An Unbiased Homomorphic System To Reduce Speckle In Images Debashis Senc , M. N. S. Swamy and M. Omair Ahmad Department of Electrical and Computer Engineering, Concordia University, Montreal, Quebec, CANADA, H3G 1M8. E-mail: c d [email protected] Abstract— In this paper, we propose an unbiased homomorphic system to reduce speckle in images. The speckle is modeled as a multiplicative noise having a lognormal distribution. First, we introduce a new filter called the mean median (MM) filter to reduce additive white Gaussian noise (AWGN) in images. Simulations show that the MM-filter is very promising for reducing AWGN in images without blurring the edges. This MM-filter is then used in a homomorphic system to reduce the speckle. Then, a technique to compensate for a biasing that occurs within the MM-filter-based homomorphic system is applied to the output of the system to get the unbiased estimate of the uncorrupted image. A study of the qualitative and quantitative performance of the proposed MM filter-based unbiased homomorphic system in reducing speckle is carried out and compared to that of a few existing ones. It is found that the proposed system has good noise reduction and edge preservation properties, and performs better than the others.

I. I NTRODUCTION Coherent imaging systems are used in various commercial, military and medical applications. These systems receive signals as a coherent sum of various reflected waves. Speckle noise is generated due to random interference between the returning coherent waves reflected from an irregular surface, and appears as strong dark and bright granulations in the image. This hinders both the manual and automatic image understanding (segmentation, recognition) capability and thus, speckle reduction system is required. A fully developed speckle has the characteristics of a random multiplicative noise and approximately follows a gamma distribution [1]. An image generated by a coherent imaging system is always associated with an attribute called the equivalent number of looks (ENL), L. For a unit-mean speckle noise the value of the ENL equals the reciprocal of the noise variance. It has been found that the lognormal distribution can also be used to approximate the shape of the probability density function (PDF) of the intensity of the speckle [2]. There have been different image restoration processes proposed in the literature to reduce speckle in images. A comprehensive review of such processes has been given in [2]. In this paper, we propose an unbiased homomorphic system employing a filter to diminish additive white Gaussian noise (AWGN), in order to obtain speckle reduction in images. We use a novel filter to reduce the AWGN referred to as the mean median (MM) filter. The MM-filter is formulated such that the edge and fine details in an image are preserved, while successfully reducing the noise from the smooth regions. We assume that the speckle has a lognormal distribution and is a unit-mean white random process. We shall now explain the algorithm of the proposed system to reduce

speckle. The speckle noise corruption is modeled as as = bs ηs

(1)

where as is the observed corrupted signal, bs is the original uncorrupted signal and ηs is the unit-mean multiplicative white noise with lognormal distribution (speckle). The problem is to get an estimate ˆbs of bs from the observed signal as . To reduce the speckle, the proposed unbiased homomorphic system uses the natural logarithm to transform the multiplicative lognormal noise into an additive Gaussian one and then processes the resulting corrupted signal using the novel MM-filter. The MM-filter uses a suitable criterion to combine the sample mean and sample median estimates to get the filtered output (see Section II). An exponential function is then applied to the output of this filter. It was shown in [3] that such an output after the exponentiation would be biased. Hence, the bias compensation technique proposed in [3] is applied to the output to get the unbiased estimate ˆbs of the original uncorrupted signal bs . The algorithm of the proposed system is given in Figure 1.

Figure 1.

Proposed MM-filter-based unbiased homomorphic system

II. T HE MM- FILTER In order to reduce speckle using a homomorphic system, it is required to diminish the AWGN after the natural logarithm operation. The sample mean filter, which is optimal in both the MMSE sense and the MLE sense, can be used to reduce the AWGN [4], [5]. But the optimality is under the condition that the uncorrupted signal is of constant amplitude within a filter window which may not be always true, for example, when the window is over the edge areas in an image. On the other hand, the sample median filter is known for its edge preserving property when applied to an image, but unlike the sample mean filter, it is not optimal in reducing AWGN. In this section, we show that a judicious combination of the mean and the median estimates, i.e., the MMfilter, could provide a better performance while reducing AWGN than the sample mean filter. Let the original signal have two different amplitudes within the filter window, say, x1 and x2 . Then, we can express the corrupted signal within the filter window as y(i) = [x1 ⊕ x2 ] + n(i)

(2)

where [x1 ⊕ x2 ] stands for either x1 or x2 depending on the value of i and n is a zero-mean AWGN. We assume that the original signal and the noise are independent of each other and hence the PDF of y is given by

another case where H = 0.7 and a = 2. In this case we obtain Q = 1.6 and the value of AREM M as

P DF (y) = P DF ([x1 ⊕ x2 ]) ∗ P DF (n)

From (6) and (7), it is evident that AREM M can have values above or below unity when the signal within the filter window is not constant. Hence, a judiciously obtained MM-filter would be superior to the sample mean filter in reducing AWGN.

