Mobile person to person speech communication has turned out to be extremely popular with more than one billion users today. Currently, mobile networks are evolving to support multimedia communications. In coming years, access to various sources of multimedia will further enrich mobile communication. New mobile phones, with graphical color displays and integrated cameras, will provide a natural platform for a large set of multimedia applications. Open software architectures in these terminals are giving new possibilities to the developer community. Evolution of the mobile multimedia will create new challenges and opportunities for related research and development activities.

Abstract Mobile person‐to‐person speech communication has turned out to be extremely popular with more than one billion users today. Currently, mobile networks are evolving to support efficient and flexible data communications. This development will also enable an efficient use of multimedia elements in mobile communication. New content types, applications and services are emerging. Multimedia Messaging Service (MMS) is one key element in this development. From the user point of view, it is the continuation of a very popular mobile messaging paradigm and it offers a framework for mobile person‐to‐person multimedia communication. New mobile phones, with graphical color displays and integrated cameras, will provide a natural platform for a large set of multimedia applications. Including multimedia capabilities for small portable devices is driving the new development on display, imaging and processor technologies. Open software architectures in these terminals are giving new possibilities to the developer community. Evolution of the mobile multimedia will create new challenges and opportunities for related research and development activities. Copyright © 2002 John Wiley & Sons, Ltd.

A new image sequence coding technique based on robust median-based predictors is presented for the transmission of image sequences over noisy channels. We analyze the robustness of median-based predictors against channel errors. A heuristic algorithm for the design of a robust predictor from a given median-based predictor is presented. It is shown that with small modifications in terms of a necessary requirement for a median-based predictor to be robust against channel errors, the robustness of a given median-based predictor can be considerably improved. Simulations on a real image sequence show significant improvement over the conventional differential pulse code modulation (DPCM) at high bit error rate (BER) using this new technique. The technique does not increase the transmission rate. It is shown that the quality of reconstructed images obtained by robust median-based predictors can be further improved by postprocessing the image using a nonlinear detail-preserving noise-smoothing filter.

A new method to enhance the quality of a telephone connection by suppressing noise and disturbances in speech signals is presented. As the noise is evenly spread over the entire signal band, a nonlinear filtering algorithm is required to improve the signal-to-noise-ratio. The system is being realized as an analog full custom ASIC using switched capacitor (SC)-substructures and CMOS-technology. The advantages of the analog implementation include low power consumption, small integration area and inexpensiveness of the chip. In this paper, the noise suppression algorithm is described and the SC-circuit implementation is presented.

In this paper, the complex matched median filter (MMF) is developed for QAM signal detection. It is shown that the MMF is robust against impulsive type noise. By combining the MMF and the linear matched filter (LMF), an extended class of matched filters is introduced. These filters combine the desirable properties of MMF and LMF and behave well in varying noise environments. Computer simulations demonstrate that the proposed detectors give a much smaller symbol error probability than the LMF when the noise has an impulsive component and produces only a slight performance degradation in the case of pure Gaussian noise.>

In this paper, nonlinear multivariate image filtering techniques are proposed to handle color images corrupted by noise. First, we briefly review the principle of reduced ordering (R-ordering) and then define three R-orderings by selecting different central locations. Considering noise attenuation, edge preservation, and detail retention, R-ordering based multivariate filters are designed by combining the R-ordering schemes. To implement color image filtering more effectively, we develop them into a locally adaptive version. The output of the adaptive filter is the closest sample to a central location that is a weighted linear combination of the mean, the marginal median, and the center sample. As a result, we study an adaptive hybrid multivariate (AHM) filter consisting of the mean filter, the marginal median filter, and the identity filter. The performance of the two adaptive filtering techniques is compared with that of some nonadaptive ones. The examples of color image filtering show that the adaptive multivariate image filtering gives a rather good performance improvement.

