1

Total variation based exact two-phase methodfor impulse noise removal

Federica Sciacchitano, Yiqiu Dong and Martin S. Andersen

Abstract—We propose a new two-phase method for reconstruc-tion of blurred images that are corrupted by impulse noise.In the first phase, we use a noise detector to identify thepixels that are contaminated by noise, and then, in the secondphase, we reconstruct the noisy pixels by solving an equalityconstrained total variation minimization problem that preservesthe exact values of the noise-free pixels. For images that areonly corrupted by impulse noise (i.e., not blurred) we applythe semismooth Newton method to a reduced problem, and ifthe image is also blurred, we solve the equality constrainedreconstruction problem using a first-order primal-dual algorithm.The proposed method features improved computational efficiencyand has the advantage of being parameter-free. Our numericalresults suggest that the method is competitive in terms of itsrestoration capabilities.

Index Terms—Image deblurring, image denoising, impulsenoise, noise detector, primal-dual first-order algorithm, semis-mooth Newton method, total variation regularization.

I. INTRODUCTION

IMPULSE noise is a common type of image degradationthat can occur during image acquisition or transmission.

Typical reasons include malfunctioning pixel elements in thecamera sensors, errors in analog-to-digital conversion, faultymemory locations, or transmission errors [1]. A characteristicproperty of impulse noise is that noise-corrupted pixels containno information about the true pixel value; only noise-freepixels carry useful information.

Several methods for impulse noise removal exist. The adap-tive median filter [10] is perhaps the most well-known method,but optimization-based methods such as `1-total variation (TV)model-based methods [6], [11], [12] are also becoming populardespite their higher computational cost. A variant of the `1-TV minimization method is the so-called two-phase methodwhich was introduced by Chan, Ho, and Nikolova [5] in 2005.The basic idea behind this method, which we will refer to asthe CHN method, is to separate noise detection and imagereconstruction. In the first phase, the method uses a noisedetector to identify which pixels are corrupted, and in thesecond phase, it reconstructs the image based on a modified `1-TV model where pixels that have been identified as corruptedare left out of the `1 data-fidelity term. In other words, onlythe noise-free pixels appear in the `1 data-fidelity term.

While the CHN method has been shown to perform well onmany test problems, the inclusion of noise-free pixels in the

F. Sciacchitano, Y. Dong and M. S. Andersen are with the Departmentof Applied Mathematics and Computer Science, Technical University ofDenmark, 2800 Kgs. Lyngby, Denmark. (email: feds@dtu.dk, yido@dtu.dk;mskan@dtu.dk)

The work of Y. Dong and M. S. Andersen is supported by Advanced GrantNo. 291405 from the European Research Council.

data-fidelity term is somewhat at odds with the assumptionthat their true values are known. If the pixels are indeednoise-free, then they can either be treated as constants oreliminated from the problem. In this paper, we investigate suchan approach and propose a modified two-phase method. Inparticular, in the first phase we use a noise detector to separatenoisy pixels from the noise-free pixels, and in the secondphase we compute the reconstruction by solving a reducedTV-minimization problem that involves only the corruptedpixels. Thus, the main difference between our method andthe CHN method is that we reconstruct only the pixels thatare identified as corrupted by noise instead of all pixels. Forthis reason, our method often leads to computational savings,and it can be viewed as an exact method in the sense that thereconstruction model matches the information about the noise-free pixels. Moreover, a notable advantage of our method isthat it is parameter-free, and hence the reconstruction resultsdo not rely on the adjustment of any parameters.

As suggested in [4], we use the adaptive median (AM)filter [10] for detecting salt-and-pepper noise, and for random-valued impulse noise, we use the adaptive center-weightedmedian (ACWM) filter [7] for noise detection. The detectorfor salt and pepper noise is able to distinguish almost all noisypixels even for noise level around 90%, while for the randomimpulse noise, the ACWM works quite well until a noise levelof 40%, since in this case the noisy pixels can be confusedas clean ones and vice-versa. We use a semismooth Newtonmethod [13] to solve the denoising problem.

