MULTICHANNEL image processing is studied in this

186 IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 7, JULY 1999 Vector Median-Rational Hybrid Filters for Multichannel Image Processing Lazhar Khriji and Moncef Gabbouj, Senior Member, IEEE Abstract In this letter, a new class of nonlinear filters called vector median-rational hybrid filters (VMRHF s) for multispectral image processing is introduced and applied to the color image filtering problem. These filters are based on rational functions (RF s) offering a number of advantages. First, a rational function is a universal approximator and a good extrapolator. Second, it can be trained by a linear adaptive algorithm. Third, it produces the best approximation (w.r.t. a given cost function) for some specific functions. The output is the result of a vector rational operation over the output of three subfilters, such as two vector median (VM) subfilters and one center weighted vector median filter (CWVMF). These filters exhibit desirable properties, such as edge and details preservation and accurate chromaticity estimation. Index Terms Color image filtering, rational functions, vector median filters, vector median-rational hybrid filters, vector rational filters. I. INTRODUCTION MULTICHANNEL image processing is studied in this work using a vector approach [7], which is more appropriate compared to traditional approaches that use instead componentwise operators. This is due to the inherent correlation that may exist between the image channels [7]. In vector approaches, each pixel value is considered as an -dimensional vector ( is the number of image channels; in the case of color images, ), whose characteristics, i.e., magnitude and direction, are examined. The vector direction signifies the chromaticity, while the magnitude is a measure of the pixel intensity. A number of vector processing filters usually involve the minimization of an appropriate error criterion [1], [9]. One class of filters considers the distance in the vector space between the image vectors; typical representative of this class is the vector median filter (VMF) [1]. A second class of filters operates on vector directions, hence the name vector directional filters (VDF s) [9]. VMF s are derived as MLE estimators for an exponential distribution [1], while VDF s are spherical estimators, when the underlying distribution is a spherical one [9]. VMF s perform well when the noise follows a long-tailed distribution Manuscript received September 16, 1998. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. R. M. Mersereau. L. Khriji is with the Department of Electrical Engineering, ENIM, CP 5019 Monastir, Tunisia (e-mail: lazhar@cs.tut.fi). M. Gabbouj is with the Signal Processing Laboratory, Tampere University of Technology, FIN-33 101 Tampere, Finland (e-mail: moncef@cs.tut.fi). Publisher Item Identifier S 1070-9908(99)04093-6. (e.g., exponential or impulsive); moreover, outliers in the image data are easily detected and eliminated by VMF s. A third class of filters operates on the input vectors using rational functions, hence the name vector rational filters (VRF s) [4]. There are several advantages to the use of this function. Similar to a polynomial function, a rational function is a universal approximator (it can approximate any continuous function arbitrarily well); however, it can achieve a desired level of accuracy with a lower complexity, and possesses better extrapolation capabilities compared to a polynomial function. Moreover, it has been demonstrated that a linear adaptive algorithm can be devised for determining optimal parameters for this structure [6]. In this letter, a novel filter structure is introduced, the vector median-rational hybrid filter (VMRHF), which constitutes a natural extension of the nonlinear rational type hybrid filters called median-rational hybrid filters (MRHF) recently introduced for one-dimensional (1-D) and two-dimensional (2-D) signal processing [3], [5], based on rational functions. The VMRHF is formed by three subfilters (in which two vector median filters and one center weighted vector median filter) and one vector rational operation. The VMRHF filters are very useful in color (and generally multichannel) image processing, since they inherit the properties of their ancestors. They constitute very accurate estimators in long- and shorttailed noise distributions and, at the same time, they preserve the local chromaticity of a color image. Moreover, they are local operators acting on relatively small windows with few number of operations, resulting in simple and fast filter structures. II. RATIONAL AND VECTOR RATIONAL FUNCTIONS A rational function is the ratio of two polynomials. To be used as a filter, it can be expressed as where are the scalar inputs to the filter and is the filter output, and are the filter parameters. The representation described in (1) is unique up to common factors in the numerator and denominator polynomials. The rational function (RF) must clearly have a finite order to be useful in solving practical problems. Like polynomial (1) 1070 9908/99$10.00 1999 IEEE

KHRIJI AND GABBOUJ: HYBRID FILTERS FOR MULTICHANNEL IMAGE PROCESSING 187 Fig. 1. Structure of VMRHF using bidirectional subfilters. (a) functions, an RF is a universal approximator [6]. Moreover, it is able to achieve substantially higher accuracy with lower complexity and possesses better extrapolation capabilities than polynomial functions. Straightforward application of the RF s to multichannel image processing would be based on processing the image channels separately. This however, fails to utilize the inherent correlation that is usually present in multichannel images. Consequently, vector processing of multichannel images is desirable [7]. The generalization of the scalar rational filter definition to vector-valued signals is given by the following definition. Definition 1: Let be the input vectors to the filter, where and, is an integer. The VRF output is given by VRF RF where is a vector-valued polynomial and is a scalar polynomial. Both are functions of the input vectors. The th component of the VRF output is written as where and is the -norm, and integer part of,., are constant, and is a function of the input vectors When the vector dimension is one, the VRF reduces to a special case of the scalar RF [4]. Fig. 2. (a) Test Lena image. Contaminated image by mixed noise [impulsive 2% in each channel and Gaussian N (0; 50)]. III. VECTOR MEDIAN-RATIONAL HYBRID FILTERS When extending the median-rational hybrid operation to vector-valued signals, we place the following requirements for the resulting vector median-rational hybrid operation. The operation should have properties similar to those of the scalar case. There exist root signals. There is good robust data smoothing ability for different i.i.d. noise distributions (Gaussian, impulsive, mixed Gaussian-impulsive), while retaining sharp edges in the signal. It reduces to the scalar filter if the vector dimension is one. 1) Vector Median-Rational Hybrid Filters: Let, represent a multichannel signal and let be a window of finite size (filter length). represents the signal dimension and the number of signal channels. The pixels in will be denoted as, and will be denoted as. are -dimensional ( ) vectors in the vector space defined by the signal channels. The VMRHF is defined as follows.

