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Optical Engineering 49�5�, 057004 �May 2010�

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mage restoration and enhancement based onunable forward-and-backward diffusion

i Wanguiqing Niuin Yuhina University of Geosciences

nstitute of Geophysics and Geomaticsumo Road 388uhan, Hubei 430074 China-mail: [email protected]

iangpei Zhanguhan Universitytate Key Laboratory of Information

Engineering in Surveying, Mapping,and Remote Sensing

29 Luoyu Roaduhan, Hubei 430079 China

uanfeng Shenuhan Universitychool of Resource and Environmental

Science29 Luoyu Roaduhan, Hubei 430079 China

Abstract. In order to improve signal-to-noise ratio �SNR� and contrast-to-noise ratio, we introduces a novel tunable forward-and-backward�TFAB� diffusion approach for image restoration and edge enhancement.In the TFAB algorithm, an alternative forward-and-backward �FAB� diffu-sion process is presented, where it is possible to better modulate allaspects of the diffusion behavior and it shows better algorithm behaviorcompared to the existing FAB diffusion approaches. In addition, there isno necessity to laboriously determine the value of the gradient threshold.We believe the TFAB diffusion to be an adaptive mechanism for imagerestoration and enhancement. Qualitative experiments, based on variousgeneral digital images and a magnetic resonance image, show signifi-cant improvements when the TFAB diffusion algorithm is used versus theexisting anisotropic diffusion and the previous FAB diffusion algorithmsfor enhancing edge features and improving image contrast. Quantitativeanalyses, based on peak SNR and the universal image quality index,confirm the superiority of the proposed algorithm. © 2010 Society of Photo-Optical Instrumentation Engineers. �DOI: 10.1117/1.3431657�

Subject terms: anisotropic diffusion; forward and backward; tunable; imageenhancement; image restoration.

Paper 090718R received Sep. 15, 2009; revised manuscript received Mar. 27,2010; accepted for publication Apr. 6, 2010; published online Jun. 1, 2010.

Introduction

igital images often suffer from a blurring effect and noiserom various sources, such as different illumination condi-ions, image quantization, compression, transmission, etc.hese sources of image degradation normally arise during

mage acquisition and processing and have a direct bearingn the visual quality of the image.1 Undoing these imper-ections to remove the image degradation is crucial forany image-processing tasks. Of particular interest to this

tudy is the work related to image denoising and sharpen-ng that aims to improve signal-to-noise ratio �SNR� andontrast-to-noise ratio �CNR�.2–7

The scale-space concept was first presented by Iijima8,9

nd Weickert et al.10 and became popular later on by theorks of Witkin11 and Koenderink.12 The theory of linear

cale-space supports edge detection and localization, whileuppressing noise by tracking features across multiplecales.11–16 In fact, the linear scale space can be expressedy a linear heat diffusion equation.12,13 However, this equa-ion was found to be problematic in that all edge featuresre smeared and distorted after a few iterations of lineariffusion. In order to remedy the difficulties encountered inhe linear scale-space theory, Perona and Malik17 developed

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an adaptive smoothing and edge-detectionscheme in whichthey replaced the linear heat diffusion equation by a selec-tive diffusion that preserves edges. This development ledsome research to focus on the development of various an-isotropic diffusion models and diverse numerical schemesto obtain steady-state solutions.18–39 Among them, one spe-cific anisotropic diffusion algorithm inspires us, theforward-and-backward �FAB� diffusion algorithm.32

In Perona and Malik’s scheme,17 the nonlinear diffusionprocess should be restricted by the “minimum-maximum”principle. This principle, to avoid creating any new minimaor maxima, was obeyed by most nonlinear diffusion pro-cesses and guaranteed stability in partial differential equa-tions �PDEs� and thus avoided the explosion of the nonlin-ear diffusion process. Instead of restricting the globalextremes for the initial signal, Gilboa et al.32 pointed outthat inverse diffusion with a negative diffusion coefficientshould be incorporated into image-sharpening and enhance-ment processes to deblur and enhance the extremes of theinitial signal �if the extremes are indeed singularities andnot generated by noise�. However, linear inverse diffusionis a highly unstable process and results in noise amplifica-tion. Thus, nonlinear diffusion methods are further ex-tended and combined with the FAB diffusion process toshow that sharpening and denoising can be reconciled inimage enhancement. Besides this pioneer work, some inter-esting work has been published on theoretical foundationsand the application of FAB diffusion to gray and colorimages.40–42

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Wang et al.: Image restoration and enhancement based on tunable forward-and-backward diffusion

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In our earlier work,39 we proposed a local varianced-ontrolled forward-and-backward �LVCFAB� diffusion pro-ess in the context of image restoration and enhancement.n essence, this diffusion scheme is based on a better be-aved diffusion coefficient, in which the transition lengthetween the maximum and minimum values of diffusionoefficient does not increase with the gradient threshold.lthough the scheme turns out to be effective for miscella-eous images, the location of the transition cannot be ad-usted, which leads to difficulty in better controlling theiffusion behavior. Furthermore, it is yet uncovered abouthe optimal strategy for estimating the two gradient thresh-lds in the FAB diffusion scheme.

