and classification. CVPR. 2008. [4]  T. F. Chan & C.-K. Wong. Total Variation Blind Deconvolution. IEEE T. Image Process, 7:370—375, 1998. [5]  B. M. Williams, K. Chen & S. P. Harding. A new constrained total variation deblurring model and its fast algorithm. Numerical Algorithms. 2014. Blurring of images occurs in many fields, causing significant problems in retinal imaging. In any dia-betic retinopathy screening programme, up to 10% of the images are ungradeable due to inade-quate clarity or poor field definition. It is important to obtain as much information as possible of retinal vessels and other structures. Deblurring is a major technique which may be developed to restore the lost true image. We present some approaches to tackle this problem. In order to derive a model for image deblurring, we consider the forward problem. We assume that the problem of blur and noise corruption of an image can be modelled as                                 z ( x , y ) = [ h ◦ u ] ( x , y ) + η ( x , y )                         (1) where we assume that the blur is linear special invariant and so can be modelled by the convolu-tion operation of a point spread function (psf) and the true, uncorrupted image:                     [ h ◦ u ] ( x , y ) = ʃ h ( x - x’ , y - y’ ) u ( x’ , y’ ) dx’ dy’ and we have  z ( x , y ) is the received image  h (x , y ) is the blur function which may be known or unknown  u (x , y ) is the true image which we later wish to reconstruct from the received data  η ( x , y ) represents additive noise  ? What if we don’t know the cause of degradation? = + ◦   Blur Identification Chan and Wong [4] proposed alternate minimisation of the blur function and image. Incorporating the ideas of [5], we reformulate this in an implicitly constrained way as the minimisation of the en-ergy functional:              f ( ψ , h ) = ʃ ( h ( x ) ◦ τa ( ψ ( x ) ) - z ( x ) ) 2 + θ ( ψ - ζ* ) 2                                     + α1 Ψ ( | τa ( ψ ( x ) ) | 2 ) + α2 Ψ ( | h ( x ) | 2 ) dx.    (3) where τa : R → Ra is a function whose range is a subset of R determined by a. This can be used to constrain intensity values to improve the quality of the result. Smoothness is considered for both the blur function h and the image  τ ( ψ )  and  α1 , α2  are regularisation parameters. Solving this problem, we simultaneously reconstruct the blur function and the image. In order to minimise this functional, we make initial estimates of the image and point spread function and, fix-ing the image, we minimise (3) with respect to the psf. Then, keeping the psf fixed, we minimise (3) with respect to the image. In the likely event that we do not know the cause of the blur, we may still be able to obtain some a priori infor-mation from the image. It may be clear that the blur was caused by motion of the camera or incorrect focus, in which case we may use semi-blind deblurring [1] using a parametric representation of the blur function. The problem of deblurring an image with no knowledge of or assumptions about the blur function is known as blind deblurring. To solve this problem, we may attempt to identify the blur using sophisticated blur identifica-tion methods [2,3] or by simultaneous recovery of the image and point spread function [4]. Much recent research has been targeted at identifying blur in an image by considering its statistics. Such methods typically attempt to distinguish blurred and sharp regions while others aim to identify the psf [4,5]. For example, log-gradient his-tograms show certain trends in blurred/ sharp regions. Inadequate quality Super-resolution Using deblurring, we aim to give a high resolution image from one or more low resolution images. Focus Fusion Given several images of the same scene, we use blur identification to au-tomatically piece togeth-er the in-focus regions to obtain a single image which is in-focus all over.   Retina Colour Fundus Imaging Blur is a major issue with oph-thalmic imaging, requiring deblurring techniques. Excellent quality 

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