In addition to the impulse noise denoising problem, wealso consider simultaneous deblurring and denoising. Again,instead of including the noise-free pixels in a data-fidelityterm, we propose a parameter-free two-phase method basedon an exact model where noise-free pixels in the blurredimage give rise to a number of equality constraints in thereconstruction problem. We solve this problem numericallyusing a primal-dual first-order algorithm [3].

The paper is organized as follows. In Section II, we reviewthe impulse noise model and propose two-phase methods fordenoising both sharp and blurred images. In Section III, wepresent the implementation details, and in Section IV, wepresent some numerical results for both sharp images andblurred images corrupted by impulse noise. We conclude thepaper in Section V.

II. THE EXACT TWO-PHASE MODELS

We start this section by introducing two impulse noisemodels, namely salt-and-pepper noise and random-valued im-pulse noise. Then, inspired by the existing denoising models

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in the literature [5], we propose (i) a two-phase methodfor denoising, and (ii) a two-phase method for simultaneousdenoising and deblurring.

Given a discrete image of size m1×m2, we define a vectoru ∈ Rm with the m = m1m2 pixels. We shall use thefollowing notation uk = ui,j , where k = (i − 1)m2 + j fori = 1, . . . ,m1 and j = 1, . . . ,m2, to identify the pixel atposition (i, j) with the kth element of u, and we define a setΩ = 1, 2, . . . ,m that contains all pixel indices.

A. Impulse Noise Models

Impulse noise can be described as a stochastic degradationprocess of the form

zk = Nr(uk) =

nk with probability ruk with probability 1− r

, k ∈ Ω

where u ∈ Rm is the original image, z ∈ Rm is the corruptedimage, and n ∈ Rm is the noise. Both images are assumedto be obtained from a two-dimensional pixel-array by meansof concatenation in the usual columnwise fashion. We referto the parameter r ∈ [0, 1] as the noise level since it can beinterpreted as the probability that a pixel is corrupted. Noticethat some pixels remain unchanged, and the pixels that arecorrupted by noise carry no information about the noise-freeimage.

The impulse noise model is called random-valued impulsenoise when for each k, the noise nk is a uniformly dis-tributed random variable with values in the gray-level range[nmin, nmax], and it is called salt-and-pepper noise when eachnk is a discrete random variable with values drawn from theset nmin, nmax with equal probability. For salt-and-peppernoise this means that corrupted pixels take the lowest or thehighest pixel value (i.e., nmin or nmax) whereas for random-valued impulse noise, the noisy pixels have values anywherein the interval from nmin to nmax. For this reason, the random-valued impulse noise is more difficult to detect than the salt-and-pepper noise.

B. Denoising Models

The `1-TV model for impulse noise denoising proposed byNikolova [11], [12] combines the TV regularization term withan `1 data-fidelity term. The resulting reconstruction problemis convex and takes the following form

minu∈Rm

‖u− z‖1 + αTV(u), (1)

where ‖u − z‖1 =∑k∈Ω |uk − zk| is the data-fidelity term,

TV(u) is a regularization term, and α > 0 is a regularizationparameter. We define the (discrete) TV as

TV(u) :=∑k∈Ω

|(∇u)k|2 =∑k∈Ω

√|(∇xu)k|2 + |(∇yu)k|2,

where the discrete gradient operator ∇ ∈ R2m×m is given by

(∇u)k =

((∇xu)k(∇yu)k

),

and ∇xu and ∇yu denote the horizontal and vertical first or-der differences, i.e., using the symmetric boundary conditionswe have

