SPECKLE: MODELING AND FILTERING. Torbjørn Eltoft. Department of Physics, University of Tromsø, 9037 Tromsø, Norway

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1 F % F C SPECKLE: MODELNG AND FLTERNG Torbjørn Eltoft Department of Physics, University of Tromsø, 93 Tromsø, Norway ASTRACT A new statistical distribution for modeling non-rayleigh amplitude statistics, which is formed as a compound distribution of a the Rician distribution and the nverse Gaussian (G), is introduced. This new model has been called the Rician nverse Gaussian (RiG) distribution. The theoretical basis of the model is briefly presented, and we give an algorithm for estimating its parameters from data. The modeling capabilities of the RiG distribution is tested by fitting the model to s generated from segments of a medical ultrasound image and a single-look SAR image. Finally, a maximum a posteriori (MAP) speckle filter has been constructed using the RiG model. The filtering performance of this filter is shown to be excellent. 1. NTRODUCTON Modeling non-rayleigh amplitude statistics in coherent imaging systems has been a research area for almost three decades, and many interesting distributions have been proposed. Some of these are the Rice-distribution, the K-distribution and the homodined K-distribution. Accurate statistical models for the amplitude statistics are important both with respect to characterization/classification of image regions [1], and with respect to speckle filtering [2, 3]. n this paper we introduce a new model for modeling the amplitude statistics of coherent imagery, which made up as a compound distribution of a the Rician distribution and the nverse Gaussian (G) [4]. We have for this reason denoted this new distribution the Rician nverse Gaussian (RiG) distribution. We give some details of the theoretical basis of this new model, describe how its parameters can be estimated, and test its ability to fit to s of segments of ultrasound images. We include comparisons with the well-known K-distribution. We also develop a maximum aposteriori speckle filter based on the RiG model, and show its filtering performance. 2. REVEW OF SPECKLE MODELS f the scattering medium is modeled as a collection of discrete scatterers, we may write the detected signal as is the number of scatterers, (1) where is the amplitude and is the phase of the! th scatterer. i: Now, if the scatterers phases are random and independent of the amplitudes, and if the number is very large, the probability density function (pdf) of, the amplitude of, will be a Rayleigh distribution "$#&% ( )+ -,/.1 32 (2) where ) is the the standard deviation of the Gaussian distributed 4 - and 5 -components of. This distribution represents the fully-developed speckle case, or the diffuse scattering signal. ii: f a specular component 6 is added to (1), the pdf of the amplitude gets Rician, i.e. "8#&% ( )+ -,. :9-;-< =2 4> % 6 )+ (3) where 4 > is the modified essel function of first kind and zero order. iii: The well-known -distribution [5, 6] results, if we allow the variance of the Rayleigh-distribution to itself being a random variable, distributed according to a A -distribution. n this case, the amplitude of is a compound variable of a Rayleigh variable and a A -distributed variable. The pdf is explicitly given as where %HG and order D, C " # % ( -C A %ED C 1F, % ( (4) is the modified essel function of second kind J D$KML N is a scale parameter. D is a shape parameter. iv: The homodyned K-distribution results when a coherent

