## Introduction

The useful information in an image is usually degraded by imperfections in the observation process, i.e. blurring due to the band-limited nature of the imaging process (like an optical system), as well as a noise process due to, for example, detector noise [e.g. photon noise in a photo multiplier tube (PMT)]. Image degradation is usually modelled as *g*(*x,y*) = *N*((*h* * *f*)(*x,y*)), with *g*(*x,y*) the blurred image, *f*(*x,y*) the unknown ideal image, *h*(*x,y*) the point spread function (PSF) and *N*( … ) the noise process. The symbol * represents the convolution operator and models the image blurring. The goal of image restoration is to recover *f*(*x,y*) as well as possible from a degraded observed image *g*(*x,y*). When the degradation parameters (in particular the PSF) are known, we are left with a classical image restoration problem (Katsaggelos, 1989; Lagendijk & Biemond, 1991; Bertero & Boccacci, 1998). However, in some cases the degradation parameters are unknown, and one has two choices: estimating the image of interest and the degradation parameters simultaneously (blind restoration, Kundur & Hatzinakos, 1996), or estimating the degradation parameters before starting the restoration process (Katsaggelos, 1989). This paper follows the latter approach.

We present a combined method for degradation estimation and image restoration that is based on steerable pyramids (Freeman & Adelson, 1991; Simoncelli *et al*., 1992; Portilla *et al*., 2003). To our knowledge, it is the first joint approach for degradation estimation and regularized deconvolution with steerable pyramids. The deblurring itself is the only step not performed with steerable pyramids (with deconvolution we refer to the whole restoration process, and with deblurring to that part of the process that sharpens the image). It uses the Richardson–Lucy (RL) algorithm, but with the regularization step in the steerable pyramid domain, it adds a new form of prior knowledge to the restoration problem.

The paper is organized as follows: in Section 2, the outline of our algorithm is discussed; it is divided into the following subsections: in 2.1 we give some background and details on the variant of the wavelet transform, namely the steerable pyramid; in 2.2, the noise reduction is discussed; in 2.3, the estimation of the image blur is explained; in 2.4 the deblurring step is explained; and in 2.5, a stopping criterion is formulated. In Section 3, some experimental results are shown and discussed. Finally, a conclusion is given in Section 4.

To illustrate how our research evolved, we refer to preliminary results about the PSF estimation in Rooms *et al*. (2001, 2002) and to results on the estimation/restoration technique in Rooms *et al*. (2003a,b). Here, the full restoration and estimation technique is described, and evaluated on synthetic and real confocal images, using colocalization analysis as an objective criterion to evaluate the restoration performance of the microscopic images.