Floating-point arithmetic offers several benefits in digital signal processing. Floating-point A/D- and D/A-converters are needed in the realization of true floating-point signal processing systems. Novel, simple floating-point A/D- and D/A-conversion methods, based on the close resemblance of floating-point and logarithmic number representations, are introduced in this paper. These methods are easy to tailor to different specifications, and they offer a simple trade-off between dynamic range and accuracy.>

The general form of positive Boolean functions (PBFs) corresponding to one weight weighted median (WM) filters, which are a subclass of WM filters, is derived in this paper. Based on the results, it is straightforward to derive the corresponding PBF from a given one weight WM filter or vice versa. The results can also be used to check whether an PBF corresponds to an one weight WM filters. The checking procedure is very simple and tractable due to the fact that linear programming is not needed.>

Earlier research has shown that the problem of optimal weighted median filtering with structural constraints can be formulated as a nonconvex nonlinear programming problem in general. However, its high computational complexity and poor performance due to its nonconvex nature prohibit it from practical applications. In this paper, we shall show that the design problem can be formulated as a convex quadratic programming problem. The new algorithm is very efficient in the sense of computational complexity. The algorithm is also efficient in the sense of its capability to approach the global minimum. Using the algorithm optimal 1-D weighted median filters preserving pulses of length 3, 4 and 5 are tabulated.>

In the paper, a multichannel edge enhancing filter (MEEF) based on the vector median is introduced for enhancing degraded edges in color images. An input multichannel signal is filtered with three subfilters, and the final output is determined by comparing the outputs of the subfilters and their vector median. Root signal and edge enhancement properties of the MEEF are examined in detail. In addition, the authors discuss line preservation and edge jitter resistance of the MEEF in noisy conditions. Finally, the MEEF is tested with: i) blurred edges caused by unfocusing or fast movement of camera, ii) serrated edges caused by interlaced scan, iii) false color edges caused by channel dispersion. The results show that the MEEF outperforms channelwise edge enhancing filtering in enhancing these degraded edges.>

Most current algorithms developed for image sequence filtering require motion information in order to obtain good results both in the still and moving parts of an image sequence. In the present paper, filters which completely preserve stationary regions in image sequences are introduced. In moving regions, the 3D filters inherently reduce to spatial filters and perform well in these areas without any motion-compensation or motion-detection. A new multivariate filtering operation called the alpha-trimmed vector median is proposed. Guidelines for the determination of optimal 3D median-related structures for color and gray-level image sequence filtering are given. Algorithms based on vector median, extended vector median, alpha-trimmed vector median, and componentwise median operations are developed. Properties of the human visual system are taken into account in the design of filters. Noise attenuation and detail preservation capability of the filters is examined. In particular, the impulsive noise attenuation capability of the filters is analyzed theoretically. Simulation results based on real image sequences are given.>

The authors extend the configuration of stack filtering to develop a new class of stack-type filters called parallel stack filters (PSFs). As a basis for the parallel stack filtering, the block threshold decomposition (BTD) is introduced, and its properties are investigated. The design of optimal PSHs under the mean absolute error (MAE) criterion is shown to be similar to the minimum MAE stack filtering theory. The only difference is that one needs now to design more than one stack filter that together construct an optimal PSF. As a result, while reviewing briefly the optimal stack filtering theory, they will put more efforts to demonstrate, via several examples, the improvement by switching from stack filtering to parallel stack filtering for the task of image noise removal.

The authors suggest two interpolative block truncation coding (BTC) image coding schemes with vector quantization and median filters as the interpolator. The first scheme is based on quincunx subsampling and the second one on every-other-row-and-every-other-column subsampling. It is shown that the schemes yield a significant reduction in bit rate at only a small performance degradation and, in general, better channel error resisting capabilities, as compared to the absolute moment BTC. The methods are further demonstrated to outperform the corresponding BTC schemes with pure vector quantization at the same bit rate and require minimal computations for the interpolation.>

A new class of nonlinear filters called neural filters based on the threshold decomposition and neural networks is introduced. Neural filters can approximate both linear finite impulse response (FIR) filters and weighted order statistic (WOS) filters which include median, rank order, and weighted median filters. An adaptive algorithm is derived for determining optimal neural filters under the mean squared error (MSE) criterion. Experimental results demonstrate that, if the input signal is corrupted by Gaussian noise, adaptive neural filters converge to linear filters and that, if corrupted by impulsive noise, optimal neural filters become WOS filters.>

Two interpolative block truncation coding (BTC) schemes with vector quantization (VQ) and median filters as the interpolator are proposed. The first scheme is based on quincunx subsampling and the second one on every-other-row-and-every-other-column subsampling. Compared to the standard BTC, the new schemes yield a significant reduction in bit rate, only a small degradation in quality, and better channel error resisting capabilities. Moreover, the methods are shown to outperform the BTC schemes with pure VQ at the same bit rate and require minimal computations for the interpolation.>

