Pattern Recognition Letters 30 (2009) 314–320

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Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

Image feature enhancement based on the time-controlledtotal variation flow formulation

Ovidiu Ghita *, Dana E. Ilea, Paul F. WhelanVision Systems Group, School of Electronic Engineering, Dublin City University, Dublin 9, Ireland

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 November 2007Received in revised form 19 June 2008Available online 1 November 2008

Communicated by W. Zhao

Keywords:TV flowAnisotropic diffusionFeature enhancementNumerical stability

0167-8655/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.patrec.2008.10.004

* Corresponding author. Tel.: +353 1 7007637; fax:E-mail address: ghitao@eeng.dcu.ie (O. Ghita).

Data smoothing and feature enhancement are two important precursors to many higher-level computervision applications such as image segmentation and scene understanding. Total variation (TV) flow algo-rithms are a distinct subcategory of diffusion-based filtering techniques that have been widely applied toreduce the level of noise in the image but not at the expense of poor feature preservation. In this paper weaddress a number of numerical aspects associated with the TV flow and in particular we are interested toredefine the TV flow regularization in order to reduce the effect of oscillations and improve the conver-gence of the implementations in the discrete domain. TV flow algorithms are implemented using iterativeschemes and one difficult problem is the selection of appropriate criteria to identify the optimal numberof iterations. In this paper we show that the application of a time-ageing procedure leads to an elegantformulation were the TV flow algorithms converge naturally to the optimal solution. To evaluate the per-formance of the proposed algorithm (referred in this paper to as time-controlled (TC)-TV flow), a largenumber of experiments on various types of natural images were conducted.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

One important step in the process of image segmentation is toreduce the errors caused by the image noise and local in-homoge-neities. Typically, the image noise is reduced by the application oflocal averaging operators such as the Gaussian, but this approachleads to the introduction of image blur and the attenuation ofimportant contextual features in the image such as edges. To com-pensate for these undesirable effects associated with linearsmoothing strategies, non-linear approaches have been developedto achieve better feature preservation (Grahs et al., 2002; Keelingand Stollberger, 2002; Sonka et al., 1998). Among non-linear tech-niques, the seminal work detailed in the paper by Perona and Malik(1990) received a special attention from the vision community(Ghita et al., 2005; Ilea and Whelan, 2007; Smolka and Plataniotis,2002; Weickert et al., 1998). Total variation (TV) flow algorithmshave been viewed by many authors as a special case of the geomet-rical-driven anisotropic diffusion (Dibos and Koepfler, 1999;Gothandaraman et al., 2001; Petrovic et al., 2004; Rudin et al.,1992; Strong and Chan, 2003). The TV flow formulation has beenevaluated from a numerical perspective by Breuß et al., 2006 andthey concluded that this data smoothing feature preserving tech-nique is well-posed and it leads to constant signals in finite times.However, in their paper they stressed that the discrete representa-tions of the standard TV flow show instability in areas defined by

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+353 1 7005508.

zero gradients and the implementations in the discrete (image) do-main should be approached with care.

In this paper we follow the work detailed in (Breuß et al., 2006)and we focus on a number of numerical aspects associated with theTV flow formulation with a view of improving the numerical stabil-ity and the convergence time. Also in line with other non-linearsmoothing strategies such as anisotropic diffusion (Ilea andWhelan, 2007; Smolka and Plataniotis, 2002; Weickert, 1998), theTV flow formulation is implemented as an iterative scheme wherethe number of iterations is typically a user defined parameter(Andreu et al., 2001; Breuß et al., 2006; Petrovic et al., 2004). In thispaper we show that the application of a time-ageing procedure tothe time step size parameter, implements a data smoothing frame-work where the evolution in time of the TV flow becomes predict-able and the algorithm will converge naturally to the optimalresult. This paper is organised as follows. In Section 2 the mathe-matical background behind the TV flow formulation is discussed.Section 3 details the numerical implementation of the TV flowand we approach the algorithmic solutions applied to improvethe numerical stability. In Section 4 a number of experimental re-sults are analysed while Section 5 concludes this paper.