(∇xu)k =

ui+1,j − ui,j if i < m1,

0 if i = m1,

and

(∇yu)k =

ui,j+1 − ui,j if j < m2,

0 if j = m2,

for k = (i−1)m2 + j with i = 1, . . . ,m1 and j = 1, . . . ,m2.The `1-TV model has some nice properties, such as con-

trast preservation, multiscale decomposition and morphologi-cal invariance [6], [11], [15]. However, as mentioned in theintroduction, the main disadvantage of this approach is thatwe have to reconstruct all the pixels of the image, includingthe ones that are noise-free. Including the noisy pixels in thedata-fidelity term introduces errors since the noise-corruptedpixels contain no information about the true image. To addressthis issue, Chan et al. [5] introduced a two-phase method (theCHN method) in which they first detect the noisy pixels (phase1) and then exclude these pixels from the data-fidelity termwhen computing a reconstruction (phase 2). Thus, in the firstphase, they use a detector (an AM filter for salt-and-pepperand an ACWM filter for random-valued impulse noise) to splitthe domain Ω into two sets: N that includes all indices ofthe corrupted pixels and U that includes the indices of thenoise-free pixels. We will henceforth assume that there are|N | = n noisy pixels and |U| = m − n noise-free pixels.In the second phase, they reconstruct the image based on thefollowing model

minu∈Rm

∑k∈U

|uk − zk|+ αTV(u). (2)

The main advantage of the CHN method is that the noisedetector improves the data-fidelity term in `1-TV model (1),and this often yields a great improvement in terms of restora-tion capabilities. However, the presence of a regularizationparameter in the model necessitates multiple reconstructionsor tests in order to find a good choice for the parameter.Moreover, the problem (2) includes all pixels of the imageas variables, including the ones that are assumed to be free ofnoise. To overcome these disadvantages of the CHN method,we propose to alter the second phase of the method such thatthe noise-free pixels are required to be equal to their knownvalues, i.e., we consider the following constrained optimizationproblem

minu∈Rm

TV(u)

s.t. uk = zk k ∈ U .(3)

In other words, instead of looking at the unconstrained mini-mization problem in (2), we are considering the constrainedversion of it. The equality constraints in this model reflect theexact prior that some pixels are known, assuming that all pixelswere correctly identified as either noise-free or corrupted inthe first phase. For this reason, the reconstruction model (2)can be seen as an approximation model since it allows thenoise-free pixels to deviate from their known value.

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Although the model (3) is parameter-free, it can be shownto be equivalent to (2) if the regularization parameter αis chosen sufficiently small. Specifically, we mention thefollowing theorem:

Theorem 1. Let u? a solution to (3) and µ? ∈ Rm−n theLagrange multiplier associated with the equality constraints,then u? is a minimizer of (2) for all α < ‖µ?‖−1

∞ .

For further details we refer the reader to [13, Thm. 17.3].Unlike the model (2), which includes all pixels as variables,

the constrained problem (3) allows us to eliminate the variablesthat correspond to noise-free pixels from the problem formu-lation. To write (3) as an unconstrained optimization problem,we define a vector uN ∈ Rn that corresponds to the corruptedpixels. With this notation, u can be expressed as

u = ΛNuN + ΛUu, (4)

where ΛU ∈ Rm×m is a diagonal matrix defined as

(ΛU )i,i =

1 if i ∈ U0 if i ∈ N

and ΛN ∈ Rm×n is a matrix with n unit vectors ej ∈ Rm,j ∈ N , as columns. Note that ΛUΛN = 0 by construction,and since ΛUu represents the intensity of the noise free pixels,based on the constraint in (3), we can substitute ΛUz into (4)and express the image u as follows

u = ΛNuN + ΛUz. (5)

The problem (3) can therefore be expressed in terms of uNas follows

minuN∈Rn

TV(ΛNuN + ΛUz). (6)

Comparing this model with (2), we see that both are uncon-strained minimization problems, but (6) has some advantages.Firstly, the minimization problem involves only n variablesinstead of m variables, and if the noise level is relatively low(i.e. n m) the reduction in the number of variables is quitesubstantial. Secondly, it is parameter-free.