A class of median type filters, called matched median filters, is defined for the estimation of the information which is carried by the amplitude of a noisy signal. These filters form the natural counterpart of linear matched filters in the class of median type filters and are maximum likelihood estimators if the noise is biexponential. Matched median filters are defined both for baseband and passband systems. Their statistical properties are analyzed and simulation results presented.>

Two 3-D median-based filtering algorithms have been developed that preserve the motion in the image sequence while attenuating noise effectively. Some observations are made on the root signals in the binary domain based on the positive Boolean functions corresponding to the filters. From the Boolean expressions the output distribution functions are derived. The performance of both filters under various noise types is examined theoretically and experimentally. The structures are simulated on a video sequencer (DVSR 100) on real image sequences. Comparisons are made with other 2- and 3-D algorithms from the literature based on mean square error, mean absolute error, and subjective criteria.>

Based on the relationship between weighted order statistic (WOS) filters and threshold logic, an algorithm is developed for determining optimal WOS filters under the mean absolute error (MAE) criterion. This algorithm requires much less computation than the adaptive stack filtering algorithm. In addition, experimental results in image restoration demonstrate that the WOS filters obtained by the proposed algorithm can even give better results than the adaptive stack filters.>

Optimal stack filtering under the mean absolute error (MAE) criterion is studied. It is first shown that this problem is equivalent to the classical a priori Bayes minimum-cost decision. Generally, a linear program (LP) with O(b2/sup b/) variables and constraints (b is the window width) is required for finding the best filter. Instead, the authors develop a suboptimal routine which renders the use of the LP obsolete, but yields reasonably good filters. Sufficient conditions under which the proposed routine results in optimal solutions are provided and shown to hold in most practical cases. Several design examples are given.>

The authors propose the use of error feedback for roundoff noise reduction in cascade-form recursive digital filters employing floating-point arithmetic. The floating-point error feedback has a similar mechanism to the fixed-point error feedback. However, the optimal error feedback solution (in the sense of minimum mean-square error) for the floating-point case is different from the corresponding fixed-point case. The formulated optimal feedback coefficients are shown to be independent of the section of ordering and zero-pole pairing, which does not hold in the fixed-point case. Finally, a suboptimal error feedback is proposed and shown to provide a constant output SNR (signal-to-noise ratio) independent of the specific characteristics of the filter except the filter's order. Several examples are given to verify these results.>

A two-step method is presented to synthesize optimal stack filters under the mean absolute error (MAE) criterion. First, the probabilities needed in the optimal filter design are estimated based on images. Second, the linear program (LP) required for finding the best filter is avoided by a 'reasonably good' suboptimal routine which only involves data comparisons. A sufficient condition under which the suboptimal routine results in optimal solutions is given and shown to hold in most practical cases. The proposed method is then applied to synthesize a family of optimal stack filters for the task of restoring an image in impulsive noise. Testing results show that the synthesized filters lead to a greatly improved image-detail restoration compared to standard median filters, although median filters remove noise better.>

A class of median-type filters called matched median filters is defined for the estimation of the information which is carried by the amplitude of a noisy signal. These filters form the natural counterpart of linear matched filters in the class of median-type filters. Matched median filters are defined both for baseband and passband systems, and their statistical properties are analyzed. It is shown that a weighted vector median filter can be used to solve the corresponding estimation problem for complex and bandpass signals.>

A robust approach for estimation of an autoregressive model (with prespecified order) which avoids the data cleaning operation used in previous robust methods is presented. The algorithm is based on the Burg lattice structure, but reflection coefficients are estimated by selecting medians from the reflection coefficient sequences defined by the forward and backward prediction error residuals. It is shown that the algorithm can be derived under the weighted L/sub 1/ norm error criterion. Simulation results demonstrate that the method is more robust than the L/sub 2/ and L/sub 1/ norm methods.>

In this paper several digital video signal processing algorithms that use the median operation are presented. Both numerical and visual evaluations of the resulting image sequences clearly show that the median operation has many desirable properties for digital image sequence processing applications.