2. TV Flow. Mathematical background

As mentioned in the previous section, the TV flow is a specialcase of the anisotropic diffusion function that has been firstproposed by Perona and Malik (1990) (PM)

O. Ghita et al. / Pattern Recognition Letters 30 (2009) 314–320 315

@u@t¼ r g

rukruk

� �� �; ð1Þ

where u(x,y,t) is the processed data at time t and r is the gradientoperator. In the PM formulation the diffusion function g is designedto control the smoothing based on the strength of the gradient. Inthe original paper (Perona and Malik, 1990), this function is imple-mented using exponential or reciprocal forms and the value of g isbounded in the interval (0, 1], g ? 0 when ru ?1. In the TV flowformulation, the functional g(ru) is replaced withru and this sim-ple modification leads to a process where the data is diffused morein image areas with small discontinuities in the intensity data andthe diffusive process is stopped in image locations where the gradi-ents have large values

@u@t¼ r ru

kruk

� �: ð2Þ

In the one-dimensional (1D) case, this process can be formulated asthe minimisation of the following functional called TV regulariza-tion (Breuß et al., 2006; Grahs et al., 2002; Strong and Chan, 2003):

TVðuÞ ¼Z

Xðu� f Þ2 þ 2t

@u@x

��������dx; uðt ¼ 0Þ ¼ f ; ð3Þ

where f is the initial data, t is a time parameter and X is the imagedomain. The main property of the function illustrated in Eq. (3) isthat it does not penalise the image discontinuities and the smooth-ing is modulated by the descent of the gradient magnitude. In Eq. (2)it is noticeable that the TV flow formulation becomes unstable whenru ? 0 and to circumvent this problem Breuß et al. (2006) proposeda numerical implementation using the following regularization:

@u@t¼ r ruffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2 þru2q

0B@

1CA; ð4Þ

where b is a small positive regularization term.

0

50

100

150

200

250

1 26 51 76 101

126

151

176

201

226

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476

0

50

100

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250

1 28 55 82 109

136

163

190

217

244

271

298

325

352

379

406

433

460

487

Fig. 1. Results of the TV flow. (a) Original signal and the noisy signal. (b and c) The result(d) The result from our TC-TV algorithm (number of iterations detected automatically u

3. Numerical implementation of the TV flow

The implementation of the TV flow in the discrete domainwould require a simple approximation of the partial derivativeswith the central differences in the (x,y,t) space. In this study, wehave extended the approach detailed in (Breuß et al., 2006) tothe two-dimensional (2D) grid G as follows:

G¼fðx;y;tÞjx¼ iDx;y¼ jDy;t¼nDt; i; j;n2N;

Dx>0;Dy>0;Dt>0g; ð5Þ

unþ1=2i;j ¼un

i;jþDtDx2

uniþ1;j�un

i;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ

uniþ1;j�un

i;j

Dx

� �2r �

uni;j�un

i�1;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ

uni;j�un

i�1;jDx

� �2r

0BB@

1CCA; ð6Þ

unþ1i;j ¼unþ1=2

i;j þ DtDy2

unþ1=2i;jþ1 �unþ1=2

i;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ

unþ1=2i;jþ1

�unþ1=2i;j

Dy

� �2s �

unþ1=2i;j �unþ1=2

i;j�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ

unþ1=2i;j

�unþ1=2i;j�1

Dy

� �2s

0BBBB@

1CCCCA;

ð7Þ

where n is the iteration index, Dx and Dy are the discrete spatialdistances of the image grid, Dt is the time size step parameterand b is the regularization term. If we consider that the datapointui,j is a local maxima or minima, we can demonstrate that the dis-crete implementation of the TV flow is stable (for more details referto Breuß et al., 2006) if Eq. (8) is upheld. This equation indicates thatthe parameters b and Dt are not mutually independent and theyneed to be carefully selected

maxDtDx2 ;

DtDy2

� �6

b2: ð8Þ

0

50

100

150

200

250

1 26 51 76 101

126

151

176

201

226

251

276

301

326

351

376

401

426

451

476

0

50

100

150

200

250

1 28 55 82 109

136

163

190

217

244

271

298

325

352

379

406

433

460

487

s from the standard S-TV flow when applied to 100 and 500 iterations, respectively.sing Eq. (12)).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1

102

203

304

405

506

607

708

809

910

1011

1112

1213

1314

1415

Iteration

Osc

illat

ion

ampl

itude

Fig. 2. Convergence of the TV flow algorithm. (a) Original image (BSDB, 2001). (b)The result obtained after the application of the S-TV flow procedure (300 iterations).(c) The result obtained after the application of the S-TV flow procedure (1500iterations). (d) The result obtained after the application of our TC-TV flow algorithm(1500 iterations). (e) The amplitude of oscillations with respect to the number ofiterations (in blue/dark – S-TV flow implementation, in red/gray – TC-TV flowimplementation). In this experiment the TV flow parameters are set to the followingvalues: Dt = 0.1, c = 0.995, b = 0.1. Note that all illustrations in this paper are bestviewed in colour. (For interpretation of the references to colour in figure legendsand in text, the reader is referred to the web version of this article.)