2 D L G T > % % component, 6, is added to the diffuse scattering signal in (1), and we allow the number of terms in the summation,, to itself being random following a negative binomial distribution. The resulting pdf, which can not be obtained in closed form, is given by the integral "$# % ( > % > % 6 > % )+ K ( F The models reviewed above have some shortcomings: the Rayleigh model certainly is a proper model for fully developed speckle, but this case does not often occur in highresolution imagery. Also, the K-model has its limitations, because, often the signal has a specular component, resulting in bad fits to the data. The homodyned model has good modeling capabilities, but its use is restricted by the lack of a tractable pdf. 3. NEW MODEL We will in the following introduce a unifying model for non-rayleigh amplitude data, which is able to overcome the problems mentioned above. n addition, its parameters can easily be estimated from data. When the amplitude of a detected signal, modeled as in (1), is non-rayleigh, this implies that the 4 - and 5 - components of the complex signal deviate from Gaussian statistics. We propose to use the Normal nverse Gaussian (NG) distribution to model these components. A NGdistribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian (4 ) as a mixing distribution []. The probability density function is explicitly given as where " % and %HG "% $(% (! #" % " % $(% 1 %&$(% 1 ) + % ) % ) + (5) (6) is the modified essel function of the second kind and order. As noted, the distribution is characterized by the four parameters %,, which are bounded such that -/.1 2.,, and )3 L4 L 3. controls the steepness of the distribution; a small means a steep distribution, a larger means a less steep distribution. determines the skewness; L - implies skewed to the left, N - implies skewed to the right, and - implies a symmetric distribution. is a location parameter, and is a dispersion parameter similar to the variance in the Gaussian distribution. Let be defined by the relation () where :9<; % - =. Hence, the conditional distribution 6 is normal ; % >J,. The marginal distribution of 6 is nverse Gaussian (G) with parameters and A). This probability density is given by " %, =C"%,EDF = #" % ) (8) The G-distribution was first introduced as the first passage time distribution of rownian motion with drift. t has recently attained increasing interest in signal processing and statistics (financial) due to its ability to model highly skewed densities, and its many analogies with the Gaussian distribution. ts mean and variance are GH 6KJ ML N and OPH 6KJ N&Q, respectively. The distribution is well documented in the literature [4]. Consider the case when the - and -components of in (1) are uncorrelated and 2-dimensional NG-distributed, i.e: 9 4R % - ES, and 9 4R % -, ET. Note that we assume that both S and UT are zero. Accordingly, and are both compound random variables of the form: > S 6 6 S and > T 6 6 T, where V % S T 9W; %XY, with Y [Z 4 ( Z -dimensional identity matrix.), and % ]\ ) % S ^ 4 is the, with. t is now easy to show that the distribution of the envelope, conditioned on 6, i.e. 6, is Rician distributed, with pdf given by " #`_ a % where \ S ^! #" % ) ^ 4> % 1 ( (9) T. From (9) we observe that the skewness parameters of the NG-distributions will have the effect of a coherent component on the amplitude statistics. Now, the amplitude will accordingly be a compound variable consisting of the Rice-distribution in (9), and the G( ) distribution, i.e. " # % ( " #`_ a % " a % (1) The integral in (1) can be evaluated to give the following closed form pdf for : "8# % ( /b D,F c " % % D,Fd D,F % 4> % ( (11) We call this distribution the Rician nverse Gaussian (RiG) distribution. As can be seen, the pdf has three parameter.

3 L 9 1 ) [1,.98, 3 ] [1,.8, 5 ] [1,.6, 2 ] [1,.8, 1 ] Fig. 1. Examples of pdfs residing to the RiG model. The numbers in the brackets correspond to,,, respectively Fig.1 displays some realizations of the pdf for some selected values of the parameters. The numbers in brackets in the upper right corner of Fig.1 give the values of,, and for each curve. We observe that by varying the parameters,, and, we may model a wide range of distributions, from a Rayleigh-shaped pdf to a Gaussian-shaped pdf. 4. PARAMETER ESTMATON The parameters of the RiG distribution can be estimated using an EM-type algorithm. n the previous section we assumed that the speckle-free image, represented by the vari- distribution with parameters able, follows an G % and. ts th-order moments are given by GH 6 J %,, N L From (12) we easily find that GH 6 J and G G 6 GH 6KJ, a are: (12) (13) (14) t is further noted that the posterior distribution 6 is given as: " a_ # % ( _ ; ;, Q &% Q! L )( " $# + Q! L ", = #"- ) EL " #.# (15) This is the pdf of a generalized inverse Gaussian (GG) []. variable specified by the parameters % ) D The moments of a GG % +/ distribution are given by [] GH 6 J - % " % (16) Let 1. We thus have that G8H 6 KJ (1) $2 G 6 1 % &% & 1 (18) Now, given N independent observations! - GG G+. From some initial values for,, and, we may estimate G8H 6KJ as and G6 6 a as N L 6 N % / 53 &% & 1 Using (13) and (14), our estimators for and are : 89 ; L 6 N < a>= L56 N (19) (2) (21) (22) respectively. The parameter is estimated using the maximum likelihood method. Using previous estimates of,, and, define A % A % as DCFEHG "8#&% is defined in (11). Our estimate for is ac- where " # % cordingly given as arg max JKA % + (23) This maximization must be performed numerically. 5. MODELNG DATA (24) Fig.2 shows a medical ultrasound image of the human heart. Two test regions are seen as marked rectangles. The s of the amplitudes of the test regions have been displayed in Fig.3, along with the best fitted RiG- and K- models. The parameters of the K-models were estimated

