Semi-adaptive, convex optimisation methodology for image denoising T. Ka ¨ rkka ¨ inen and K. Majava Abstract: An optimisation methodology based on a semi-adaptive, convex (SAC) formulation for the image denoising problem is proposed for recovering both sharp edges and smooth subsurfaces from a given noisy image. Basic steps to realise an image denoising algorithm with proper restoration properties and practical computational efficiency with automatic determination of free parameters are described. A set of example images is used to illustrate the proposed approach. 1 Introduction Image denoising is a fundamental task in image processing. In various applications of computer vision, image proces- sing is usually started by removing or reducing noise and other distortions from the digital image. In many appli- cations, especially if edge detection or segmentation is required, the crucial task in the denoising process is the preservation of sharp edges [1, 2]. A common drawback of standard denoising methods, like filtering using Fourier or wavelet transforms or statistical methods [3, 4], is that they are linear and hence smear sharp edges. Recently, several adaptive non-linear denoising methods based on wavelets have been proposed to better preserve the edges of the image ([5, 6] and the references therein). Other efforts in this direction have been made, for example in [7], where an adaptive median filter was proposed and in [8], where the method presented was based on the psychophysical phenomenon of human visual contrast sensitivity. In this paper, we discuss optimisation-based techniques for image denoising. They have proven to be very efficient in preserving edges, especially the method based on total (or bounded) variation (TV) [9]. A well-known drawback of this method is, however, that the denoised image contains a staircase-like structure, which is not optimal for images with smooth subsurfaces. A lot of effort has recently been made in order to decrease the staircase effect of the TV method [10-15], and that is also the purpose of this paper. Our approach, based on a semi-adaptive, convex (SAC) optimisation formulation, performs a kind of post- processing of the TV result, which reduces the staircase effect while preserving the sharp edges. Let us assume that a noisy image, denoted by z, results from a degradation of the form z ¼ z  þ  in O where z  is the true image,  represents a random noise, and O  R d is the image domain. A basic family of optimisation-based image denoising techniques can be given as follows. To recover a smoothed image u from the noisy data z, consider the minimisation of the following cost functional: JðuÞ¼ 1 2 Z O ju  zj 2 dx þ g s Z O jHuj s dx ð1Þ where 1  s  2: Here, O  R 2 is a rectangle and g > 0 is a regularisation parameter. In the case s ¼ 1; the term R O jHuj dx denotes the total variation of u and hence yields the TV denoising model. Several solution algorithms have been proposed for solving the TV problem [16-18]. In [19], the global TV denoising model was localised and a digital TV filter was proposed. The TV term does not penalise discontinuities in u, thus allowing one to recover sharp edges of the original image. However, the theoretical analysis in [20, 21] and numerical experiments, [11, 12, 16, 22] show that the denoised image due to the TV formulation contains a staircase-like structure, which is not optimal for images with smooth subsurfaces. On the other hand, having s > 1 recovers smooth surfaces better but smears sharp edges of the image [11, 23, 24]. Hence, some form of adaptivity is needed for an improved image restoration capability. Adaptive optimisation formulations for image restoration have been considered [10-14]. As soon as adaptivity is required, formulations tend to become much more compli- cated. Adaptive formulations are often non-convex or non- smooth (or even both), which makes the solution process complicated, and in many cases the number of unknowns and free parameters in these formulations is increased. Closest to our approach is the adaptive formulation [10], where the idea was to use a TV-like regularisation (s ¼ 1 in (1)) near edges, smooth regularisation ðs ¼ 2Þ in flat regions, and values 1 < s < 2; between. The exponent s was chosen to be a gradient-driven function s ¼ sðjHujÞ; which made the formulation non-convex and hence its solution non-unique. The SAC method proposed in this paper is a simplifica- tion of the method [10]. The SAC formulation is not purely adaptive but always strictly convex and sufficiently smooth that well-known solution methods, especially the conjugate gradient method, can be applied as part of the solution process. The principal part of the algorithm, however, relies on the fast active-set method for solving the TV-regularised image restoration problem, as described in [16]. Concerning the proposed approach, we also emphasise that using the heuristic determination of the regularisation parameter, as q IEE, 2005 IEE Proceedings online no. 20041148 doi: 10.1049/ip-vis:20041148 The authors are with the Department of Mathematical Information Technology, University of Jyva ¨skyla ¨, P.O. Box 35 (Agora), FIN-40014 University of Jyva ¨skyla ¨, Finland E-mail: majkir@mit.jyu.fi Paper first received 10th February and in revised form 6th September 2004 IEE Proc.-Vis. Image Signal Process., Vol. 152, No. 5, October 2005 553