INTERNATIONAL JOURNAL OF c© 2014 Institute for ScientificNUMERICAL ANALYSIS AND MODELING Computing and InformationVolume 5, Number 3, Pages 255–268

IMAGE DENOISING USING LLT MODEL AND ITERATED

TOTAL VARIATION REFINEMENT

FENLIN YANG, KE CHEN, BO YU, AND ZHIGANG YAN

Abstract. Developing a variational model that is capable of restoring both smooth (no edges)and non-smooth (with edges) images is still a valid challenge in the image processing. In thispaper, we present two methods for image denoising problems based on the use of the LLT model(see [14]) and iterated total variation refinement. The idea of our methods is, first make use ofthe LLT model to get a smooth primal sketch, and then get some meaningful signal by iteratedtotal variation refinement from the removed noise image. Numerical experiments show that ourmethod is able to maintain some important information such as small details in the image, andat the same time to get a better visualization.

Key words. Image denoising, staircasing effect, primal sketch, hierarchical decomposition, iter-ated regularization.

1. Introduction

Image denoising is an essential and fundamental pre-processing phase for furtherimage processing tasks such as edge detection, pattern recognition, and objecttracking, etc. The task is to extract a “quality” image u from the observed imagef by the degradation model

f = u+ η,(1)

where η is an additive noise term.There are many different variational techniques proposed to obtain an estimate

of u [10, 14, 16, 25]. One effective and well-known method is the total variation-based (TV) model by Rudin, Osher and Fatemi [20], which minimizes the totalvariation over the space of bounded variation Ω,

minu

α

∫

Ω

|∇u|dxdy +1

2‖u− f‖2.(2)

Here | · | is the Euclidean norm in R2, ‖ · ‖ is the norm in L2(Ω), α is a positive pa-

rameter controlling the trade-off between goodness of fit-to-the-data and variabilityin u. On the one hand, since the Euler-Lagrange partial differential equation (PDE)is the second order PDE

g1(u) = −α∇ ·

(

∇u√

|∇u|2 + β

)

+ (u− f) = 0, (x, y) ∈ Ω,(3)

with homogeneous Neumann boundary condition ∂u/∂~n = 0, where β is a smallpositive parameter, and ~n is the normal vector of the boundary, there are many fastmethods for (3) up to now (see [6, 7, 9, 17, 18, 20, 23]). On the other hand, thismodel can preserve shape edges and boundaries with a high quality recovery. Butfor images without edges (jumps), the solution to this model has the undesirable

Received by the editors February 8, 2014.2000 Mathematics Subject Classification. 35R35, 49J40, 60G40.The research was supported by the Natural Science Foundation of Hunan Province (13JJB014)

and the National Natural Science Foundation of China (11171051,91230103).

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256 F. YANG, K. CHEN, B. YU, AND Z. YAN

staircasing effect. Some effort has been made to remedy this unfavorable property[3, 4, 8, 11, 12, 14, 15, 17, 19, 21, 27].

In [14], Lysaker, Lundervold and Tai (LLT) proposed a second order functionalminimization by the following formula:

minu

α

∫

Ω

|D2u|dxdy +1

2‖u− f‖2,(4)

where |D2u| =√

u2xx + u2

xy + u2yx + u2

yy is a second-order convex functional. The

corresponding Euler-Lagrange PDE for (4) is

g2(u) = α

[

( uxx

|D2u|β

)

xx+( uxy

|D2u|β

)

yx+( uyx

|D2u|β

)

xy+( uyy

|D2u|β

)

yy

]

(5)

+(u− f) = 0,

where |D2u|β =√

u2xx + u2

xy + u2yx + u2

yy + β (β > 0). It is known that the high-

order PDEs can recover smoother surfaces. However such models cannot preservesharp features such as jumps; it is a challenge for a single model to restore bothsmooth and non-smooth images.