In this paper, we further develop our previous heuristicdea from an alternative perspective in order to come upith a systematic and tunable FAB diffusion algorithm. Un-

ike our earlier work, we explore an alternative FAB diffu-ion process based on a suitable sigmoid function, in which

ig. 1 The GSZ FAB diffusion coefficient Eq. �6� and the corre-ponding flux, plotted as a function of the gradient magnitude �kf30, kb=80, w=10, �=0.25�.



both the location of the transition from isotropic to orientedflux and the transition length can be easily tuned. As aresult, our tunable FAB �TFAB� algorithm is more effectiveat controlling the behavior of the diffusion function whencompared to the existing FAB diffusion approaches. In ad-dition, it is not necessary to determine the value of thegradient because since there is no such parameter in thediffusion coefficient. We believe the alternative FAB diffu-sion algorithm to be a novel mechanism for image restora-tion and enhancement.

The remainder of this paper is organized as follows:Section 2 presents the proposed TFAB diffusion algorithm;Section 3 describes simulations including comparative re-sults between several existing anisotropic diffusionschemes and our proposed algorithm; and Section 4 statesour concluding remarks.

(a)

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Fig. 2 The LVCFAB diffusion coefficient Eq. �7� and the correspond-ing flux, plotted as a function of the gradient magnitude �kf=30, kb=80, �1=0.2, �2=0.05, �=0.25�.

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TFAB Diffusion Algorithm

.1 FAB Diffusion

n image restoration, the problem of finding the true images modeled as follows:43

0 = I + n , �1�

here I0 is the noisy output result of this linear model that

s given to us. I=LI is a blurred version of the originalmage I, L is a linear operator representing the blur, usually

convolution. The additive noise is given by n with thessumed known mean and variance �2. This, in general, isn ill-posed problem. Edge-preserving regularization meth-ds are proposed to restore the image,43,44 by minimizinghe following energy functional:

(a)

(b)

ig. 3 Diffusion coefficient Eq. �8� and the corresponding flux, plot-ed as a function of the gradient magnitude ��=5�10−4, �=10, 2�.



E�I� = ��

�I − I0�2dx + � · ��

��I��dx , �2�

where � is the image domain and I=LI. The first term onthe right-hand side of Eq. �2� is the quadratic data fidelityterm, and the second term represents a prior assumptionabout the true image I. Selecting a suitable regularizer � isparamount in restoring/enhancing the edges and is an openproblem. Nonconvex regularizing functions have also beenused in the past17,43–45 in spite of the absence of existenceresults. The equivalence of these methods to the anisotropicPDE-based models can be seen via the Euler–Lagrangeequation. Starting with the pioneering work of

Fig. 4 Tunable diffusion coefficients with the different parameter ��=10, =2�.

Fig. 5 Tunable diffusion coefficients with the different parameter ��=5�10−4, =2�.

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Perona–Malik,17 the following class of second-order PDEsare used extensively used in image restoration and in otherearly vision problems:

�I�x,y,t��t

= div�c��I�x,y,t�� I�x,y,t�� , �3�

where � is the gradient operator and div is the divergenceoperator. c� · � is a non-negative monotonically decreasingfunction of local spatial gradient. If c� · � is constant, thenisotropic diffusion is enacted. In this case, all locations inthe image, including the edges, are equally smoothed. This

(b)

(d)

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al image with 8-bit gray levels; �b� GaussianlyGaussian noise with 50; �c� CLMC anisotropic

.05�; �e� GSZ FAB diffusion �kf=0.2�MAG, kb200, �1=0.2, �2=0.02, =0.2, �=0.2�; and �g�

ig. 6 Tunable diffusion coefficients with the different parameter �=5�10−4, �=10�.

(a)

(c)

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Fig. 7 Enhanced images for Parrots image: �a� Originblurred Lena image ��=1�, contaminated by zero-meandiffusion �=0.05�; �d� MB anisotropic diffusion �=0=0.4�MAG, �=0.2�; �f� LVCFAB diffusion �kf=50, kb=TFAB diffusion ��=0.006, �=3, =2� �15 iterations�.

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s an undesirable effect because the process cannot main-ain the natural boundaries of objects. One common form of� · � is

��I�x,y,t�� =1

1 + ��I�x,y,t�/k�1+� , where � � 0, �4�

here the parameter k serves as a gradient threshold: amaller gradient is diffused and positions of a larger gradi-nt are treated as edges.

The diffusion coefficient �4� is chosen to be nonincreas-ng functions of the image gradient. This scheme selec-ively smoothes regions without large gradients. However,n the FAB diffusion process, the points of extremes aremphasized in signal enhancement, image sharpening, andestoration. The emphasized extremes occur if these points

(a)

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Fig. 8 A regional enlarged portion of homogennoisy image, and results corresponding to Fig.



are indeed represented by singularities and do not emergeas the result of noise. It was observed by Gilboa et al. thatif we want to emphasize large gradients we should like tomove “mass” from the lower part of a “slope” upwards.32

This process can be viewed as “moving back in time” alongthe scale space, or reversing the diffusion process. Math-ematically, this can be accomplished simply by changingthe sign of the diffusion coefficient

�I�x,y,t��t

= div�− c��I�x,y,t�� I�x,y,t�� . �5�

However, we cannot simply use an inverse anisotropic dif-fusion process for image enhancement because it is highlyunstable. There is a major problem associated with thebackward diffusion: noise amplification. To remedy this

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ones of original noise-free image, blurred andg�.