4 ) Fig. 2. Ultrasound image (log-compressed). The marked areas define the image regions which the RiG-model was tested on Fig. 4. Single-look SAR image (displayed log-compressed). The marked areas define the image regions which were tested against the K and the RiG models..4 RiG model K model.12.1 RiG model K model.2.18 RiG K.3 RiG K Fig. 3. Fitted RiG-model (solid), (thin solid) and fitted K-model (dotted-dashed) corresponding to test regions Fig. 5. Fitted RiG-model (solid), (thin solid) and fitted K-model (dotted-dashed) corresponding to test regions using a moment based estimator, or by matching the higherorder moments of the data. We note that the RiG-model fits the s of the data much better than the K-model. Fig.3 also demonstrates the high flexibility of the the RiG model. Fig.4 shows a single-look SAR image of a mountainous area in southern Norway. The same conclusion drawn from the analysis of the the ultrasound data, is valid for the SAR data. The RiG model also has a very good fit to the s of the SAR data, as can be noted from fig MAP-FLTERNG The problem in ayesian de-speckling is the following: We want to estimate the amplitude, where is the spatial location, which is most likely to occur in a specklefree image, given an observed amplitude value. This is known as MAP-filtering, and neglecting the spatial coor- dinates the estimation is stated in mathematical terms as " a_ # % arg max " a_ # % (25) is generally not known, and one has to restate the expression in terms of known pdfs using ayes rule " a_ # % " #`_ a % G1"Ea % "$#&% ( (26) f we assume that is RiG distributed, (15) explicitly gives us " a_ # %, and the de-speckled is simply obtained by maximizing( the pdf of (15) with respect to. This gives R % (2) is explicitly given in terms of the parameters and Hence, of the RiG model. When MAP filtering is performed, the RiG-parameters are estimated from the observed data in a local window around each pixel. The estimated

5 speckle-free amplitude is obtained from (2) with and replaced by their local estimates, and equal to the observed amplitude. Fig. (6) shows the original ultrasound image, which shows the interior of the human heart. This image has been filtered using the proposed model in an MAP filtering approach. Note that no thresholding is applied, as is usual in the traditional A -MAP filter. All pixel-values in the output image has been estimated using (2). The filtered image is shown in Fig. () Figs. (8) and (9) show the original and the filtered SAR images, respectively. [6] C. J. Oliver, A model for non-rayleigh scattering statistics, Optica acta., vol. 31, no. 6, pp. 1 22, [] O. E. arndorff-nielsen, Normal nverse Gaussian Distributions and Stochastic Volatility Modelling, Scand. J. Statist., vol. 24, pp. 1 13, CONCLUSON We have presented a new statistical distribution for modeling amplitude data, which do not follow a Rayleigh distribution. We call this distribution the Rician nverse Gaussian distribution. The model has three parameters, which gives the distribution rich flexibility. These parameters may be estimated from real data using an EM-type algorithm. The experimental analysis shows that the new pdf is well suited for modeling medical ultrasound and SAR data. We have also developed a MAP filter based on the new distribution. This is easy to implement, due to the fact that the a posterior distribution is given in closed form. The filter output is hence given explicitly. Our initial analysis of the performance of this speckle filter seems very promising. 8. REFERENCES [1] R. M. Cramblitt and M. R. ell, Surface characterization using frequency diverse scattering measurements and regularity models, EEE Trans. Signal Processing, vol. 44, no. 3, pp , March [2] D. T. Kuan, A. A. Sawchuck, T. C. Strand, and P. Chavel, Adaptive noise smoothing filter for images with signal dependent noise, EEE Trans. Pattern Anal. and Machine ntell., vol., pp , [3] S. Foucher, G.. énié, and J.-M. oucher, Multiscale MAP Filtering of SAR mages, EEE Trans. mage Pros., vol. 1, no. 1, pp. 49 6, Jan. 21. [4] M. T. Wasan, First passage time distribution of brownian motion with positive drift (inverse gaussian distribution), Queen s papers in pure and applied matematics -no.19, Queen s University, [5] E. Jakeman and P. N. Pusey, Significance of K distributions in scattering experiments, Phys. Rev. Lett., vol. 4, pp , 198.

6 Fig. 6. Original ultrasound image (log-compressed). Fig. 8. Original single-look SAR image Fig.. MAP-filtered version of the image in Fig.(6). Window size (, ) Fig. 9. MAP-filtered version of the image in Fig.(8). Window size (,.