Before we discuss how to solve (6), we first consider an ex-tension of our denoising approach to simultaneous deblurringand denoising.

C. Deblurring and Denoising Models

Suppose the observed image z is not only corrupted byimpulse noise but also blurred by a known linear blurringoperator K ∈ Rm×m, i.e., we define z = Nr(Ku). Tosolve this deblurring and denoising problem, Chan et al.[4] proposed a two-phase method (the CDH method) thatextends the CHN method. In the first phase, they identify thecorrupted pixels, and then, in the second phase, they computea reconstruction by solving the following problem

minu∈Rm

∑k∈U

|(Ku)k − zk|+ αTV(u). (7)

Note that only the noise-free pixels are included in thedata-fidelity term. As in the denoising problem (3), we can

formulate a constrained minimization problem

minu∈Rm

TV(u)

s.t. (Ku)k = zk k ∈ U(8)

which implies that the value of noise-free pixels in the blurredimage are treated as known constants. However, unlike in thedenoising case, the blur operator K introduces coupling andmakes it difficult to eliminate the equality constraints. In thenext section, we address how to solve the problem numerically.

III. THE ALGORITHMS

We now present methods for solving the denoising problem(6) as well as the deblurring and denoising problem (8).

A. Solving the Denoising Problem

Since the objective function in (6) is not differentiableeverywhere, we introduce the following smooth approximationof TV,

TVγ(u) =∑k∈Ω

Φγ(|(∇u)k|2),

where the function Φγ is the Huber function which is definedas

Φγ(t) =

|t| − γ

2 if |t| ≥ γ1

2γ |t|2 else,

with parameter γ > 0. Other smooth approximations maybe used instead, such as e.g.

√t2 + γ2. The gradient of

TVγ(ΛNuN + ΛUz) can be expressed as

F γ(uN ) = −Λ>N divDγ(uN )−1∇(ΛNuN + ΛUz),

where div ∈ Rm×2m represents the divergence, Dγ(uN ) ∈R2m×2m is defined as

Dγ(uN ) =

(Nγ(uN ) 0

0 Nγ(uN )

),

and Nγ(uN ) ∈ Rm×m is a diagonal matrix with diagonalmax

(|∇(ΛNuN + ΛUz)|2, γ

). The divergence satisfies the

equation div = −∇>, where ∇> is the transpose of the gra-dient operator. Hence, the explicit formula of the divergencecan be found using the definition of transpose

〈−divp,v〉Rm = 〈p,∇v〉R2m ,

for every p ∈ R2m and v ∈ Rm, where 〈·, ·〉Rm and〈·, ·〉R2m denote the standard scalar products in Rm and R2m,respectively.

It follows from the first-order optimality condition associa-ted with (6) that the solution to the smooth approximationshould satisfy the following equation

Λ>N divDγ(uN )−1∇(ΛNuN + ΛUz) = 0. (9)

Before describing our algorithm, we give the definition ofgeneralized differentiability and semismoothness of a mappingF : Rs → Rt, with s, t ∈ N. The mapping F is calledgeneralized differentiable in an open set V ⊂ Rs if there existsGF : Rs → Rs×t such that

lim‖δv‖→0

1

‖δv‖‖F (v + δv)− F (v)−GF (v + δv)δv‖ = 0,

4

i− 1

i

i+ 1

j − 1 j j + 1

Fig. 1. Neighborhood of a noisy pixel ui,j (black) containing six noise-freepixels (white).

for every v ∈ V; see e.g. [9]. This definition is equivalentto the semismoothness of locally Lipschitz maps F in [14].Thus, we can define the semismooth Newton’s method as thegeneralized version of Newton’s method for semismooth maps.In particular, given the current iterate ulN , the semismoothNewton iteration is

GF (ulN )δu,l = −F γ(ulN ), (10)

with δu,l = ul+1N − ulN .