In this paper we analyze the statistical properties of discrete morphological filters. These properties are important when we are applying morphological filters to noisy images. Analytical expressions for the output distribution of basic discrete morphological filters are derived using the theory of stack filters.

In the vector median approach the samples of the vector-valued input signal are processed as vectors as opposed to componentwise scalar processing. The VM-type filters utilize the correlation between different components in video signal. This makes the VM-filters attractive in video signal processing. In this paper the performance of vector median filters for color video signals is investigated. Vector median FIR median hybrid filters (VFMH) for cross luma and cross color cancellation are introduced. In the VFMH filter the vector median operation is combined with linear substructures resulting in improved cross color and cross luma attenuation and in very good noise attenuation.

Two nonlinear algorithms for processing vector-valued signals are introduced. The algorithms, called vector median operations, are derived from two multidimensional probability density functions using the maximum-likelihood-estimate approach. The underlying probability densities are exponential, and the resulting operations have properties very similar to those of the median filter. In the vector median approach, the samples of the vector-valued input signal are processed as vectors. The operation inherently utilizes the correlation between the signal components, giving the filters some desirable properties. General properties as well as the root signals of the vector median filters are studied. The vector median operation is combined with linear filtering, resulting in filters with improved noise attenuation and filters with very good edge response. An efficient algorithm for implementing long vector median filters is presented. The noise attenuation of the filters is discussed, and an application to velocity filtering is shown.

The authors introduce vector median-type (VM) filters for complex narrowband signals and investigate basic properties of the VM operation. They present and analyze the root structures of the filters. Sinusoidal signals modulated by root signals of standard median filters are shown to be root signals of VM filters. For actual implementation, the authors use vector FIR (finite impulse response)-median hybrid (VFMH) filters. They show that these filters preserve abrupt changes in amplitude and phase. The filters are shown to remove impulsive and Gaussian noise effectively. Examples are given of VM and VFMH performance for complex signals.>

The authors introduce two classes of finite impulse response (FIR)-median hybrid (FMH) filters called in-place growing FIR-median hybrid (IPG-FMH), and recursive in-place growing FIR-median hybrid (RIPG-FMH) filters. This extended family of filters exhibits a small bias error in the vicinity of a step edge and yet provides good noise attenuation. The new filters are compared with standard median, recursive median, FIR-median hybrid, and recursive FIR-median hybrid filters. Examples of each filter type are given and the behavior of the filters is examined by means of simulations. It is shown that the IPG-FMH filter can be expressed as a weighted FMH (WFMH) filter.>

A class of linear-phase digital filters is discussed for which the filter coefficients are simple functions of the parameters determining the desired frequency response. In the case of lowpass filters, the 6-dB cutoff frequency is the parameter considered. High-pass, bandpass, bandstop, and notch filters can be obtained with simple transformations from a low-pass prototype. The filters are variable over nearly the whole frequency range from zero to half the sampling frequency. Due to direct control of filter coefficients, the length of the filter does not change when it is made variable.>

FIR (finite-impulse response) median hybrid (FMH)-filter-based algorithms for real-time detection are developed. The algorithms are designed so that trends are gradually refined as data become available. Special attention is paid to the detection of sharp edges in trends. The algorithms are based on five- or three-point median operations taken over the outputs of linear subfilters or some other auxiliary outputs. The noise attenuation and the edge preserving abilities of several FMH trend-detection filters are analyzed. The results of the detection show that the in-place growing FMH-filter-based trend detector has significant advantages over the other methods. The trend filtering concept can also be successfully applied to the filtering of the beginning and end of a finite-length data sequence.>

In this paper, we introduce a new class of FIR Median Hybrid (FMH) filters which contain linear FIR substructures to estimate the current signal value using forward and backward prediction. The output of the overall filter is the median of the predicted values and the actual signal value in the middle of the filter window. Predictors maximizing the signal to noise ratio on signal sections described by an 1th order polynomial are derived. The ramp enhancement filters are shown to attenuate the noise on a ramp signal better than the Standard Median (SM) filters. The new predictive FMH filters are shown to have root signals which do not exist for the SM filters, e.g. triangular waves. By combining the level and the ramp enhancement FMH filters, we obtain a filter which attenuates noise on constant and ramp signals.