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rawback of the linear inverse diffusion process, Gilboa etl.32 proposed that two forces of diffusion working simul-aneously on the signal are needed. First, a backward forces used at medium gradients, where singularities are ex-ected, and the second, a forward force, is implemented foruppressing oscillations and reducing noise. The forwardnd backward forces are combined into one coupled FABiffusion process with a diffusion coefficient that possessesoth positive and negative values. Thus, a diffusion coeffi-ient �see Fig. 1� that controls the Gilboa–Sochen–ZeeviGSZ� FAB diffusion process was proposed32

��I�x,y,t�� =1

1 + ��I�x,y,t�/k �n

(a)

(c)

(e)

Fig. 9 Regional enlarged portion of edge feaimage, and results corresponding to Fig. 7�c�–7

f



−�

1 + ��I�x,y,t� − kb�/w�2m , �6�

where kf is similar to the role of the parameter k in the PMdiffusion equation; kb and w define the range of backwarddiffusion, and are determined by the value of the gradientthat is emphasized; � controls the ratio between the for-ward and backward diffusion; and the exponent parameters�n ,m� are chosen as �n=4, m=1�. Equation �6� is locallyadjusted according to image features, such as edges, tex-tures, and moments. The GSZ FAB diffusion process cantherefore enhance features while locally denoising thesmoother segments of images.

However, the transition length between the maximumand minimum coefficient values varies with the gradientthreshold, which makes controlling the evolution procedure

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f original noise-free image, blurred and noisy

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ifficult.39 Thus, we proposed the LVCFAB diffusion coef-

(a)

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Fig. 10 Enhanced images for Couple image: �ablurred image ��=1�, contaminated by zero-mdiffusion �=0.05�; �d� MB anisotropic diffusion=0.4�MAG, �=0.2�; �f� LVCFAB diffusion �kf=5TFAB diffusion ��=0.006, �=8, =2� �15 iterati



ficient �see Fig. 2� as follows:39

(b)

(d)

(g)

nal image with 8-bit gray levels; �b� Gaussianlyaussian noise with 50; �c� CLMC anisotropic.05�; �e� GSZ FAB diffusion �kf=0.2�MAG, kb200, �1=0.2, �2=0.02, =0.2, �=0.2�; and �g�

��I�x,y,t�� =1 − tanh��1 · ��I�x,y,t� − kf�� − � · �1 − tanh2��2 · �kb − �I�x,y,t��

2, �7�

here �1 and �2 control the steepness for the min-maxransition region of forward diffusion and backward diffu-

sion, respectively. These two parameters are vital to theFAB diffusion behavior, and the transition width from iso-

(f)

� Origiean G�=0

0, kb=ons�.

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ropic to oriented flux can be altered by modulating them.n addition, the LVCFAB diffusion process can preserve theransition length from isotropic to oriented flux and, thus, its better at controlling the diffusion behavior than that ofhe GSZ FAB diffusion.

.2 Alternative FAB (AFAB) Diffusion

n this section, an AFAB diffusion is presented to be able toodulate everything: the location of the transition from iso-

ropic to oriented diffusion and the corresponding transitionength. In this paper, we define an AFAB diffusion coeffi-

(a)

(c)

(e)

Fig. 11 Regional enlarged portion of interest oand results corresponding to Fig. 10�c�–10�g�.



cient �see Fig. 3� that is based on a suitable sigmoid func-tion as follows:

c��I�x,y,t��

=e−��2

+ e−��I�x,y, t��2+ · �e−��I�x,y, t��2

− 1�

e−��2+ e��I�x,y, t��2 ,

�8�

where � denotes the length of transition from isotropic tooriented flux, � governs the transition location of FAB dif-fusion, and dominates the transition coefficient. Figs. 1–3show plots of the diffusion coefficients and respective

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al noise-free image, blurred and noisy image,

(f)

f origin

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uxes of the threeFAB diffusion coefficients. It is apparenthat the proposed diffusion coefficient has a different for-ulation to that of the GSZ FAB and LVCFAB diffusion

oefficients. However, they all combine two opposingorces into the diffusion process: one backward force atedium gradients for sharpening, and a second, a forward

orce at low gradients for smoothing and restoration. Fur-hermore, they have the same property: negative diffusionoefficients are explicitly employed in a certain gradientange. Meanwhile, there is no gradient threshold k in Eq.8�, are used which is helpful in reducing the complexity ofhe parameter selection procedure.

(a)

(c)

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Fig. 12 Enhanced images for Lena image: �a�blurred Lena image ��=1�, contaminated by zerdiffusion �=0.05�; �d� MB anisotropic diffusion=0.4�MAG, �=0.2�; �f� LVCFAB diffusion �kf=5TFAB diffusion ��=0.006, �=8, =2� �15 iterati



2.3 Parameter Analysis

To better control the diffusion coefficient, we analyze theeffect of varying the different parameters and show how thetunability is related to diffusion behavior. From Figs. 1–3,we observe remarkably that the plot of the FAB diffusioncoefficient will always drop and rise to zero, so thatsmoothing is performed when the diffusivity function ispositive and sharpening occurs for negative diffusion coef-ficient values. In Eq. �8�, � determines the location of thisdrop and rise occurs. As � increases, the drop part of thelocation will descend, while the rise part will ascend

(b)

(d)

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al image with 8-bit gray levels. �b� GaussianlyGaussian noise with 50; �c� CLMC anisotropic

.05�; �e� GSZ FAB diffusion �kf=0.2�MAG, kb200, �1=0.2, �2=0.04, =0.2, �=0.2�; and �g�

(f)

Origino-mean

�=00, kb=

ons�.

