an unbiased risk estimator for multiplicative noise ...tblu/monsite/pdfs/panisetti1401.pdf · iv....

An Unbiased Risk Estimator for MultiplicativeNoise – Application to 1-D Signal DenoisingBala Kishore Panisetti

Department of Electrical EngineeringIndian Institute of ScienceBangalore - 560012, India

Email: [email protected]

Thierry Blu1Department of Electrical Engineering

The Chinese University of Hong KongHong Kong


Chandra Sekhar SeelamantulaDepartment of Electrical Engineering

Indian Institute of ScienceBangalore - 560012, India


Abstract—The effect of multiplicative noise on a signal whencompared with that of additive noise is very large. In this paper,we address the problem of suppressing multiplicative noise inone-dimensional signals. To deal with signals that are corruptedwith multiplicative noise, we propose a denoising algorithm basedon minimization of an unbiased estimator (MURE) of mean-square error (MSE). We derive an expression for an unbiasedestimate of the MSE. The proposed denoising is carried out inwavelet domain (soft thresholding) by considering time-domainMURE. The parameters of thresholding function are obtainedby minimizing the unbiased estimator MURE. We show that theparameters for optimal MURE are very close to the optimalparameters considering the oracle MSE. Experiments show thatthe SNR improvement for the proposed denoising algorithm iscompetitive with a state-of-the-art method.

Index Terms—multiplicative noise, denoising, risk estimation,thresholding.

I. INTRODUCTION

A. Background

Signal denoising is a widely studied problem. Most ofthe literature deals with the additive noise model: given anoriginal signal x, one typically assumes that it is corruptedby additive noise w. The problem is then to recover x fromthe measurements y = x + w. Many approaches have beenproposed in the literature to suppress additive noise.

In this paper, we are concerned with a different denoisingproblem. The assumption is that the original signal x hasbeen corrupted by multiplicative noise w: the goal is toestimate x from the measurement y = xw. Multiplicativenoise is encountered in coherent imaging systems such as laserDoppler imaging, synthetic aperture radar (SAR) imaging,synthetic aperture sonar (SAS) imaging, ultrasound imaging,etc.

In general, the noise in the multiplicative noise model isdescribed by a non-Gaussian probability density function withGamma being the common employed model [1], [2].

B. Related Literature

Several methods have been proposed in the literature [3]–[13] to reduce the effect of multiplicative noise. One way is totransform the multiplicative noise in the signal to an additive

1Th. Blu was supported by the General Research Fund CUHK410012 fromthe Hong Kong Research Grant Council.

one by employing a log transformation. Thereafter, additivesuppression methods can be employed for restoring the signal.Finally, taking the exponential of the processed signal, weobtain the restored signal. However, such a straightforwardmethod does not lead to satisfactory results. Most widely usedfiltering approaches to suppress multiplicative noise includeKuan filter [3], Lee filter [4], Frost filter [5] and adaptivespeckle filtering by Lopes et al. [6]. Variational approachesusing total variation (TV) were proposed by Rudin et al. [7],Aubert and Aujol [8], Shi and Osher [9], and Huang et al. [10].Durand et al. proposed an approach by combining curveletthresholding and variational method [11]. Bioucas-Dias andFigueiredo proposed a method based on variable splitting andconstrained optimization [13].

C. This Paper

We propose a denoising method based on minimization ofunbiased risk estimator. Since the oracle MSE (or risk) isunknown, we derive an expression for the unbiased estimatorof MSE, namely multiplicative noise unbiased risk estimator(MURE). We find the optimal parameters of the denoisingfunction by minimizing MURE, which yields a good estimateof the original signal. Unbiased risk estimation approacheshave been developed for additive noise [14]–[16]. To the bestof our knowledge, risk estimators for multiplicative noisehave not been reported in the literature. We compare theperformance of the proposed method with a state-of-the-artmethod.

II. PROBLEM STATEMENT

Consider the multiplicative noise signal model,

yi = xiwi, for i = 1, 2, · · · , N,

where yi is the ith element of y, y ∈ RN is the noise-contaminated observation of original (unknown) signal x ∈RN , {wi, i = 1, 2, · · · , N} are identical and independentnoise random variables with probability density function de-noted by q(wi).

Proceedings of the 19th International Conference on Digital Signal Processing 20-23 August 2014

978-1-4799-4612-9/14/$31.00 © 2014 IEEE 497 DSP 2014

The multiplicative noise is Gamma distributed with proba-bility density function given as

q(w) =N∏i=1

q(wi), where w = [w1, w2, · · · , wN ] ∈ RN ,

q(wi) =ab

Γ(b)wb−1

i exp−wia, (1)

where a and 1b are the scale and shape parameters, respectively.

q(wi) has mean E [q(wi)] = a/b, and variance

σ2w = E

{[q(wi)− E(q(wi))]

2

}=

a

b2.