In this paper, we introduce a new type of nonlinear filters, the Adaptive Median Hybrid (AMH) filters, for the suppression and detection of short duration interferences. In the AMH filters, adaptive filter substructures are used to estimate the current signal value from the future and past signal values. The output of the overall filter is the median of the adaptive filter outputs and the current signal value. This kind of nonlinear filter structure is shown to adapt and preserve rapid changes in signal characteristics well. However, it filters out short duration interferences. By examining the difference between the original and filtered data, interferences can be detected. We introduce two types of AMH filters, the AMH filter with separate adaptive substructures (SAMH) and the AMH filter with coupled substructures (CAMH), which have different convergence properties and implementation. We use both synthetic and real data (speech and electroencephalogram (EEG)) to show the applicability of the proposed filters.

A new class of median type filters for image processing is proposed. In the filters, linear FIR substructures are used in conjunction with the median operation. The root signals and noise attenuation properties of the FIR-median hybrid filters are analyzed and compared to representative edge preserving filtering operations. The concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the FIR-median hybrid filters. In the multilevel filters there exists a tradeoff between noise attenuation and detail preservation. The analysis and examples indicate that FIR-median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the K-nearest neighbor averaging filters.

In this paper we analyze the root structures of the standard median (SM) filters, the recursive median (RM) filters and of the new FIR Median Hybrid (FMH) filters which contain nonrecursive (FIR) substructures. It is shown that with infinite length signals the SM and the RM filters have many oscillatory binary roots. With finite signal lenght the roots of the SM and the RM filters may depend very much on the first and the last values of the sample sequence which are appended to the beginning and the end of the signal. In these situations the new FMH filters are shown to behave better than the SM and the RM filters. If the length of the filter is changed properly almost all the abnormal root structures can be avoided.

A new class of median type filters for image processing is proposed. In the filters linear FIR substructures are used in conjunction with the median operation. The root signals and noise attenuation properties of the FIR-median hybrid (FMH) and multilevel FMH filters are compared with representative edge preserving filtering operations. The concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the FMH filters. In the multilevel filters there exists a trade-off between noise attenuation and detail preservation. The analysis and examples indicate that FMH filters preserve details better and are computationally much more efficient than the conventional median and the K-nearest neighbor averaging filters.

This paper introduces a new class of uniform digital filter banks based on DFT and recursive Nth-band filters. We consider both even and odd types of channel stacking arrangements. Due to the use of the DFT, the basic filter bank is complex. Real filter banks are obtained by adding the output signals from two symmetrically located complex channels. The filter banks are designed to satisfy the power complementarity and allpass complementarity constraints with high accuracy. Furthermore, nearly alias-free analyzer-synthesizer system can be obtained in the critically sampled case.

In this paper we introduce a new generalized class of Median Filters (MF) which we call Smoothed Median Filters (SMF). The input signal x(n) is filtered with M linear phase FIR filters and the output of the SMF is the median of the outputs of the FIR filters. The statistical properties of the SMF's are analyzed for input signals with Gaussian, double exponential, and uniform density functions. It is shown that SMF's have essentially the same statistical properties and same type of root signals as the conventional MF's. SMF's preserve corner points and ramps and attenuate impulsive type noise components effectively. An interesting subclass of the SMF's requiring only two scaling multipliers and three data sorting operations irrespective of the filter length is introduced.

The class of digital filter pairs satisfying two types of complementary properties simultaneously is reexamined. The pair is required to have complementary square magnitude characteristics with respect to a constant and transfer functions complementary with respect to an allpass function. This filter class, called doubly complementary filters is then extended to have M filters with complex outputs. Basic properties of the filters are discussed. It is shown that all the filters in the bank can be implemented as linear combinations of the same low order allpass filters. The individual filters can have different bandwidths and tolerances. The filter bank can also produce [M/2]+1 real outputs with allpass complementary responses that can be arbitrarily added up to form a variety of allpass complementary filters. The filters have low coefficient sensitivities in their passbands if the allpass sections are implemented with structures that are canonic with respect to the coefficients.