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see Fig. 4�. This rule is helpful for setting the proper valuef � to control the location of the transition from isotropico oriented flux. Meanwhile, both the GSZ FAB and LVC-AB diffusion coefficients have a given transition locationnd, therefore are not adjustable. The function curves of theroposed FAB diffusion coefficient with different � valuesre depicted in Fig. 5. It is noted that � is closely related toow fast the drop happens. Moreover, the plot of the diffu-ion coefficient in Fig. 5 declines faster as � increases.bviously, � is a crucial parameter that can offer muchexibility to achieve specific FAB diffusion characteristics.n our previous work, we employed the two similar param-ters to the LVCFAB diffusion process to control the shapef the diffusion function where the diffusion coefficient isositive and negative. However, the GSZ FAB scheme haso such variables.39 can be used to control the ratio be-

(a)

(c)

(e)

Fig. 13 Intensity values of row 138: �a� in the oand in the results given by �c� CLMC anisotropdiffusion, �f� LVCFAB diffusion, and �g� TFAB di

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tween the FAB diffusion. It can be observed from Fig. 6that if �1, then the diffusion scheme with the diffusioncoefficient �8� serves as a tunable FAB diffusion process.On the contrary, if �1, then the proposed diffusionscheme transforms to the traditional anisotropic diffusionprocess, where the diffusion coefficient is always positive.

2.4 TFAB Diffusion Algorithm

In this section, we summarize the idea of the TFAB diffu-sion coefficient mentioned above into a complete and adap-tive image-restoration and -enhancement algorithm. Toachieve this goal, we propose to filter the data by the fol-lowing algorithm of FAB diffusion.

(b)

(d)

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ena image, �b� in the blurred and noisy image,ion, �d� MB anisotropic diffusion, �e� GSZ FAB

.

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riginal Lic diffusffusion

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1. Initialization

a. Input a given image I. I�x ,y ,0� denotes theoriginal intensity of pixel �x ,y�.

b. Set parameters �, �, and for the proposedFAB diffusion; and T for the maximal num-ber of iterations.

2. Iterate until t=T.

a. For each pixel �x ,y�, the diffusion coeffi-cient c��I�x ,y , t�� is computed by Eq. �8�.

b. The four-nearest-neighbor diffusion dis-cretization equation is performed to updateI�x ,y , t�. Our FAB diffusion algorithm is adiscretization on a 3�3 lattice. In order toaccomplish the assumptions mentionedabove, we propose the following anisotropicdiffusion equation:

I�x,y,t + 1� = I�x,y,t� + � · �cN · �NI

+ cS · �SI + cE · �EI

+ cW · �WI� ,

where N, S, E, and W are the mnemonicsubscripts for four directions �i.e., North,South, East, and West�. The subscripts onthe parenthesis are applied to all the termsenclosed. � is the time step: 0��1 /4 forthe numerical scheme to be stable. As de-fined in the original paper,46 the spatial gra-dient at pixel �x ,y� is the first derivative ofits image intensity function.

Experimentsn this section, we first describe the methodology used inur simulations and then reveal comparative filtering re-ults for blurred and noisy images. Moreover, we providedetail discussion on the impact of parameters and the per-ormance of the proposed algorithm at different noise lev-ls. Finally, we demonstrate that as a useful tool for earlyision, the proposed algorithm effectively extracts fine edgetructures from medical images.

In order to demonstrate the effectiveness, we compareur algorithm to two traditional anisotropic diffusion algo-ithms: Catte-Lions-Morel-Coll �CLMC� anisotropiciffusion18 and Monteil-Beghdadi �MB� anisotropiciffusion.47 We also apply two existing FAB diffusion al-orithms: GSZ FAB diffusion32 and LVCFAB diffusion.39

he ultimate goal of image filtering is to facilitate the sub-equent processing for computer vision. To demonstrate thealidity of the proposed algorithm in an early vision task,e apply the above diffusion algorithms to enhance a medi-

al image for an application-based evaluation. As a definingharacteristic, iterative operations are inevitably involvedn anisotropic diffusion. Therefore, implementation of anterative algorithm depends greatly on the termination time,hich causes what we often refer to as the terminationroblem. Although there still does not exist a widely ac-epted analytical method, several heuristic methods have



been proposed to determine the stopping time to overcomeinstability in anisotropic diffusion.26,40,48,49 As far as sim-plicity is concerned, the nonlinear cooling method is mostsuitable for applications as a general denoising scheme.Gilboa et al.48 proposed a threshold-freezing nonlinearcooling method, using the cooling rate we refer to as , andapplied it to the anisotropic diffusion scheme. In our simu-lations, we adopted this strategy in the CLMC and MBanisotropic diffusion and GSZ FAB diffusion. Meanwhile,the gradient threshold of the GSZ FAB diffusion is deter-mined by calculating the mean absolute gradient �MAG�from the original paper.32

Table 1 PSNR and quality value for images of Parrots, Lena, Cam-eraman, and Couple.