The generalized derivative of F γ is given by

GF (uN ) = Λ>N div(Dγ(uN ))−1∇ΛN

+1

2h(uN )w> +

1

2w(h(uN ))>,

where h(uN ) = Λ>N div∇(ΛNuN + ΛUz) and w ∈∂((Dγ(uN ))−1), with ∂((Dγ(uN ))−1) indicates the gene-ralized derivative of (Dγ(uN ))−1 [9]. Thus, w is given byw = Λ>N w where

w =

−div(Dγ(uN ))−3∇(ΛNuN + ΛUz) if m(uN ) > γ

0 otherwise

and m(uN ) = |∇(ΛNuN + ΛUz)|2.A solution δu,l in (10) may not exist or may not be unique,

since it is not ensured that GF is positive definitive. For thisreason, we add a small multiple of the identity matrix to GF ,

GεF = GF + εI, (11)

where ε is a small positive constant. Thus, substituting GFwith GεF in (10), the semismooth Newton iteration is given by

GεF (ulN )δu,l = −F γ(ulN ), (12)

with δu,l = ul+1N − ulN . Since the regularized matrix is

positive definitive and symmetric, we can solve (12) usingthe conjugate gradient method [13].

In our numerical experiments, we have tested our implemen-tation of the algorithm with different values of ε ∈ [10−3, 1],and we always obtained good reconstructions which suggeststhat our method is robust with respect to the choice of ε. In thenumerical experiments reported in Section IV, we fix ε = 0.5.

1) Preprocessing: The problem (6) is very structured andpartially separable. It follows from the definition of the discretegradient operator ∇ that a noisy pixel with a noise-freeneighborhood as shown in Fig. 1 is completely independent ofother noisy pixels and hence can be computed independently.Specifically, if we regard u as a matrix instead of a vector

and consider a noisy pixel ui,j with six noise-free neighboringpixels

ui,j−1, ui+1,j−1, ui−1,j , ui+1,j , ui−1,j+1, ui,j+1,

then it follows from (6) that ui,j can be obtained by solvingan unconstrained univariate optimization problem

ui,j = arg minv

Hi,j(v) (13)

where

Hi,j(v) =

∥∥∥∥(ui+1,j − vui,j+1 − v

)∥∥∥∥2

+∥∥∥∥(ui+1,j−1 − ui,j−1

v − ui,j−1

)∥∥∥∥2

+

∥∥∥∥( v − ui−1,j

ui−1,j+1 − ui−1,j

)∥∥∥∥2

.

The function Hi,j(v) is clearly convex, and hence the minimi-zation problem (13) can be solved using e.g. the golden sectionsearch method. To this end, it is worth noticing that each ofthe three terms of Hi,j(v) is coercive, and hence we can easilyderive a lower bound and an upper bound on ui,j . Indeed, thethree terms of Hi,j(v) have minimizers (ui+1,j + ui,j+1)/2,ui,j−1, and ui−1,j , so the interval [a, b] with

a = min(ui+1,j + ui,j+1)/2, ui,j−1, ui−1,j

and

b = max(ui+1,j + ui,j+1)/2, ui,j−1, ui−1,j

must contain a solution.When the noise-level is high, there may not be many pixels

with a noise-free neighborhood as shown in Fig. 1. However, itmay still be possible to separate the problem (6) into a numberof independent subproblems. Extraction of such subproblemscan easily be automated using morphological image processingand image analysis, but in this paper, we will only considercorrupted pixels with a neighborhood as in Fig. 1 and solvefor these independently.

2) Algorithm: Our denoising algorithm is summarized inAlgorithm 1.

Algorithm 1 Impulse noise denoising1: Detect the noise-free pixels using a noise detector.2: If the estimated noise level r ≤ 30%, do preprocessing:

• find pixels with noise-free neighborhood;• solve (13) to find optimal pixel value.