In this paper we discuss a new class of hybrid IIR/ FIR digital filters. These filters present a maximally flat group delay in the given passband and an equiripple nature in the magnitude response both in the passband and in the stopband(s). The all-pole IIR component is synthesized so that its group delay exhibits a maximally flat behaviour in the pass-band whereas the linear phase FIR component is used to obtain the desired equiripple nature in the amplitude response both in the passband and stopbands. The filter types handled are lowpass, band-pass and highpass. By suitably optimizing the maximum value in the group delay and the order of the IIR component we obtain filters requiring significantly fewer multipliers than equivalent FIR and group delay equalized elliptic designs when the passband is narrow (of the order of .1π or smaller) and a rather sharp cutoff is required. The performance of these filters is, however, slightly worse that obtainable with corresponding filters having equiripple group delay [1]. Several comparisons between different filter types are presented.

This paper presents a method for finding efficient multiprocessor realizations of digital filters. It provides a framework for studying the tradeoffs between the sampling rate and the number of arithmetic units or processors. A simple method for determining the minimal sampling period as a function of addition and multiplication times is introduced. Also the effect of delays in data transfers on the sampling period can be analyzed for different hard-ware configurations. Some examples are presented to show the use of the method in obtaining efficient multiprocessor realizations.

In this paper we discuss a new class of hybrid IIR/ FIR digital filters. These filters present equiripple behaviour both in the magnitude and group delay responses. The all-pole IIR component is synthesized so that its group delay exhibits an equiripple variation in the passband whereas the linear phase FIR component is used to obtain the desired equiripple nature in the amplitude response both in the passband and stopbands. The filter types handled are lowpass, bandpass and highpass. By suitably optimizing the ripple in the group delay and the order of the IIR component we obtain filters requiring significantly fewer multipliers than equivalent FIR and group delay equalized elliptic designs when the pass-band is narrow (of order .1π or smaller) and a rather sharp cutoff is required. Examples illustrating this feature are presented in the paper.

In this paper we discuss the use of linear transformations on z+1/z in designing magnitude squared functionsH(z)H(\frac{1}{z}). It is shown how the design of lowpass, highpass and nonsymmetric bandpass recursive digital filters with given zeroes is transformed to the problem of designing a magnitude squared function having its passband stretched onto the whole unit circle. This magnitude squared function for equal ripple and maximally flat responses is then obtained analytically using the theory of rational approximations.

We discuss the design of digital filters with maximally flat or equiripple passband behavior and transfer functions of the formH(z)= K \frac{(z+1)^{q}\Pi\min{i=1}\max{r}(z-e^{j\omega_{i}})(z-e^{-j\omega_{i}})}{D_{n}(z)}, where q+2r, the number of finite-plane zeros, is allowed to vary from 0 to n, the filter order. Analytic expressions are given for the magnitude squared function. Identification of the stopband edge frequency is treated in detail. The hard-ware requirements of these filters are compared with those of Chebychev and elliptic designs.

Explicit design procedures are given for maximally flat and equal ripple lowpass and highpass transfer functions of the form H(z) = (z ± 1)q/Dn(z) , (q = 1, 2 ..... n - 1). Efficient bandpass structures are obtained directly from the realization of H(z) using the Crochiere - Penfield method. The one-multiplier lattices of Gray and Markel realize the lowpass, highpass and bandpass transfer functions with n + 1 multipliers, if q = 1. If one-multiplier lattices with ε = +1 are used in the realization, the coefficient sensitivity peaks down in the passband of the filter.

This paper presents results on the analysis and digital hardware realization of a non-Gaussian impulse noise source which can be used to model the impulsive noise present in wide-band communications systems. The impulse noise appearing in practical systems is characterized by bursts of much higher amplitudes than would be predicted by a normal- or Gaussian-distribution law. A need exists for a better characterization of this non-Gaussian impulsive noise as well as actual noise generators which can be used for evaluation of present and future communications equipment under realistic system-operating conditions. The impulse noise model and the realization of the noise generator described in this paper are capable of producing Gaussian background noise, lognormal impulsive noise, and burst timing to set prescribed starting and ending times of the bursts and impulse density within each of the burst periods. The techniques used in this paper are based on first generating a basic pseudorandom binary sequence (PRBS) and then from this deriving the Gaussian background noise by adding samples from the PRBS generator. The lognormal noise is then derived from the same sequence by using an incremental exponential generator. The burst timing signals which realize an approximate Poisson distribution are obtained by observing the occurrences of specific subsequences from the PRBS generator.