Scheme Image PSNR �dB� UIQI

Blurred andnoisy

Parrots 27.07 0.5351

Lena 34.98 0.6910

Cameraman 25.29 0.4066

Couple 26.97 0.6866

CLMC Parrots 28.17 0.7313

Lena 35.70 0.7646


Couple 27.55 0.6934

MB Parrots 28.27 0.7303

Lena 35.88 0.7638


Couple 27.62 0.6955

GSZ FAB Parrots 28.37 0.7348

Lena 36.43 0.8174


Couple 27.81 0.7391

LVCFAB Parrots 28.35 0.7462

Lena 36.42 0.8142


Couple 27.90 0.7239

TFAB Parrots 28.42 0.7514

Lena 36.72 0.8214


Couple 27.97 0.7556

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(a) 0.002β =(b) The histogram of the original Lenaimage and diffusivity ( 0.002β = )

(c) The histogram of the filtered Lenaimage and diffusivity ( 0.002β = )

(d) 0.004β =(e) The histogram of the original Lenaimage and diffusivity ( 0.004β = )

(f) The histogram of the filtered Lenaimage and diffusivity ( 0.004β = )

(g) 0.006β =(h) The histogram of the original Lenaimage and diffusivity ( 0.006β = )

(i) The histogram of the filtered Lenaimage and diffusivity ( 0.006β = )

(j) 0.008β =(k) The histogram of the original Lenaimage and diffusivity ( 0.008β = )

(l) The histogram of the filtered Lenaimage and diffusivity ( 0.008β = )

(m) 0.01β =(n) The histogram of the original Lenaimage and diffusivity ( 0.01β = )

(o) The histogram of the filtered Lenaimage and diffusivity ( 0.01β = )

Fig. 14 Impact of � on the TFAB diffusion method’s performance ��=8, =2�.

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.1 General Imageshe performance of the proposed algorithm is evaluatedsing four 256�256 standard images with 256 gray-scalealues. The image of Parrots is used as an example of theiecewise-constant image. Lena and Cameraman are twoxamples with both textures and smooth regions. Couple isn example with different edge features. Figure 7�a� showshe original noise-free Parrots image. For the test, we gen-rate a blurred and noisy version of the image, as shown inig. 7�b�. The results yielded by CLMC and MB aniso-

ropic diffusion are depicted in Figs. 7�c� and 7�d�, respec-ively. We can observe that the two algorithms result in theoss of important information from the original image,hough the noise is entirely removed. A better combinationf smoothing and sharpening is given by the GSZ FAB andVCFAB diffusion processes. Figs. 7�e� and 7�f� show theorresponding results. Finally, the image yielded by ourlgorithm is represented in Fig. 7�g�. The noise is readilyemoved, and this is due to tunable forward diffusion.

eanwhile, edge features, including most of the fine de-ails, are sharply reproduced with the TFAB diffusion algo-ithm. From the perspective of vision quality, the resultsiven by the GSZ FAB, the LVCFAB, and the TFAB dif-usion are comparable because the three processes simulta-eously enhance, sharpen, and denoise images. Neverthe-ess, the degree of enhancement of the TFAB diffusionould be tuned more appropriately to account for improvinghe SNR and CNR.

In order to obtain a better evaluation of the visual qual-ty of the enhanced images, two regional enlarged portionsf interest in Fig. 7 �see Fig. 7�a��, and the correspondingnhanced results are zoomed in Figs. 8 and 9. From theseetails, we can easily observe that all the diffusion algo-ithms mentioned above could reduce noise. Nevertheless,he CLMC and MB anisotropic diffusion algorithms blurignificant edge features. The homogeneous zones and edgeeatures processed by the GSZ FAB, LVCFAB, and TFABiffusion algorithms are depicted in Figs. 8�e�–8�g� and�e�–9�g�, respectively. It is quite obvious that the GSZAB and LVCFAB diffusion algorithms achieve a goodompromise between sharpening and denoising, while theroposed algorithm exhibits the best edge-enhanced diffu-ion behavior �for a comparison, see the original portions inigs. 8�a� and 9�a�, respectively�.

Figure 10�a� shows the image of Couple. Figure 10�b� ishe blurred and noisy version of Fig. 10�a�, generated bydding Gaussian blur and noise. We apply five diffusion

Table 2 PSNR and quality value for Lena image with different �.

Parameter PSNR �dB� UIQI

�=1 36.73 0.8174

�=4 36.79 0.8192

�=9 36.72 0.8203

�=16 36.05 0.7967

�=25 35.18 0.7507



algorithms to the blurred and noisy image and illustrateresulting images in Figs. 10�c�–10�g�. In order to have aclearer view, one region of interest in Fig. 10 is displayedin Fig. 11 for the five algorithms. It can be seen that noiseis thoroughly removed in the enhanced images; however,many important features are missing or blurred with theCLMC and MB anisotropic diffusion algorithms used forcomparison. In contrast, our algorithm delivers the best vi-sual quality among the three FAB diffusion algorithms;most fine details are enhanced well during the evolutionaryprocess.

We also present in Fig. 12 the image of Lena to show theeffects of the five diffusion algorithms applied in this study.The resulting images are presented in Figs. 12�c�–12�g�. Itis observed that the TFAB diffusion produces the best im-age, judged by subjective image quality, compared to theother four algorithms. In order to appraise the nonlinearbehavior of the five anisotropic diffusion algorithms, theintensity values of a row are graphically depicted in Fig.13. The original noise-free row number 138 �from top tobottom� is shown in Fig. 13�a�. The corresponding row inthe blurred and noisy image is represented in Fig. 13�b�.According to the previous observation, the CLMC and MBanisotropic diffusion algorithms blur significant edge fea-tures and the results are shown in Figs. 13�c� and 13�d�.The data processed by the GSZ FAB, LVCFAB, and TFABdiffusion algorithms are depicted in Figs. 13�e�–13�g�, re-spectively. It is obvious that the GSZ FAB and LVCFABdiffusion algorithms can reconcile the balance betweensharpening and denoising, while the proposed algorithm ex-hibits the best edge-enhanced diffusion behavior �for acomparison, see the original image in Fig. 13�a��. It is evi-dent from the comparative results that our algorithm out-performs the others in terms of sharpening and restorationof the blurred and noisy image.