188 IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 7, JULY 1999 Definition 2: The output vector of the VMRHF is the result of a vector rational function operating on three suboutputs, where ( ) is a center weighted vector median subfilter (3) (. a) where is an -vector norm, is a constant weight vector for the subfilter outputs. and are some positive constants. The parameter is used to control the amount of the nonlinear effect. In this approach, we have chosen a very simple weight vector that satisfies the condition:. In our study,. The subfilters and are chosen so that an acceptable compromise is obtained between noise reduction and edge and chromaticity preservation. It is easy to observe that this VMRHF differs from a linear lowpass filter mainly in the scaling, which is introduced on the and terms. Indeed, the scaling factor is proportional to the output of an edge-sensing term characterized by -vector norm of the vector difference between and. The weight of the vector median-operation output term is modified accordingly in order to keep the gain constant. The behavior of the proposed VMRHF structure for different positive values of parameter is as follows. 1), the form of the filter is given as a linear lowpass combination of the three nonlinear subfilters: where the coefficients,, and are some constants. 2), the output of the filter is identical to the central subfilter output and the vector rational function has no effect: (4) (5) 3) For intermediate values of, the term perceives the presence of a transition and accordingly reduces the smoothing effect of the operator. Therefore, the VMRHF operates as a linear lowpass filter between three nonlinear suboperators, the coefficients of which are modulated by the edge-sensitive component. The proposed structure of the VMRHF is shown in Fig. 1 using two bidirectional vector median subfilters. The center weighted vector median filter has the following filter weights corresponding to the plus-shaped mask (c) Fig. 3. Comparative results from the original image in Fig. 2 (left) contaminated by (a) Gaussian, impulsive, and (c) mixed noise (salt and pepper 2% in each component). IV. EXPERIMENTAL RESULTS The performance of VMRHF has been compared against that of some widely known multidimensional nonlinear filters, such as VMF and DDF, using RGB color images as test data. The noise attenuation properties of the different filters are examined by simulations using color image Lena [see Fig. 2].

KHRIJI AND GABBOUJ: HYBRID FILTERS FOR MULTICHANNEL IMAGE PROCESSING 189 (a) (c) Fig. 4. Results on the Lena image [Fig. 2]. (a),, and (c) are the processed images by the DDF, VMF, and VMRHF, respectively. The test image has been contaminated using various noise distributions in order to assess the performance of the filters under the following different scenarios. 1) Gaussian noise: Zero mean additive Gaussian noise with standard deviation,( is variable). 2) Impulsive noise: Each image channel is corrupted independently using salt and pepper noise. We assume that both salt and pepper are equally likely to occur. 3) Mixed Gaussian-impulsive noise: The impulsive noise (salt and pepper 2% in each image channel) is mixed with Gaussian noise with various standard deviations. The original image, as well as its noisy versions, are represented in the RGB color space and filtering is also performed in the same color space. A number of different objective measures can be utilized for quantitative comparison of the performance of the different filters. All of them provide some measure of closeness between two digital images by exploiting the differences in the statistical distributions of the pixel values [2]. The most widely used measures are the mean absolute error (MAE), the mean square error (MSE), and the normalized color difference (NCD). The latter measure is used to quantify the perceptual error between images in the perceptually uniform color space [8]. The results obtained are shown in the form of plots in Fig. 3(a) (c) for the three noise models: Gaussian, impulsive, and Gaussian mixed with impulsive, respectively. As can be seen from the plots, the performance of the new VMRHF is superior to that of VMF and DDF. Moreover, consistent results have been obtained when using a variety of other color images and the same evaluation procedure. The filtered images are presented in Fig. 4 for subjective comparison. Fig. 4(a) (c) are the filtered images of the corrupted image in Fig. 2 (right) by mixed noise [impulsive 2% in each channel, Gaussian ], using DDF, VMF, and VMRHF, respectively. All the filters considered operate using a 3 3 square processing window. A comparison of the images clearly favors the new VMRHF over their counterparts (VMF and DDF). The proposed VMRHF can effectively remove impulses, smooth out nominal noise and keep edges, details and color uniformity unchanged. Considering the number of computations required for the implementation of the VMRHF, it should be noted that it is comparable to that of VMF. The vector rational operation does not introduce significant additional computational cost, (a small look-up table for the denominator, one multiplication, three additions and one division per output sample). V. CONCLUSION A new class of nonlinear vector rational type hybrid filters for multichannel image processing is introduced in this paper. The vector median-rational hybrid filter is a vector rational operation over three subfilters, in which the central one is a central weighted vector median filter. These filters combine the good behavior of rational functions and vector median filters. Simulation results and subjective evaluation of the filtered images indicate that the VMRHF outperform all other filters used in the study. Moreover, as it can be seen from the processed images, VMRHF preserve well the chromaticity of the color image. REFERENCES [1] J. Astola, P. Haavisto, and Y. Neuvo, Vector median filter, Proc. IEEE, vol. 78, pp. 678 689, Apr. 1990.

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