3: Initialize u0N ∈ Rn as the value obtained from the

previous steps and set l = 0.4: Compute δu,l, by solving the equation (12), and updateul+1N = ulN + δu,l. (Note that, with preprocessing, a part

of noisy pixels have been restored in step 2. In this case,the size of uN in (12) is further reduced.)

5: Stop if stopping criteria are satisfied; otherwise set l =l + 1 and go to step 3.

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B. Solving the Deblurring and Denoising Problem

A saddle-point formulation of the problem (8) is given by

maxb∈B

minu∈C

b>∇u, (14)

where b ∈ R2m is a dual variable, and the sets C and B aredefined as

C = u ∈ Rm |ΛUKu = ΛUz

andB = b ∈ R2m | ‖b‖∞ ≤ 1

with projection operators PB and PC . The norm ‖b‖∞ denotesthe discrete maximum norm, defined as

‖b‖∞ = maxk|bk|2 = max

k

√|(bx)k|2 + |(by)k|2,

withb =

(bxby

)and bx, by ∈ Rm.

The Chambolle–Pock algorithm [3] for solving the convex–concave saddle-point problem (14) is summarized in Algo-rithm 2.

Algorithm 2 Chambolle–Pock algorithm for deblurring anddenoising

1: Detect the noise-free pixels using a noise detector.2: Initialize u0 with image from step 1 and set l = 0.3: Set θ ∈ [0, 1] and τ, σ > 0 such that τσ‖∇‖2 < 1.4: Compute ul+1:

ul+1 = PC(ul + τdivbl). (15)

5: Compute ul+1:

ul+1 = ul+1 + θ(ul+1 − ul). (16)

6: Compute bl+1:

bl+1 = PB(bl + σ∇ul+1). (17)

7: Stop if stopping criteria are satisfied; otherwise set l =l + 1 and go to step 4.

The projection operator PC in (15) can be evaluated bysolving the following least-norm problem

minu‖u−wl‖22

s.t. ΛUKu = ΛUz,(18)

where wl = ul + τdivbl. This problem has the closed-formsolution

ul+1 = wl −K>Λ>U (ΛUKK>Λ>U )−1ΛU (Kwl − z).

Furthermore, the projection PB in (17) is a pointwise Eu-clidean projection onto L2 balls, i.e.

bl+1 =bl + σ∇ul+1

max(1, |bl + σ∇ul+1|2).

The Chambolle-Pock primal-dual algorithm ensures conver-gence if θ = 1 and

τσ‖∇‖22 < 1. (19)

TABLE ISNRS OF RESTORATIONS OF “PARROT” CORRUPTED BY DIFFERENT

LEVELS OF SALT-AND-PEPPER NOISE.

Noise AM filter CHN method Our method20% 23.24 27.78 27.8240% 19.63 23.13 23.2060% 16.22 20.16 20.1380% 12.92 15.64 15.68

For more details we refer the reader to [3]. We note that thebound on the norm of the linear operator ∇ is [2]

‖∇‖22 = ‖div‖22 < 8,

and hence the algorithm converges if 8τσ < 1. In ournumerical experiments, we use

τ =α

3and σ =

1

3α,

with α > 0. In this way, the convergence of the algorithmis ensured. In our experiments described in the next section,we fixed α = 0.3 which generally worked well. Thus,the proposed method for deblurring and denoising is alsoparameter-free.

IV. NUMERICAL RESULTS

In this section, we present some reconstructions for three256× 256 gray-level images (“Peppers”, “Parrot” and “Cam-eraman”) and a 1012× 1012 gray-level image (“Pirate”); seeFig. 2. The quality of the images is compared in terms of thesignal-to-noise ratio (SNR) which is defined as

SNR(u?) = 20 log10

‖u‖2‖u− u?‖2

,

where u and u? represent the original image and the recon-structed image.