Zhu and Chan [27] want to find a piecewise smooth surface to approximatethe image surface by incorporating the corresponding geometric quantities – meancurvature into the processing of denoising

minu

α

∫

Ω

Φ(

κ)

dxdy +1

2‖u− f‖2.(6)

The functional Φ is defined either as Φ(κ) = |κ|, Φ(κ) = κ2 or a combination ofboth, here κ is the mean curvature of the image which is defined by

κ = ∇ ·∇u

|∇u|.(7)

Although the mean curvature model can avoid the staircase effect, the fourth orderpartial differential equations (PDE) arising from minimization of this model is

gβ(u) = α∇ ·

(

1

|∇u|β

(

I2 −∇u∇uT

|∇u|2β

)

∇Φ′(

κβ

)

)

+ (u− f) = 0,(8)

∇u · ~n = 0 (x, y) ∈ ∂Ω,

where I2 ∈ R2×2 is the identity matrix and κβ = ∇ · (∇u/

√

|∇u|2 + β), the con-struction of stable numerical schemes for the above PDE is very difficult due tohigh nonlinearity and stiffness. In [26], Yang, Chen and Yu used a homotopy ideato devise a feasible method. But an equation of type (8) has to be solved severaltimes.

In recent years, among others, researchers have turned to the combination TVmodel and LLT model (see [16, 10]). Lysaker and Tai [16] suggested a convexcombination of the respective two solutions from (3) and (5). Specifically, withw0 = f , a new iteration wk+1 is generated by the convex combination

wk+1 = θkvk+1 + (1− θk)uk+1 k = 0, 1, 2 · · · ,(9)

where vk+1 and uk+1 are respectively obtained by the kth time marching iterationof TV model and LLT model with wk as their old iteration. Here the parameter θk

IMAGE DENOISING 257

which is applied to control the combination depends on ∇wk as follows:

θk =

1, if |∇wk| ≥ c,

0.5 cos(2π|∇wk|/c) + 0.5, elsewhere,(10)

where c is some constant parameter in the interval [0, 1]. We remark that the TGVmethod by Bredies, Kunisch and Pock [2] may be viewed also as a combinationmethod. Other simpler combination methods include Blomgren, Chan, and Mulet[1], and a weighted H1 seminorm regularization method for Fredholm integral e-quations of the first kind by Lin and Yang [13].

The above convex combination solution (9) reduces to the TV solution in regionswhere |∇u| is large (near edges) or to the LLT solution where |∇u| is 0 (flat regions).It would be better to use the TV solution when |∇u| ≈ 0 i.e. not exactly 0 andalso one may wish to solve a single PDE (from a combined optimization) instead ofsolving two separate PDEs. This is the idea taken up in Chang, Tai and Xing [10]who proposed a new combination of the TV model and the LLT model in the form

minu

[

α(

∫

Ω

θ|∇u|dxdy +

∫

Ω

(1− θ)|D2u|dxdy)

+1

2‖u− f‖2

]

,(11)

and its Euler-Lagrange equation is

α

− θ∇ ·

(

∇u√

|∇u|2 + β

)

+ (1− θ)

[

( uxx

|D2u|β

)

xx+( uxy

|D2u|β

)

yx(12)

+( uyx

|D2u|β

)

xy+( uyy

|D2u|β

)

yy

]

+ (u− f) = 0.

Here the variable parameter θ is chosen as:

θ =

1, if |∇u| ≤ C0 and |∇u| ≥ C1,

Cd, if C0 + 5 ≤ |∇u| ≤ C1 − 5,

1−(|∇u| − C0)(1 − Cd)

5, if C0 ≤ |∇u| ≤ C0 + 5,

1 +(|∇u| − C1)(1 − Cd)

5, if C1 − 5 ≤ |∇u| ≤ C1.

(13)

Note that this varying θ suggests a global iteration scheme which is exactly theimplementation of [10]. In computation, the parameters C0 = 0, and Cd = 0.05 arefixed. The parameter C1 is taken as 50 for most images and is properly modifiedfor some images.

Numerical results show these algorithms can inherit the advantages of the TVmodel and the LLT model, and avoid the disadvantages of both models in somedegree. However, as far as the restored smooth parts are concerned, theirs are notas well as that for the LLT model, and the restored edges are not as well as thatfor TV model.

In this paper, we intend to restore effectively both smooth images (with no clearjumps) and blocky images (of piecewise constant intensities) by taking the mostmeaningful signals depicted by f . Note that the LLT model can recover smoothersurfaces, so our procedure would be to decompose the f into a primal sketch u andthe additive noise η by the LLT model, and then find some signal from the η byiterated total variational refinement.