316 O. Ghita et al. / Pattern Recognition Letters 30 (2009) 314–320

3.1. Time-controlled TV flow formulation

From the two parameters that control the smoothing process (band Dt), the time size Dt has a larger influence on the intensity ofthe smoothing process and this observation is motivated by thefact that this parameter has a multiplicative effect. Surprisingly,in the original implementation (Breuß et al., 2006) this parameteris kept constant during the iterative process and as a result withthe increase in the number of iterations the amplitude of oscilla-tions does not decrease monotonically. This can be demonstratedif we consider again that a datapoint ui,j is a local maxima or min-ima and we set the regularization parameter b to a value closer tozero. Thus, the maximal value that is applied to update the value ofthe datapoint unþ1

i;j in Eqs. (6) and (7) is

max uniþ1;j � un

i;j

��� ���; uni;j � un

i�1;j

��� ���; uni;jþ1 � un

i;j

��� ���; uni;j � un

i;j�1

��� ���n o6

2DtDx

:

ð9Þ

Eq. (9) indicates that the oscillations are bounded to a finite value(for the sake of simplicity we assume that Dx = Dy), but there isno guarantee that they will follow a monotonic decrease. In fact,in our experiments we found that the amplitude of oscillationsstarts to decrease only after the TV flow process is applied for alarge number of iterations. Nonetheless, this approach is not appro-priate for two reasons. Firstly, with the increase in the number ofiterations the computational load required to smooth the imagedata is high. Secondly, this precludes the application of numericalsolutions to identify the optimal number of iterations.

To address these problems we propose an elegant solutionwhere a time-controlled (TC) ageing procedure is applied to de-crease the value of the parameter Dt after each iteration (this ap-proach is referred to as TC-TV flow). The time-controlled ageingprocedure has the role of cooling the smoothing process with theincrease in the number of iterations and this can be implementedusing either the linear (Eq. (10)) or exponential (Eq. (11)) function

Dtnþ1 ¼ Dt0; n ¼ 0;Dtnþ1 ¼ Dtn � c; n > 0 and c 2 ð0;1�;

(ð10Þ

Dtnþ1 ¼ Dt0 � e�nc; c > 0; ð11Þ

where Dt0 is the initial value of the time size parameter (at iterationzero). In our experiments the implementation detailed in Eq. (10)proved to be more stable with respect to the selection of the cparameter and in this paper the reported results are obtained whenthe ageing procedure is implemented using the linear form. Fig. 1illustrates the results when the standard TV (S-TV) flow and ourTC-TV flow algorithms were applied to a 1D noisy test signal. Forclarity purposes, the results obtained after the application of theTV flow algorithms were superimposed on the original noiselesssignal.

In Eqs. (10) and (11) the parameter c controls to what extent thesmoothing procedure is cooled and it has the effect of lowering theamplitude of the oscillations at each iteration (see Eq. (9)). Thisprocess is illustrated in Fig. 2.

The application of the ageing procedure has another beneficialcontribution, namely it allows us to derive a numerical solutionfor the identification of the optimal number of iterations. In Eq.(9) it can be observed that the absolute maximal updating valueis bounded to 2Dt0/Dx, thus, taking in consideration that the oscil-lations can be either positive or negative we can set the conver-gence criteria in the following way:

maxi;j2X

unþ1i;j � un

i;j

��� ���� �<

Dt0

Dx; ð12Þ

where X is the image domain. The selection of the parameters Dt0, band c is usually carried out based on experimentation.

For instance, if the parameter Dt0 is set to a large value, thealgorithm will converge faster but the results returned by the TVflow will show a significant amount of blur. Conversely, if theparameter Dt0 is set to a low value such as 0.05 or lower thenthe algorithm will preserve better the features in the image, butthis advantage is obtained at the expense of a high computationalcost since the algorithm will require a large number of iterations toconverge. Fig. 3 illustrates the results obtained when the S-TV andTC-TV flow algorithms were applied to the image depicted inFig. 2a and the time size parameter (Dt0) is set to 0.05. From theexperimental results depicted in Figs. 2 and 3 it can be observedthat the selection of the time size parameter plays an importantrole for the S-TV algorithm while the results returned by theTC-TV flow are virtually unaffected by the selection of this param-eter (see the results depicted in Figs. 2d and 3d). Fig. 3 also illus-trates the results obtained after the TC-TV flow is applied for1500 iterations and for the number of iterations calculated