We choose a = b = k. Let f : RN → RN be the denoisingfunction that yields denoised estimate of original signal, x̂ =f(y). We have taken a parametric form for f . The goal is tofind the optimal parameters of denoising function f such that

MSE =1

NE{‖x̂− x‖2

}is minimized.

The ground truth signal is not known. Hence, instead ofminimizing the oracle MSE, one needs to obtain an unbiasedestimator of the MSE and minimize it.

III. MULTIPLICATIVE NOISE UNBIASED RISK ESTIMATOR(MURE)

Notation. Given a 1-D function f , we define M by thefollowing operator

Mf(y) = k

∫ 1

0

f(sy)sk−1 ds.

For a multivariate function f(y) = f(y1, y2, . . . , yN ), wedefine the operator Mi which applies the operator M to theith component of f(y) only.

This notation is extended straightforwardly to multivariatevector functions f(y) = [f1(y), f2(y), . . . , fN (y)]T accordingto

Mf(y) = [M1f1(y),M2f2(y), . . . ,MNfN (y)]T

Lemma 1. Consider a 1-D function f such that E {|f(y)|}is finite. If y = xw where w is a multiplicative Gammadistributed random variable, with mean 1 and variance 1/k,then

E {xf(y)} = E {yMf(y)} (2)

Here, the expectations are taken over the realizations of w.

Proof: Letting y = wx, we have

E {yMf(y)} =

∫ ∞0

kwx dw

∫ 1

0

f(swx)q(w)sk−1 ds

= kx

∫ 1

0

sk−1 ds

∫ ∞0

wf(swx)q(w) dw

= kx

∫ 1

0

sk−1 ds

∫ ∞0

w

s2f(wx)q

(ws

)dw

= kx

∫ ∞0

f(wx) dw

∫ 1

0

wk

s2kk

Γ(k)e−kw/s ds

= kx

∫ ∞0

f(wx)kkwk

Γ(k)

[e−kw/s

kw

]10+

dw

= x

∫ ∞0

f(wx)kkwk−1

Γ(k)e−kw dw

= x

∫ ∞0

f(wx)q(w) dw = E {xf(y)}

Here, the hypothesis E {|f(y)|} <∞ was used to validate theintegration sign exchanges, and the limit at 0+.

Using Lemma 1, We derive an expression for the unbiasedestimate of the risk as follows:

Theorem 1. An unbiased estimate (or risk) of the MSE isgiven by the expression

MURE(f) =1

N

(k

k + 1‖y‖2 + ‖f(y)‖2 − 2yTMf(y)

).

(3)

Proof:Since ‖f(y) − x‖2 = ‖x‖2 + ‖f(y)‖2 − 2xTf(y) we need

to find two functions of y alone that are unbiased estimatesof ‖x‖2 and xTf(y).• First, xTf(y) = x1f1(y)+x2f2(y)+· · ·xNfN (y): From

Lemma 1, we have that

E {xifi(y)} = E {yiMif(y)}

This shows that

E {xTf(y)} = E {y1M1f(y)}+ E {y2M2f(y)}+ · · · E {yNMNf(y)}

= E {yTMf(y)}

and so that yTMf(y) is an unbiased estimate of xTf(y).• Second, ‖x‖2 = x21 + x22 + · · ·x2N : looking at the ith

component of y, yi = xiwi, we have that E{y2i}

=x2i E

{w2

i

}= x2i

(1 + 1/k

). Hence

x2i =k

k + 1E{y2i},

and finally ‖x‖2 =k

k + 1‖y‖2

Therefore, the unbiased estimate of MSE is

MURE(f) =1

N

(k

k + 1‖y‖2 + ‖f(y)‖2 − 2yTMf(y)

)


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IV. DENOISING PROCESS

The denoising process that we have employed is asfollows: First, we decompose the observed signal intodifferent subbands using undecimated wavelet transform.D and R represents a wavelet decmposition and reconstruc-tion matrices satisfying the perfect reconstruction conditionRD = I. Typically D = [D1

TD2T . . . DJ

T ]T and R =[R1R2 . . .RJ], Dj,Rj ∈ RN×N , j = 1 to J implement aJ-band filterbank of undecimated analysis and synthesis filters.

Fig. 1. Denoising process

We suppress noise by proper thresholding of wavelet co-efficients in highpass subbands. After thresholding, we applythe corresponding reconstruction filter to revert to the timedomain. The key idea of the denoising process is to performtransform-domain denoising by considering the time-domainMURE.