The rest of this paper is organized as follows. In Section 2 we first reviewthe iterated total variation refinement. In Section 3 we describe our method in

258 F. YANG, K. CHEN, B. YU, AND Z. YAN

detail. Finally, we give the numerical results of the implementation of the proposedalgorithms on several tests in Section 4.

2. Iterated total variation refinement

The ideal result of the denoising method would be decompose f into the truesignal u and the additive noise η without any signal. In practice, this is not fullyattainable. Take the TV model for example, the removed noise is treated as anerror, some details, such as texture will be swept as an error. In more recent yearssome effort has been made to extract more meaningful signals from the noise partη (see [19, 22]).

2.1. The hierarchical decomposition. Tadmor, Nezzar and Vese [22] proposea multiscale image decomposition procedure which uses a hierarchical, adaptiverepresentation. The starting point is obtained by a variational decomposition of animage

[u0, η0] = arg minf=u+η

α0

∫

Ω

|∇u|dxdy +1

2‖u− f‖2

,(14)

f = u0 + η0.

Remarkably, u0 extracts the main features of f while η0 captures small texturesand oscillatory details. So taking u0 to contain only geometric information, theydecompose

[u1, η1] = arg minη0=u+η

α1

∫

Ω

|∇u|dxdy +1

2‖u− η0‖

2

,(15)

η0 = u1 + η1,

where u1 is assumed to be again a geometric part and η1 contains less geometricinformation than η0. Proceed with successive application of the dyadic refinement

[uj+1, ηj+1] = arg minηj=u+η

αj+1

∫

Ω

|∇u|dxdy +1

2‖u− ηj‖

2

,(16)

ηj = uj+1 + ηj+1,

where j = 1, 2, · · · . After k steps, we get the following hierarchical decompositionof f :

f = u0 + η0

= u0 + u1 + η1

= · · · · · ·

= u0 + u1 + · · ·+ uk + ηk.

Here ηk is the noise residue and the denoised image is given by

u = u0 + u1 + u2 + · · ·+ uk.

IMAGE DENOISING 259

2.2. Iterated regularization. In order to extract some signals from the removednoise image. Osher et al. [19] proposed to added the removed noise back to theoriginal noise image f , and the sum then processed by the TVmodel. The procedureis

step 1. Set η0 = 0, k = 0.step 2. compute uk+1 as a minimizer of the modified TV model,

uk+1 = argminu

α

∫

Ω

|∇u|dxdy +1

2‖f + ηk − u‖2

.

step 3. update

ηk+1 = ηk + f − uk+1.

set k := k + 1, then return to step 2.

It is enough to proceed iteratively until the result gets noiser or the distance ‖uk −u‖2 gets smaller than σ2, where σ is the standard deviation of the added noise.

3. Algorithms based on the LLT model and iterated total variation re-

finement

In the ideal denoising case, ideal methods would be restoring effectively bothblocky images (of piecewise constant intensities) and smooth images (with no clearjumps). As in the above discussion, we know a smooth primal sketch u can beobtained by the LLT model, and the removed noise f − u is texture plus noise.The idea of our methods is using the LLT model to get a smooth cartoon, and thenextract some textures from the removed noise by iterated total variation refinement.In this section, we will introduce two algorithms for image denoising.

3.1. Algorithm 1. The hierarchical decomposition is one of the efficient algo-rithms for extracting texture and small details from the removed noise image. Ouralgorithm 1 is constructed by using it to get more meaningful signals from the re-moved noise part, and the primal sketch is obtained by the LLT model. The detailsof our algorithm are given in the following

1. First, solve the original LLT model

u0 = argminu

α0

∫

Ω

|D2u|dxdy +1

2‖u− f‖2

to obtain the decomposition f = u0 + η0.2. Proceed with successive application of the dyadic refinement

uj+1 = argminu

αθj+1

∫

Ω

|∇u|dxdy +1

2‖u− ηj‖

2

, j = 0, 1, 2, · · · ,

and obtain the decomposition ηj = uj+1 + ηj+1.3. After k steps, we get the denoised image

u = u0 + u1 + · · ·+ uk.