Fig. 3. Results of the TV flow when applied to the image depicted in Fig. 2a. (a–c) The results after the application of the S-TV flow procedure for 50, 300 and 1500 iterations,respectively. (d and e) The results after the application of our TC-TV flow algorithm for 1500 iterations and for the number of iterations calculated using Eq. (12), respectively.In this experiment the TV flow parameters are set to the following values: Dt = 0.05, c = 0.995, b = 0.1.

Fig. 4. Performance of the TC-TV flow when the ageing parameter c is varied. (a) c = 0.99. (b) c = 0.995. (c) c = 0.9995. To isolate the effect of the ageing parameter c on theperformance of the TC-TV flow in this experiment the time size and regularization parameters were maintained constant and set to the following values: Dt0 = 0.1, b = 0.1.

Fig. 5. TV flow results. (a) Original image (BSDB, 2001). (b–d) Results returned by the S-TV flow algorithm when executed for 250, 500 and 1000 iterations, respectively. (e)Result returned by the TC-TV flow algorithm (the number of iterations selected automatically using Eq. (12)). (f and g) Close-up details from (b) S-TV flow (250 iterations) and(e) TC-TV flow, respectively.

O. Ghita et al. / Pattern Recognition Letters 30 (2009) 314–320 317

318 O. Ghita et al. / Pattern Recognition Letters 30 (2009) 314–320

automatically using Eq. (12). It can be observed that the results de-picted in Fig. 3d and e show similar levels of smoothing and this ismotivated by the fact that due to the monotonic descent of the age-ing function illustrated in Eq. (10) the smoothing effect has a neg-ligible effect after the algorithm reaches the number of iterationscalculated using Eq. (12).

The regularization parameter b has a lower influence on theperformance of the TV flow and is typically set to a low value in

Fig. 6. TV flow results. (a) Original image. (b–d) Results returned by the S-TV flow algorithby the TC-TV flow algorithm. (f and g) Close-up details from (b) S-TV flow (250 iteratio

Fig. 7. TV flow results. (a) Natural image corrupted with Gaussian noise (standard deviatalgorithm when executed for 250, 500 and 1000 iterations, respectively. (e) Result retu

the interval [0.05, 0.2]. The parameter c controls the descent ofthe time size parameter and to avoid a step decrease of Dt that willresult in a premature termination of the TC-TV flow it has to be setto a positive value close to 1 (c 6 1). Fig. 4 shows a number of re-sults after the application of the TC-TV flow to the image depictedin Fig. 2a when the ageing parameter c is varied. The results de-picted in Fig. 4 indicate that a selection of the ageing parameterc close to 1 (see Fig. 4c) would generate a smoothing framework

m when executed for 250, 500 and 1000 iterations, respectively. (e) Result returnedns) and (e) TC-TV flow, respectively.

ion 20 gray-levels on each colour channel). (b–d) Results returned by the S-TV flowrned by the TC-TV flow algorithm.

Table 1Computational overhead associated with the standard (S-TV) and proposed TC-TVflow algorithms.

Image Size S-TV flow (500 iteration) (s) TC-TV flow (s)

Fig. 2a 256 � 171 51.58 39.53Fig. 6a 256 � 256 80.30 62.68Fig. 7a 256 � 171 51.80 38.84Fig. 8a1 256 � 171 52.18 34.43Fig. 8a2 256 � 171 51.74 39.14Fig. 8a3 257 � 269 82.74 62.50Fig. 8a4 256 � 171 51.80 40.45

O. Ghita et al. / Pattern Recognition Letters 30 (2009) 314–320 319

that is similar with the S-TV flow (the monotonic descent of theageing function is very slow). On the other hand, a selection ofthe ageing parameter c to a value lower than 0.99 would forcethe TC-TV flow algorithm to a premature termination as illustratedin Fig. 4a. In our experiments we have observed that theoptimal values for the ageing parameter c are in the interval[0.991, 0.999].

Fig. 8. Additional results. Column (a) – original images (BSDB, 2001). Column (b) – resulshift filtering algorithm (rs, rr) = (8,4).