A. Undecimated wavelet decomposition

The wavelet coefficient vector wj = {wjm}Nm=1 in the jth

subband is given by,

wj = Djy, j = 1 to J,

Dj = [djm,l]1≤m,l≤N , Rj = [rjm,l]1≤m,l≤N.

B. Subband thresholding in the highpass subbands

To suppress the noise, thresholding is performed on thewavelet coefficients using suitable thresholding function ineach subband except the lowpass subband.

Since the unbiased estimator given by (3) contains integra-tion term, if we consider soft-thresholding function, the com-putation of MURE becomes difficult. To reduce computationalcomplexity, we have considered the thresholding function,which reliably approximates the soft-thresholding function.The transfer characteristic of the thresholding function withinput w is given by,

θ(w, T ) = w

[1− exp

(−(w

γ

)2)]

where γ = T ×√w and T , w are the thresholding parameter,

and subband variance estimator, respectively.

The unbiased estimate of variance of mth wavelet coefficientof jth subband is given by ,

wjm =

1

k + 1

N∑l=1

dj2

m,l y2l .

4 5 6 7 8 9 1020

30

40

50

60

70

80

THRESHOLD PARAMETER

MS

E/M

UR

E

MSE

MURE

Fig. 2. MSE/MURE (MURE is scaled by ∼5) versus thresholding parameterT for Square signal with Gamma noise (scale parameter a = 10, shapeparameter 1

b= 0.1).

The Taylor’s series approximation of θ(w, T ) is

θ(w, T ) = w

[(w

γ

)2

− 1

2!

(w

γ

)4

+1

3!

(w

γ

)6

− · · ·

](4)

The polynomial representation on the right-hand side of(4) makes the computation of the integration term in MUREeasier.

C. Wavelet reconstruction

After thresholding, we apply the corresponding undecimatedwavelet reconstruction filter, which yields the denoised esti-mate of original signal x. The denoised estimate f(y) is,

f(y) =

J∑j=1

RjΘ(wj , T ),

where Θ(wj , T ) = [θ(wj1, T ), θ(wj

2, T ), · · · , θ(wjN , T )]T .

The parameter of the thresholding function is T . We choosethe optimal T such that MURE is minimized.

V. RESULTS AND PERFORMANCE COMPARISON

In simulations, we used Haar wavelet basis and four-levelwavelet decomposition. We find the optimal thresholding pa-rameter T that minimizes the unbiased risk estimator (MURE)by exhaustive search method.

Simulations are performed on different standard signals withGamma distributed multiplicative noise with varying scaleparameter. Our experiments showed that the optimization ofMURE gives minima close to those obtained by optimizationof MSE. From Figure 2, we observe that although MURE isnot so close to the oracle MSE for some values of the thresholdparameter T , since MURE minima is so close to oracle MSEminima with respect to threshold parameter, minimization ofMURE is nearly equivalent to the minimization of oracle MSE.

Figures 3, 4, and 5 show the noisy signals and correspondingrestored signals using MURE-denoising method.

We compare the results obtained by the proposed methodwith a state-of-the-art method. We compare the performance


978-1-4799-4612-9/14/$31.00 © 2014 IEEE 499 DSP 2014

0 20 40 60 80 100 120 140 160 180 2001.5

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2.5

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3.5

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4.5

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plit

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(a) Original signal

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(c) Restored signal (SNR = 20.86 dB)

Fig. 3. Denoising performance on a sinusoidal signal

of the proposed denoising method on noisy sinusoidal signal,square signal and triangular signal with the variational ap-proach [8]. Table I shows a comparison of results obtained bythe proposed method with the variational approach. The SNRimprovement for the proposed method is competitive with thestate-of-the-art method.

0 1000 2000 3000 4000 5000 60000.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Sample Index

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plit

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Fig. 4. Denoising performance on a square wave

VI. CONCLUSIONS

We developed a new risk estimation framework for sup-pressing multiplicative noise in one-dimensional signals. Start-ing with the mean-square error as the risk function, we havedeveloped an unbiased estimator that approximates the risk andenables efficient optimization of the denoising function. The


978-1-4799-4612-9/14/$31.00 © 2014 IEEE 500 DSP 2014

0 1000 2000 3000 4000 5000 6000

−1

−0.5

0

0.5

1

Sample Index

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0 1000 2000 3000 4000 5000 6000−8

−6

−4

−2

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0 1000 2000 3000 4000 5000 6000−1.5