Here α0 and α are regularization parameters, and θ (0 < θ ≤ 1) is the weightparameter to tune α. Since the cartoon u0 is obtained by the LLT model, it takesfull advantages of the LLT model. Moreover, the iterated total variation refinementusing the hierarchical decomposition extracts the refine geometric part of the imagefrom the noise residue.

260 F. YANG, K. CHEN, B. YU, AND Z. YAN

3.2. Algorithm 2. The iterated regularization method leads to another efficientalgorithms for extracting texture and small details from the removed noise image.Our algorithm 2 is constructed by using this method to get more meaningful signalsfrom the removed noise part, and the primal sketch is obtained by the LLT model.The details of our algorithm are given in the following

1. First, solve the original LLT model

u0 = argminu

α0

∫

Ω

|D2u|dxdy +1

2‖u− f‖2

to obtain the decomposition f = u0 + η0.2. For k = 0, 1, 2, · · · , compute uk+1 as a minimizer of the modified TV model,

uk+1 = argminu

α

∫

Ω

|∇u|dxdy +1

2‖f + ηk − u‖2

,

and update

ηk+1 = ηk + f − uk+1.

3. The denoised image is given by u = u0 + uk+1.

Here we replace the hierarchical decomposition by iterated regularization methodto get the most meaningful signals from the noise residue. In [19], Osher, et al.clarify that the sequence uk converges monotonically to η0 in L2, as k → ∞. Sowe use k = 1 in the following tests.

4. Numerical experiments and discussions

In this section we present some of the results we have obtained by the compar-isons of our algorithms with other classical denoising methods. We use the signalto noise ratio (SNR)

SNR = 10 log10

n∑

i=1

n∑

j=1

u2i,j

n∑

i=1

n∑

j=1

(ui,j − ui,j)2,

and the difference between a digital image and its denoised version are used tomeasure the quality of the restored images.

4.1. Comparisons of our algorithms with the TV model and the LLT

model. Below we shall refer to the method of our proposed methods with the TVmodel and LLT model. Since the TV model does well in “blocky” images and LLTmodel works almost perfectly for smooth images, we choose the standard “Lena”image as the test image which is composed of flat subregions, subregions with asmooth change in intensity value and jumps. The original and noisy images areshown in Figure 1. From the restored results of Figure 2 and Figure 3, we see thatthe recovered images by the LLT model and our algorithms are visually better thanthe TV model, and images denoised by the TV model and our methods preservesthe edges better than the LLT model.

To highlight our algorithms can restore effectively both smooth images (with noclear jumps) and blocky images (of piecewise constant intensities), we extract theflat subregions and the smooth subregions of the original, noisy and restored imagesof “Lena” (see Figures 4-5). It is remarkable that both the recovered flat subregionsand the recovered smooth subregions by our algorithms are qualified as well as the

IMAGE DENOISING 261

TV model and the LLT model. We can also see the strengths and weakness aboutboth ROF model and LLT model.

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Figure 1. Left Plot: The original “Lena” image. Right Plot:Noisy image of “Lena” (SNR=20.97).

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Figure 2. Left Plot: Image recovered by TV model withSNR=25.39. Right Plot: By LLT model with SNR=24.81.

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Figure 3. Left Plot: Image recovered by algorithm 1 withSNR=25.84. Right Plot: By algorithm 2 with SNR=25.89.

262 F. YANG, K. CHEN, B. YU, AND Z. YAN

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Figure 4. (a) The flat subregion of original “Lena” image. (b)The noisy image. (c) TV model. (d) LLT model. (e) Algorithm

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Figure 5. (a) The smooth subregion of original “Lena” image.(b) The noisy image. (c) TV model. (d) LLT model. (e)

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IMAGE DENOISING 263

4.2. Comparison of algorithm 1 with the mean curvature model. Ournext test uses an image containing both a human face and some textures (seeFigure 6). The challenge with this image is to maintain both texture details andsmooth transitions in the human face during processing. As a high order model,Mean curvature model is known to yield satisfactory results for restoring smalldetails and enhancing the recovery of smooth subsurfaces contained in the image.