In order to objectively evaluate the performance of thedifferent diffusion algorithms, we adopt the peak SNR�PSNR� and universal image quality index �UIQI�. ThePSNR is used to estimate the effectiveness of noise reduc-tion:

PSNR = 10 log10 �i,j

2552

�i,j

�I�i, j,0� − I�i, j,T��2�dB, �9�

where I�0� is the original image and I�T� denotes the recov-ered image. Recently, UIQI has been widely used to betterevaluate image quality50

Table 3 PSNR and quality value for Lena image with different �.


�=0.002 35.89 0.7809

�=0.004 36.57 0.8076

�=0.006 36.72 0.8214

�=0.008 36.70 0.8126

�=0.01 36.38 0.7955

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O

(a) 1γ =(b) The histogram of the original Lena

image and diffusivity ( 1γ = )(c) The histogram of the filtered Lena

image and diffusivity ( 1γ = )

(d) 4γ =(e) The histogram of the original Lena

image and diffusivity ( 4γ = )(f) The histogram of the filtered Lenaimage and diffusivity ( 4γ = )

(g) 9γ =(h) The histogram of the original Lena

image and diffusivity ( 9γ = )(i) The histogram of the filtered Lenaimage and diffusivity ( 9γ = )

(j) 16γ =(k) The histogram of the original Lenaimage and diffusivity ( 16γ = )

(l) The histogram of the filtered Lenaimage and diffusivity ( 16γ = )

(m) 25γ =(n) The histogram of the original Lenaimage and diffusivity ( 25γ = )

(o) The histogram of the filtered Lenaimage and diffusivity ( 25γ = )

Fig. 15 Impact of � on the TFAB diffusion method’s performance ��=0.006, =2�.

ptical Engineering May 2010/Vol. 49�5�057004-14

Downloaded from SPIE Digital Library on 26 Sep 2010 to 202.114.65.9. Terms of Use: http://spiedl.org/terms

Q

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=1

M�M=1

j

Qj , �10�

here M is the total step number and Qj denotes the localuality index computed within the moving window. In thisaper, a sliding window of size 8�8 is applied to estimaten entire image. The list of PSNR and UIQI values that are

(a) 1ξ = (b) The histogramimage and dif

(d) 2ξ = (e) The histogramimage and diff

(g) 3ξ = (h) The histogramimage and diff

Fig. 16 Impact of on the TFAB diffu



reported by the different algorithms, which were performedon four blurred ��=1� and noisy ��2=50� versions of testimages used in our experiments, are found in Table 1. Re-markably, the statistical results shown in Table 1 definitelyindicate that all three FAB diffusion algorithms are betterthan the existing anisotropic diffusion algorithms. Also, the

original Lenay ( 1ξ = )

(c) The histogram of the filtered Lenaimage and diffusivity ( 1ξ = )


(f) The histogram of the filtered Lenaimage and diffusivity ( 2ξ = )


(i) The histogram of the filtered Lenaimage and diffusivity ( 3ξ = )

ethod’s performance ��=0.006, �=8�.

of thefusivit

of theusivit

of theusivit

sion m

May 2010/Vol. 49�5�5


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etter performance of the proposed TFAB diffusion is ap-arent.

.2 Impact of Parametersn Section 2.3, we provided detailed explanations on howhe parameters in Eq. �8� can be used to tune the diffusion.n this section, we perform the three-dimensional analysiso evaluate the parameters’ impact on the method’s perfor-ance using the image of Lena. To this end, we apply the

roposed algorithm to Fig. 12�b� using different settings offor evaluation. The resultant images filtered by our

ethod are shown in Figs. 14�a�, 14�d�, 14�g�, 14�j�, and4�m�, respectively. In Fig. 14, we believe that if � is toomall, then the diffusion evolution results in deblurring fineetails, such as the hat, its decoration and the hair �see Fig.4�a��. On the other hand, if � is too large, there are somescillations in the face �see Figs. 14�j� and 14�m��. In ad-ition, we demonstrate the plots of original and enhancedmage histograms and diffusivity in Figs. 14�b�, 14�c�,4�e�, 14�f�, 14�h�, 14�i�, 14�k�, 14�l�, 14�n�, and 14�o�,espectively. From these plots, we can easily observe that �overns the range of gradient magnitudes to be enhanceduring the evolution procedure �see the position of diffu-ivity as c�s� 0�. The list of PSNR and UIQI values by

Table 4 PSNR and quality value for Lena image with different .


=1 36.60 0.8081

=2 36.72 0.8214

=3 35.93 0.7954

able 5 Performance �PSNR �dB� & UIQI� of the proposed TFABespect to different noise variances.

Scheme Image

25 50

PSNR�dB� UIQI

PSNR�dB� U

Blurredand

noisy

Parrots 28.04 0.6188 27.07 0.5

Lena 36.65 0.7661 34.98 0.6

Cameraman 25.89 0.4506 25.29 0.4

Couple 27.88 0.7427 26.97 0.6

TFAB Parrots 28.63 0.7834 28.42 0.7

Lena 37.46 0.8506 36.72 0.8

Cameraman 26.21 0.5579 25.97 0.5

Couple 28.29 0.7778 27.97 0.7



our algorithm with different settings of � are demonstratedin Table 2. We note from Table 3 that it is optimal for � tobe in the range of 0.004–0.006. The more edge features theimage contains, the larger � should be. On the contrary, themore smooth regions the image contains, the smaller �should be. Meanwhile, it is evident from image histogramsthat the gradient magnitudes become lower and lower dur-ing the diffusing process. As time advances, only smootherand smoother regions are being filtered, whereas large gra-dient magnitudes can be enhanced due to backward diffu-sion. It can be also concluded that we can perform empiri-cal analysis to select the optimal set of parameters bycomparing the original and resultant image histograms. Weexhibit the impact of � on our algorithm in Fig. 15 byenhancing the Lena image. It is obvious that � controls thedegree of inverse diffusion �see the degree of bending ofdiffusivity as c�s� 0�. The list of PSNR and UIQI valuesby our algorithm with different settings of � are given inTable 3. Because inverse diffusion can easily cause noiseamplification, which decreases the image quality, � should