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−0.5

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Fig. 5. Denoising performance on a triangular wave

result has been derived specifically for the case of Gammadistributed noise. However, suitable risk estimators may bederived for other distributions as well. We have considered un-decimated wavelet decomposition based denoising, where thedenoising function parameters on every subband are optimizedfor using the risk estimator. Performance analysis on noisy

Signal SNRin(dB) SNRout(dB) SNRout(dB)[8] MURE

sinusoidal 4.69 19.76 21.03

sinusoidal 7.20 21.70 23.14

sinusoidal 9.98 24.13 25.24

sinusoidal 14.75 27.41 28.15

square 4.90 19.56 21.06

square 6.85 20.42 22.39

square 9.90 23.36 24.52

square 14.63 27.82 27.46

triangular 4.76 18.25 20.43

triangular 7.0 20.61 22.16

triangular 9.90 23.03 23.24

triangular 12.89 25.44 24.73

TABLE ICOMPARISON OF THE PROPOSED DENOISING METHOD WITH THE

VARIATIONAL APPROACH [8].

1-D signals showed that the proposed method is capable ofenhancing the signal-to-noise ratio significantly. Comparisonswith a state-of-the-art technique showed that the proposedtechnique has about 0.5 to 1.5 dB higher output SNR. Athorough comparison with other techniques, and performanceassessment on real signals is currently being carried out.

ACKNOWLEDGMENT

The authors would like to thank Prof. J. F. Aujol andProf. G. Aubert for providing their MATLAB software forthe variational approach, which facilitated the comparisonsreported in this paper.

REFERENCES

[1] J. Goodman. “Some fundamental properties of speckle,” Journal of theOptical Society of America, vol. 66, no. 1, pp. 1145−1150, 1976.

[2] J. Goodman, Speckle Phenomena in Optics: Theory and Applications,Greenwood Village, CO: Roberts, 2007.

[3] D. T. Kuan, A. A. Sawchuk, T. C. Strand, and P. Chavel, “Adaptiverestoration of images with speckle,” IEEE Trans. Acoust. Speech SignalProcess., vol. ASSP-35, pp. 373−383, Feb. 1987.

[4] J. S. Lee, “Digital image enhancement and noise filtering by use of localstatistics,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-2, pp.165−168, Mar 1980.

[5] V. S. Frost, J. A. Stiles, K. S. Sanmugan, and J. C. Holtzman, “Amodel for radar images and its application to adaptive digital filteringof multiplicative noise,” IEEE Trans. Pattern Anal. Machine Intell., vol.PAMI-4, pp. 157−165, 1982.

[6] A. Lopes, E. Nezry, R. Touzi, and H. Laur, “Structure detection andstatistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens.,vol. 14, pp. 1735−1758, 1993.

[7] L. Rudin, P. Lions, and S. Osher, “Multiplicative denoising and deblur-ring: Theory and Algorithms,” Geometric Level Set Methods in Imaging,Vision, and Graphics, pp. 103−119, Springer, 2003.

[8] G. Aubert and J. Aujol, “A variational approach to remove multiplicativenoise,” SIAM Journal on Applied Mathematics, vol. 68, pp. 925−946,2008.

[9] J. Shi and S. Osher, “A nonlinear inverse scale method for a convexmultiplicative noise model,” SIAM J. Imaging Sci., vol. 1, pp. 294−321,2008.


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[10] Y. Huang, M. Ng, and Y. Wen, “A new total variation method formultiplicative noise removal,” SIAM Journal on Imaging Sciences, vol.8, pp. 11−49, 1991.

[11] S. Durand, J. Fadili and M. Nikolova, “Multiplicative noise removal us-ing `1 fidelity on frame coefficients,” Journal of Mathematical Imagingand Vision, vol. 36, no. 3, pp. 201−226, 2009.

[12] C. Liu, R. Szeliski, S. Kang, C. Zitnick, and W. Freeman, “Automaticestimation and removal of noise from a single image,” IEEE Trans.Pattern Anal. Machine Intell., vol. 30, pp. 299−314, 2008.

[13] J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal usingvariable splitting and constrained optimization,” IEEE Trans. ImageProcess., vol. 19, no. 7, pp. 1720−1730, July 2010.

[14] C. Stein, “Estimation of the mean of a multivariate normal distribution,”Analysis of Statistics, vol. 9, no. 6, pp. 2225−2238, Nov. 2005.

[15] T. Blu and F. Luisier, “The SURE-LET approach to image denoising,”IEEE Trans. Image Process., vol. 16, no. 11, pp. 2778−2786, Nov. 2007.

[16] Y. Eldar, “Generalized SURE for exponential families: Applicationsto regularization,” IEEE Trans. Signal Process., vol. 57, no. 2, pp.471−481, 2009.


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