4. Experiments and results

In this study we focus our investigation on the adaptivesmoothing of colour images. To achieve this we have extendedthe smoothing model depicted in Eqs. (6) and (7) to cover datawhere each datapoint is formed by a three-dimensional (3D) vectorwhose elements are defined by the colour components. One of themain issues associated with the formulation detailed in Section 2 isthe poor convergence of the standard TV flow algorithms and tocompensate for this problem we have implemented a cooling pro-cedure that acts upon the time size parameter to force the ampli-tude of oscillations to follow a monotonic descent with theincrease in the number of iterations. As indicated in Section 3 theintroduction of this ageing procedure allows us to identify numer-ically the number of iterations (see Eq. (12)) and in our experi-ments we evaluate the effectiveness of this procedure. Figs. 5–7depict a number of experimental results when the standard S-TVand our TC-TV flow implementations were applied both to noise-less data and to images that are artificially corrupted with noise(Gaussian noise with a standard deviation of 20 gray-levels on each

ts returned by the TC-TV flow algorithm. Column (c) – results returned by the mean

320 O. Ghita et al. / Pattern Recognition Letters 30 (2009) 314–320

colour channel). In the experimental results shown in Figs. 5–8 thenumber of iteration for TC-TV flow is determined automaticallyusing Eq. (12).

The results depicted in Figs. 5 and 6 indicate that the standardS-TV flow introduces an undesired level of blur while our TC-TVflow algorithm is able to achieve superior results. This undesiredblur is obvious in image regions defined by the cart’s wheel (seeFig. 5f (S-TV flow) and Fig. 5g (TC-TV flow)) and in the image areasrepresenting the fur in Fig. 6 (see the close up details shown inFig. 6f (S-TV flow) and Fig. 6g (TC-TV flow)). The improved perfor-mance of the TC-TV flow algorithm when compared with the stan-dard algorithm was expected since the strength of the smoothingprocess is decreasing with the increase in the number of iterations.We have also found in our experiments that the TV flow algorithmsproduce robust results even in cases when they are applied toimages corrupted with a high level of image noise. Fig. 7 depictsthe results returned by the standard and TC-TV flow implementa-tions when applied to an image corrupted with Gaussian noisewith standard deviation of 20 gray-levels on each colour channel.Another set of experiments was conducted to evaluate the compu-tational overhead associated with the TV flow algorithms analysedin this study and the results are depicted in Table 1. While thecomputational time required to compute the standard S-TV flowfor one iteration is constant, the computational load when thealgorithm is executed for 250 and 1000 iterations can be obtainedby multiplying the computational time required to execute thealgorithm for 500 iteration with a factor of 0.5 and 2, respectively.The experiments have been conducted using a Pentium 1.7 GHz PC,500 MB RAM memory and running Windows 2000.

Fig. 8 presents some additional results that illustrate the perfor-mance of our TC-TV flow algorithm when compared with that of-fered by the mean shift filtering (Comaniciu and Meer, 2002).These experiments were conducted on a set of standard testimages from Berkeley database (BSDB, 2001). In all experimentsthe TC-TV flow parameters Dt0, b and c were set to 0.1, 0.1 and0.995, respectively. The parameters that select the space (rs) andthe range (rr) resolutions of the mean shift filtering were set to 8and 4, respectively, as recommended in (Comaniciu and Meer,2002). The experimental results depicted in Fig. 8 indicate thatthe TC-TV flow algorithm outperforms the mean shift filteringwhen the results are analysed with respect to noise reductionand feature preservation.

5. Conclusions

The aim of this paper was to address some numerical aspectsrelated to the stability of the discrete TV flow implementations.In this sense, we briefly detailed the methodology to implementthe TV flow algorithms in the discrete domain and we have deviseda number of improvements that increased the overall stability ofthe TV flow scheme. As indicated in other papers that have ad-

dressed the numerical stability of the TV flow algorithms (Andreuet al., 2001; Breuß et al., 2006; Grahs et al., 2002), the most difficultproblems associated with the implementation in the discrete do-main is to control the level of oscillations and to select the optimalset of parameters. In this paper we demonstrated that the applica-tion of an ageing procedure that adjusts the value of the time sizeparameter with the increase in the number of iterations, leads toan efficient computational framework where the amplitude ofoscillations decrease monotonically. We have also demonstratedthat the application of the ageing procedure allows the develop-ment of a numerical solution to identify the optimal number ofiterations. The experimental data indicated that the TV flow formu-lations are robust pre-processing schemes that can be successfullyincluded in the development of a large number of algorithms rang-ing from adaptive data compression to image segmentation.

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