hm for the images of Parrot, Lena, Cameraman, and Couple with

noise variance

100 225 400

PSNR�dB� UIQI

PSNR�dB� UIQI

PSNR�dB� UIQI

25.64 0.4458 23.36 0.3483 21.41 0.2794

32.93 0.6024 30.10 0.4952 27.80 0.4092

24.34 0.3646 22.62 0.3143 20.97 0.2764

25.58 0.6165 23.30 0.5104 21.33 0.4289

28.14 0.7009 27.42 0.6534 27.08 0.6102

35.96 0.7835 34.23 0.7100 33.20 0.6576

25.73 0.4714 25.08 0.4118 24.82 0.3832

27.56 0.7308 26.34 0.6491 25.90 0.6019

Table 6 Parameter settings of the proposed TFAB algorithm withrespect to different noise variances.

Noise variance Parameters

25 �=0.01, �=6, =2

50 �=0.006, �=8, =2

100 �=0.004, �=6, =2

225 �=0.002, �=8, =2

400 �=0.001, �=6, =2

algorit

IQI

351

910

066

866

514

214

224

556

May 2010/Vol. 49�5�6


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ot be fixed too large. In our experiments, we suggest thatbe selected within the bound: ��15. As mentioned in

ection 2.3, determines the ratio between forward diffu-ion and backward diffusion. Thus, it is significant to ex-

(a) Blurred and noisy image ( 252 =σ ) (b) Enhanced image ( 252 =σ )

(c) Blurred and noisy image ( 502 =σ ) (d) Enhanced image ( 502 =σ )

(e) Blurred and noisy image ( 1002 =σ ) (f) Enhanced image ( 1002 =σ )

(g) Blurred and noisy image ( 2252 =σ ) (h) Enhanced image ( 2252 =σ )

(i) Blurred and noisy image ( 4002 =σ ) (j) Enhanced image ( 4002 =σ )

ig. 17 Enhanced Gaussianly blurred Lena image ��=1� using ourlgorithm with respect to different noise variances �10 iterations�.



plore the impact of on our method. The resultant imagesare illustrated in Fig. 16 by modulating while fixing � and� in the proposed algorithm. It is clear that if is too small,then our method evolves as a traditional anisotropic diffu-sion process and results in an imperceptible enhancementeffect �see Fig. 16�a��. On the contrary, if is too large, ourmethod is highly unstable and often leads to noise amplifi-cation �see Fig. 16�g��. The statistical results in Table 4confirm that should be carefully selected. Our simulationson various images including those not reported here, con-firm that the parameter should be in the range of 1–3, sothat it can guarantee the realization of the tunable FABdiffusion instead of the classic anisotropic diffusion or in-verse diffusion.

The aim of this paper is to present a TFAB diffusionalgorithm and to apply it in image enhancement as well asimage sharpening. We focus on enhancing and sharpeningblurry signals, while still allowing some additive noise to

(a) (b)

(c) (d)

(e) (f)

Fig. 18 Comparison of the five diffusion methods using a MR im-age: �a� Brain MR image; �b–f� filtered images corresponding to theimage in �a� resulting from CLMC anisotropic diffusion �=0.05�, MBanisotropic diffusion �=0.05�, GSZ FAB diffusion �kf=0.25�MAG,kb=2�MAG, �=0.25�, LVCFAB diffusion �kf=50, kb=200, �1=0.1,�2=0.02, =0.25, �=0.5�, and the proposed TFAB diffusion ��=0.006, �=8, =4� �10 iterations�.

May 2010/Vol. 49�5�7


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nterfere with the process. To demonstrate the effectivenessf the proposed TFAB algorithm in noise reduction, theSNR and UIQI values of the original and enhanced im-ges for our algorithm with respect to different noise vari-nces are listed in Table 5. In addition, the parameter set-ings of the proposed algorithm at different noise levels areisted in Table 6. From the PSNR and UIQI values, we canee that the proposed TFAB algorithm can effectivelychieve noise reduction and greatly improve image quality.o compare the visual quality of the result from our algo-ithm, its enhanced images with respect to different noiseariances are shown in Fig. 17. By comparing the noisymage and enhanced image at different noise levels, we canee that sharpening and denoising can be reconciled by theroposed tunable mechanism that controls the orientation,ype, and extent of the diffusion process. Meanwhile, it isbvious from Table 6 that � is sensitive to noise levels andhould be lower as noise variance increases.

.3 Medical Imagesn medical images, low SNR and CNR often degrade thenformation and affect several image-processing tasks, such

(a) (b)

(c) (d)

(e) (f)

ig. 19 A regional enlarged portion of interest of original image andesults corresponding to Fig. 18�c�–18�g�.



as segmentation, classification, and registration. Therefore,it is of considerable interest to improve SNR and CNR toreduce the deterioration of image information. In this sec-tion, we present an example from magnetic resonance�MR� patient studies.