(3)

In (3), the index i has been dropped for simplicity. P DF (.) stands for the PDF of the corresponding random variable and ∗ stands for convolution. Without loss of generality, we assume the noise variance to be unity 2 . Now, and shift the mean of the PDF of y by − x1 +x 2 the PDF of y is expressed as
−(y − a)2  H + ... fy (y) = √ exp 2σ 2 2π
−(y + a)2  1−H √ exp 2σ 2 2π

V (mean, f ) = (1 + a2 )4f 2 (Q) V (median, f )

In this subsection, three criteria are presented for combining the mean and median estimates to remove AWGN. The first two criteria are based on an unbiased weighting of the sample median and sample mean estimates as given below τ=

where a = and H is a weight equivalent to the ratio of the number of x1 valued elements to the total number of elements within the filter window. We shall now consider the concept of asymptotic relative efficiency (ARE) [4] to show that the MMfilter would perform better than the sample mean filter, when reduction of an AWGN is considered. The measure ARE is used to analytically compare the performance of two estimators in estimating the same parameter of a distribution. We use the ARE to compare the performance of the sample mean and sample median estimates and refer such an ARE as AREM M . If AREM M < 1, then the sample mean estimate is better or otherwise sample median filtering prevails. If it is shown that AREM M corresponding to the distribution in (4) can have values above or below unity, it would essentially indicate the better estimating ability of the MM-filter, which performs estimation by judiciously combining the sample mean and sample median estimates. Using the various definitions given in [4] and by simple algebraic manipulations, we get the expression of AREM M for f = fy (y) as (5)

where V (κ, f ) stands for the asymptotic variance [4] corresponding to the estimator κ of a parameter of the PDF f and Q is determined from the expression F (Q) = 12 , F being the cumulative distribution function corresponding to f . The values of Q and a may have the range (−∞,∞). We now consider the case where H = 0.5. In such a case we obtain Q = 0 for any value of a. The AREM M obtain for this case can be expressed as 2 AREM M = (1 + a2 ) exp(−a2 ) (6) π From (6), it can be easily shown that the value of AREM M corresponding to this case is always less than unity with the maximum value as 0.6367. Let us consider

(7)

A. Criteria to combine the mean and median estimates

(4)

x1 −x2 2

AREM M =

AREM M = 1.33

ατmean + βτmedian α+β

(8)

where τmean represents the coefficients of the sample mean estimator, τmedian those of the sample median estimator, and α and β are the weights used to obtain the coefficients τ of the MM-filter. The weights corresponding to the first criterion (Cr1) are given as α = σ 2 and β =

1 −1 σ2

(9)

where σ 2 (σ 2 will never be greater than 1 when the MMfilter is used in the proposed system to reduce speckle) is the variance of the AWGN. For the second criterion (Cr2) the weights are given by
1  σ2 σ2 −1 2 (10) α = σ 2 2 and β = 2 σI σ σI where σI2 is the variance of the observed corrupted image. The third criterion (Cr3) is based on some modifications to the edge-adaptive Wiener filter equation. The AWGN corruption is modeled as a = b + η. The filter equation thus obtained is given by

σ2 − v2  x,y ˆb(x, y) = (1 − ∆x,y ) µ + ... 2 σx,y  (a(x, y) − µ) + ∆x,y med[R] (11) 2 are the local mean and the local where µ and σx,y variance of a calculated within the filter window, v is the variance of the corrupting noise η and med[R] stands for the median value among the elements of an array R given by

R = [(a(n1 , n2 ) − µ)]∀(n1 , n2 ) ∈ Λx,y

(12)

In the above, Λx,y represents the filter window centered  2 2  σ −v , at the position (x, y), The value of ∆x,y = x,y 2 σx,y with the notation [Z x,y ] signifying the normalization with respect to the maximum value amongst the values of Zn1 ,n2 , where (n1 , n2 ) corresponds to the pixel positions within the window Λx,y .

TABLE I. MSE AND FOM FOR THE VARIOUS FILTERS IN AWGN ( VARIANCE = 0.025) IN THE ‘ CROWD ’ IMAGE

REDUCING

AWGN variance σ 2 = 0.025 Noisy image Image recovered by (1) Image recovered by (2) Image recovered by (3) Image recovered by (4) Image recovered by (5)

MSE 1629.6 191.6 192.85 219.48 219.56 176.855

FOM 4.2043 16.295 36.953 24.696 24.726 33.77

(a) Original

(b) Noisy (σ 2 = 0.025)

(c) Recovered by (1)

between the desired and the actual response, and the figure of merit (FOM) employed in [2], which gives the measure of the amount of edge preserved, are used as the measure in the tables. For noisy images, the measurement is carried out between the noisy image and the uncorrupted image. Smaller MSE signifies a better noise reduction property, whereas a large FOM signifies a better edge preservation feature. Figure 2 gives the qualitative performance of the various filters considered. It is evident from the figure that the proposed MM-filter, using any one of the three criteria performs better edge preservation than the other two, while achieving good noise reduction. This shows that the MM-filter does a better balancing between edge preservation and noise reduction. Quantitatively, the MM-filter is as good as the other two. III. S IMULATIONS

(d) Recovered by (3)

(e) Recovered by (4)

(f) Recovered by (5)

(g) Recovered by (2)

