A numerically efficient minimum variance filter for noise reduction in RGB color images Leopoldo Jetto and Valentina Orsini Abstract— This paper shows the application of a new 2D minimum variance filter to noisy RGB color images. The filter is derived from an image model that can be obtained without any onerous identification procedure of the signal generation process. Moreover the basic modeling assumptions are such that, when applied to RGB color images, they allow the implementation of the algorithm as three independent single-channel 2D minimum variance filters. Each filter has a 2D point-to point recursive structure. This results in a restoration algorithm whose main merits are in terms of simplicity and numerical efficiency. I. I NTRODUCTION Several optimal methods have been proposed for reducing the measurement noise affecting multichannel images and, in particular, color images. These methods include least squares techniques (see e.g.[1] and ref- erences therein), the Wiener filter (see e.g. [2],[3] and references therein) and the Kalman filter (see e.g. [4] -[6], and references therein). All the above methods have the two following common features: i) the restora- tion process incorporates both the within-channel and between-channel correlation of the image, ii) the signal generation process is a priori known or identifiable from the available data. All these papers use a statistical description of the image, so that the first feature allows the filter to work on the basis of an image model carrying a greater statistical information with respect to statistical single channel filters. The second feature concerns the way of obtaining such information. This is a crucial point. The assumption that it is a priori available is unrealistic because the noise-free image is not accessible, hence it must be drawn from noisy data. This means that the parameters of the statistical model can not be estimated within a given precision, thus obtaining a degraded filter performance. More- over, the related multi-channel adaptive filters based on identification-estimation algorithms have an increased complexity and a high computational cost. More re- cently, an adaptive restoration algorithm for noisy RGB images has been proposed in [7]. The algorithm is based on three independent adaptive Kalman filters, and to avoid the crucial problem of estimating the image model, a set of a priori fixed typical parameter values is used. Clearly, there is not any guarantee of fit L. Jetto and V. Orsini are with Universit` a Politecnica delle Marche, Dipartimento di Ingegneria Informatica, Gestionale e dell’Automazione, Via Brecce Bianche, 60131 Ancona, Italy. L.Jetto{vorsini}@univpm.it between the actual image to be filtered and the set of standard parameters. The purpose of this paper is to propose a reduc- ing noise minimum variance filter which has not the above mentioned drawbacks. The filtering algorithm is based on an image model analytically obtained from some general basic assumptions which are an extension of those ones originally proposed in [8]-[10] for monochromatic images. Hence no onerous identi- fication procedure of the signal generation process is required. Moreover the basic modeling assumptions are such that, when applied to RGB color images, they allow the implementation of the filtering algorithm as three independent single-channel 2D minimum vari- ance filters. The present approach results in a computationally simple restoration process whose efficacy in noise re- duction is illustrated by the application to real images. II. THE RGB IMAGE MODEL ASSUMPTIONS The minimum variance filter proposed here derives from an RGB image model directly obtained from the following basic assumptions. Smoothness assumption: each monochromatic single- channel signal of an RGB image is modelled as the union of open, disjoint, homogeneous 2D subregions whose interior is regular enough to be well approximated by a 2D surface of class C ¯ n . Stochastic assumption: all the derivatives of order ¯ n +1 of each monochromatic 2D signal are modelled by means of zero-mean, independent, 2D white Gaussian random fields. Within channel inhomogeneity assumption: for each monochromatic single-channel signal, the random fields rep- resenting the image process relative to different, smooth, homogeneous subregions are independent. Between channel inhomogeneity assumption: the ran- dom fields representing the image process relative to different channels are also considered to be relative to different, smooth, homogeneous subregions . The first two hypotheses are based on the practical consideration that, save at edge points, the signal of most monochromatic images does not show relevant discontinuities, so that it can be well approximated by a 2D surface which is almost everywhere smooth. The points where the smoothness property fails repre- sent the boundary of each smooth (or homogeneous) subregion, namely those locations where sharp signal discontinuities of arbitrary amplitude occur (i.e. the 16th Mediterranean Conference on Control and Automation Congress Centre, Ajaccio, France June 25-27, 2008 978-1-4244-2505-1/08/$20.00 ©2008 IEEE 1652