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Figure 6. Left Plot: The original “barbara” image. Right Plot:Noisy image of “barbara” (SNR=22.04)

In this section, we conduct numerical experiments to compare this method withour algorithm 1. The difference images tells us that both of methods can restoretextures on the scarf in a proper way, but the background and human feature likea hand, shoulder, and face are visually better by our algorithm (see Figure 7 and8).

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Figure 7. Left Plot: Image recovered by mean curvature modelwith SNR=24.89. Right Plot: The difference image.

264 F. YANG, K. CHEN, B. YU, AND Z. YAN

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Figure 8. Left Plot: Image recovered by algorithm 1 withSNR=25.27. Right Plot: The difference image.

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Figure 9. Left Plot: The original “Aircraft” image. Right Plot:Noisy image of “Aircraft” (SNR=21.76)

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Figure 10. Left Plot: Image recovered by the convexcombination method with SNR=26.69. Right Plot: The

difference image.

4.3. Comparison of Algorithm 1 with the convex combination method.

Image restoration combining total variation minimization and a second-order func-tional [10] can restore effectively both the blocky subregion (of piecewise constant

IMAGE DENOISING 265

intensities) and smooth subregion (with no clear jumps) of an image. The abovetwo numerical examples show our method also inherit the advantages of the TVmodel and the LLT model. The third example concerns a “Aircraft” image, whichis corrupted with zero mean Gaussian random noise (see Figure 9). Both the con-vex combination method [10] and our method obtain a good visualization, but thedifference images tells us that our method do a better work on preserving smalldetails (see Figures 10 and 11).

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Figure 11. Left Plot: Image recovered by algorithm 1 withSNR=28.71. Right Plot: The difference image.

4.4. Comparisons of our algorithm 1 with Other methods. The final exam-ple concerns a “Pepper” image which contains some smooth transitions (see Figure9). The purpose of this test is to show our algorithm is qualified with maintainingthe smooth transitions. Here we refer to some improved TV models such as thesplit bregman anisotropic and isotropic total variation denoising methods [12] andthe spatially dependent parameter selection method [3], and high order models likethe mean curvature model and TGV model. Numerical results (see Figures 13-15)show the visualization by the TGV method and our method is better, and the SNRobtained by our method is the highest of all compared.

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Figure 12. Left Plot: The original “Pepper” image. Right Plot:The noisy image.

266 F. YANG, K. CHEN, B. YU, AND Z. YAN

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Figure 13. Left Plot: Image recovered by split Bregmananisotropic total variation denoising method with SNR=27.34.Right Plot: By split Bregman isotropic total variation denoising

method with SNR=27.88.

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Figure 14. Left Plot: Image recovered by the spatiallydependent parameter selection method for TV model withSNR=27.98. Right Plot: By mean curvature model with

SNR=27.99.

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Figure 15. Left Plot: Image recovered by the TGV methodwith SNR=27.90. Right Plot: By algorithm 1 with SNR=28.29.

IMAGE DENOISING 267

5. Conclusions

Image denoising combining total variation minimization and a second-order func-tional can restore effectively both the blocky subregion (of piecewise constant in-tensities) and smooth subregion (with no clear jumps) of an image. In this paper,we proposed two new methods to inherit the advantages of the TV model and theLLT model. Our methods are constructed by using the iterated total variationrefinement to get more meaningful signals from the removed noise part, and theprimal sketch is obtained by the LLT model. The interpretation of our method issimple; first we make full use of the advantages of the LLT model by extractinga smooth primal sketch, and, thereafter, we try to find some meaningful signal-s which is filter out like noise from the removed noise part by the iterated totalvariation refinement. With these approaches, some important information such assmall details are preserved. If we reject using the LLT model in the first step, othermodels which can get a smooth primal sketch is all right. Numerical experimentssubstantiate that our composed methods inherit the advantages of the TV modeland the LLT model better than the convex combination methods.

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College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, PR ChinaE-mail : yangfenlinlin@126.com

Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, TheUniversity of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom.

E-mail : k.chen@liverpool.ac.uk.URL: http://www.liv.ac.uk/∼cmchenke/cmit

School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024,PR China

E-mail : yubo@dlut.edu.cnURL: http://math.dlut.edu.cn/

Accounting School, Jiangxi University of Finance and Economics, Nanchang, Jiangxi330013,PR China

E-mail : zhigangyan@126.com