We present in Fig. 18 an example of the performance ofthe different diffusion algorithms for MR image enhancing.The original MR image is depicted in Fig. 18�a� with thesize of 256�256. The results for the five algorithms areshown in Figs. 18�b�–18�f�. As seen from all five enhancedimages, the smoothness in homogeneous regions, such aswhite matter, seems to be visually the same in all images,while the TFAB diffusion algorithm achieves greater con-trast and produces more reliable edges, which is especiallyuseful for segmentation and classification purposes neces-sary in medical image applications.

Figure 19 shows a regional enlarged portion of interestfrom Fig. 18�a� and the diffusive filtered results of thisportion using the five diffusion algorithms. As expected, thefive algorithms remove noise present in Fig. 19�a� and si-multaneously smooth the homogeneous regions. Accordingto the visual analyses of the image quality, the results gen-erated by the three FAB diffusion processes are compa-rable. However, the boundary contrast appears to be higherusing the proposed TFAB algorithm when compared toother algorithms. In addition, the TFAB diffusion enhancesboundary sharpness and fine structures better than otherdiffusion methods.

4 ConclusionDigital image acquisition techniques often suffer from lowSNR and CNR, which degrade the information contained inthe digital image and thus reduce its potential utility forindustry use. We have presented a novel TFAB diffusionalgorithm for image restoration and enhancement to im-prove on the SNR and CNR that preclude the current utilityof digital images for industry. The primary advantage of theproposed diffusion algorithm is that tenability of the im-proved diffusion coefficient offers user flexibility to adjustedge-enhancing performance. At the same time, it is notnecessary to consider the rational time-consuming strategyfor estimating the gradient threshold. The proposed algo-rithm was tested on various digital images, including fourgeneral images and a medical image. The results from oursimulations show an improvement in visual effect andquantitative analyses over the preexisting algorithms.

AcknowledgmentsThis work was supported, in part, by the following: theNational High Technology R&D �863� Program of Chinaunder Grant 2007AA12Z160; National Natural ScienceFoundation of China under Grants No. 40901205, No.40771139, and No. 40930532, Foundation of the State KeyLaboratory of Information Engineering in Surveying, Map-ping and Remote Sensing under Grant No. 08R02, Founda-tion of Key Laboratory of Mapping from Space of StateBureau of Surveying and Mapping under Grant No.200809, Key Laboratory of Mine Spatial Information Tech-nologies �Henan Polytechnic University, Henan Bureau ofSurveying and Mapping�, State Bureau of Surveying andMapping, and the Special Fund for Basic Scientific Re-search of Central Colleges, China University of Geo-

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ciences, Wuhan, under Grant No. GUGL090210. The au-hors would also like to thank the anonymous reviewers forheir valuable comments and suggestions which signifi-antly improved the quality of this paper.

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Yi Wang received his BS, MS, and PhD inphotogrammetry and remote sensing fromWuhan University, Wuhan, in 2002, 2004,and 2007, respectively. He is currently anassociate professor with the Institute ofGeophysics and Geomatics, China Univer-sity of Geosciences. His research interestsinclude remote sensing image processingusing partial differential equations, and par-ticularly, pattern recognition.

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http://dx.doi.org/10.1109/83.841940

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Ruiqing Niu received his BS, MS, and PhDfrom China University of Geosciences, Wu-han, in 1992, 2001, 2005, respectively. Heis currently an associate professor with theInstitute of Geophysics and Geomatics,China University of Geosciences. His re-search interests include geovisualization,environmental remote sensing, and geologi-cal remote sensing.

Xin Yu received his PhD in pattern recogni-tion and intelligent system from HuazhongUniversity of Science and Technology, Wu-han, China, in 2007. He is currently a Lec-turer with the Institute of Geography andGeomatics, China University of Geo-sciences. His research interests includecomputer vision, pattern recognition, andtheir applications in remote sensing and ge-ology.

Liangpei Zhang received his BS in physicsfrom Hunan Normal University, Changsha,China, in 1982, MS in optics from the Xi’anInstitute of Optics and Precision Mechanicsof the Chinese Academy of Sciences, Xi’an,in 1988, and PhD in photogrammetry andremote sensing from Wuhan University, Wu-han, China, in 1998. From 1997 to 2000, hewas a professor with the School of the LandSciences, Wuhan University. In August2000, he joined the State Key Laboratory of

nformation Engineering in Surveying, Mapping, and Remote Sens-ng, Wuhan University, as a professor and head of the Remoteensing Section. He has published more than 120 technical papers.is research interests include hyperspectral remote sensing, high-

esolution remote sensing, image processing, and artificial intelli-ence. Dr. Zhang has served as co-chair of the SPIE Series Con-

erences on Multispectral Image Processing and Pattern



Recognition �MIPPR� and the Conference on Asia Remote Sensingin 1999; editor of the MIPPR01, MIPPR05, Geoinformatics Sympo-siums; associate editor of the Geo-spatial Information Science Jour-nal; on the Chinese National Committee for the InternationalGeosphere-Biosphere Programme; and as executive member forthe China Society of Image and Graphics.

Huanfeng Shen received his BS, MS, andPhD in photogrammetry and remote sens-ing from Wuhan University, Wuhan, in 2002,2004, and 2007, respectively. He is cur-rently an associate professor at the Schoolof Resource and Environmental Science,Wuhan University. His main research inter-est is image superresolution enhancement.

May 2010/Vol. 49�5�0


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