Figure 2. Qualitative performance of the various filters in reducing AWGN in the ‘crowd’ image

B. Performance of the MM-filter In this subsection, qualitative and quantitative results of the MM-filter in reducing AWGN from corrupted images are presented and compared to that of a few known filters. The filters considered are (1) The sample mean filter, (2) The edge-adaptive Wiener filter, (3-5) The MM-filter based on Cr1, Cr2 and Cr3, respectively. We will use the serial number of the filters in the list to represent them in tables and figures. Table I gives the quantitative results of the various filters considered in reducing AWGN with a variance σ 2 = 0.025 (normalized with respect to maximum grayscale value, i.e, 255). The mean square error (MSE)

In this section, a comparative study of the performance of some of the existing filters and that of the proposed system to reduce speckle is carried out. Both the quantitative and qualitative results are considered. The filters considered are (i) Lee filter, (ii) Filter by Kuan et al., (iii) Filter by Frost et al., (iv) Gamma Filter, (v) Edge-adaptive Wiener filter based homomorphic system, (vi-viii) MM-filter-based unbiased homomorphic system, with the MM-filter being based on Cr1, Cr2 and Cr3, respectively. We will use the serial number of the filters in the list to represent them in tables and figures. The MSE, the FOM, and the complexity of the filters in terms of processing time (PT) and the number of computations (C) are used as the quantitative measures of performance. The number of computations (C) is calculated as, C = no. of additions + no. of multiplications + no. of comparisons + no. of other operations. TABLE II.

MSE

FOM FOR THE VARIOUS FILTERS IN (ENL=10) IN THE ‘ PEPPER ’ IMAGE

AND

REDUCING SPECKLE

L = 10 Noisy image Image recovered by (i) Image recovered by (ii) Image recovered by (iii) Image recovered by (iv) Image recovered by (v) Image recovered by (vi) Image recovered by (vii) Image recovered by (viii)

MSE 1006.3 1377.1 1164.2 163.67 181.45 143.16 95.032 96.139 228.24

FOM 11.412 15.125 14.99 23.368 41.302 41.102 52.722 52.545 28.631

TABLE III.

C OMPLEXITY OF THE VARIOUS FILTERS IN TERMS OF 512 X 512 IMAGE AND THE

PROCESSING TIME ( IN SECONDS ) FOR A

NUMBER OF COMPUTATIONS PER INPUT SAMPLE

Filters (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Processing Time 46 sec 47 sec 1150 sec 10 sec 60 sec 7 sec 7 sec 17 sec

No. of Computations 96 100 906 107 111 93 95 145

Table II displays the quantitative results. Figure 3 gives the qualitative performance of the various filters considered. It is evident from the qualitative and quantitative results, that the proposed system achieves both effective noise reduction and good edge preservation, unlike the others. Table III shows that the proposed system has low complexity. We have carried out experiments for various other values of L and on images obtained directly from various coherent imaging systems such as the synthetic aperture radar system, the ultrasound imaging system. The results of our experiments showed that the proposed system is very promising in reducing speckle from images, while preserving the fine details in the images.

(a) Original

(b) Noisy (L = 10)

(c) Recovered by (vi)

(d) Recovered by (vii)

(e) Recovered by (viii)

(f) Recovered by (v)

(g) Recovered by (i)

(h) Recovered by (ii)

(i) Recovered by (iii)

(j) Recovered by (iv)

IV. C ONCLUSION In this paper, we have proposed a new filter referred to as the MM-filter in order to reduce AWGN. This MM-filter is then used within an unbiased homomorphic system to reduce speckle. A study of the quantitative and qualitative results of the MM-filter in reducing AWGN and the MM-filter-based unbiased homomorphic system in reducing speckle have been carried out and compared with that of a few corresponding existing filters. The proposed MM-filter and the MM-filter-based unbiased homomorphic system have been found to perform better than the other corresponding filters. Hence, the proposed speckle reduction system promises to be an effective preprocessing step for various high-level image processing operations such as classification, recognition. V. R EFERENCES [1] J. W. Goodman, Speckle Phenomenon in Optics: Theory and Applications, Work in progress, Version 2, 2004. [2] L. Gagnon and A. Jouan, Speckle Filtering of SAR Images- A Comparative Study Between Complex Wavelet-Based and Standard Filters, Dept of R&D, Lockheed Martin, Canada, 1997. [3] D. Sen, M. N. S. Swamy, and M. O. Ahmad, “Order StatisticsBased Unbiased Homomorphic System to Reduce Multiplicative Noise,” in the proceedings of 13th European Signal Processing Conference, 2005. [4] I. Pitas and A. N. Venetsanopoulos, “Nonlinear Digital Filters: Principles and Applications,” in The Kluwer International Series in Engineering and Computer Science, Jonathan Allen, Ed. Kluwer Academic Publisher, 1990. [5] J. Astola and P. Kousmanen, “Fundamentals of Nonlinear Digital Filtering,” in The Electronic Engineering Systems Series, J. K. Fidler and Phil Mars, Eds. CRC Press, 1997.

Figure 3. Qualitative performance of the various filters in reducing speckle in the ‘pepper’ image