In our first experiment, we consider denoising without blur-ring. Recall that in the first phase of our algorithm, we detectthe noisy pixels using the adaptive median (AM) filter [10]for salt-and-pepper noise and the adaptive center-weightedmedian (ACWM) filter [7] for random-valued impulse noise.In the second phase, we solve the minimization problem (6)to denoise the corrupted pixels.

In Fig. 3, we show the restored images from salt-and-peppernoise using (i) the adaptive median filter [10], (ii) the CHNmethod [5], and (iii) our method. For the CHN method, weadjusted the regularization parameter through numerical testsand show the best results. Comparing the SNRs listed in TableI, we see that the proposed method outperforms the AM filterand is competitive when compared to the CHN method. Takinginto account that the proposed method is parameter-free andonly restores the noisy pixels, it is more practical and moreefficient than the CHN method; for a comparison of the CPUtime, see Table III.

In Fig. 4, we present the restorations based on different testimages with 50% salt-and-pepper noise. In order to show theinfluence of the regularization parameter α on the restorationresults in the CHN method, we also give the result using avalue of α that differs slightly from be best value. From Table

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Fig. 2. Original images:“Peppers”, “Parrot”, “Cameraman” and “Pirate”.

Fig. 3. First row: noisy images “Parrot” with salt-and-pepper noise with noise level 20%, 40%, 60%, and 80% (left to right). Second row: results obtainedwith AM filter. Third row: results obtained with CHN method. Fourth row: results obtained with proposed method.

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Fig. 4. Results of the CHN method with α = 3 (first row), with α = 4 (second row) and our method (third row) for the noisy images “Peppers”, “Parrot”and “Cameraman” with 50% salt-and-pepper noise.

TABLE IISNRS OF RESTORATIONS OF CORRUPTED IMAGES WITH 50%

SALT-AND-PEPPER NOISE.

Image CHN α = 3 CHN α = 4 Our methodPeppers 19.06 19.42 19.44Parrot 20.00 21.60 21.59

Cameraman 19.74 21.53 21.54

II and Fig. 4, it is clear that the images obtained with theproposed method are of similar quality as those obtained usingthe CHN method with the best α. However, the performanceof the CHN method strongly depends on the selection of α.With a value of α that differs only slightly from the bestchoice, the restoration results can be much worse. Hence,being parameter-free, our method is more practical.

To compare the computational cost, we list the CPU timeof the CHN method and the proposed method in Table III. Allof the numerical experiments were carried out in in MATLABR2014a on a PC equipped with a 3.20GHz CPU and 8Gbmemory. The results are based on the 1024× 1024 test image“Pirate” and represent the average computation time basedon ten noise realizations. Note that since the first phase ofthe CHN method and our method is same and the maincomputational load is in the second phase, we only give theCPU time associated with the second phase in Table III. Toshow the effect of the preprocessing step in our method, wereport the CPU times for restoration with and without thepreprocessing when the noise level is at most 30%. Based

TABLE IIICOMPARISON OF CPU TIME (IN SECONDS) FOR “PIRATE” CORRUPTED BY

SALT-AND-PEPPER NOISE.

Noise CHN method Our methodNo preprocessing Preprocessing

10% 177.9 54.7 33.720% 202.1 83.4 69.630% 199.9 110.3 101.340% 284.0 142.8 -50% 293.9 195.2 -60% 353.3 245.1 -70% 359.3 313.0 -80% 425.8 375.5 -

on the results in Table III, we find that the CHN method isslower than our method, especially when the noise level is low.Moreover, the results verify that preprocessing is beneficialwhen the noise-level is low. Furthermore, the computationtimes for the CHN method do not include the overhead oftuning the regularization parameter, and hence the results donot reflect the added advantage of our method being parameter-free.

In Fig. 5, we show the results when restoring the image“Parrot” corrupted by 30% random-valued impulse noise. Inthe first phase, we use the ACWM filter [7] as noise detector.It is clear that in this case, our method still provides resultssimilar to those obtained with the CHN method, and bothmethods outperform the ACWM filter.

As a final experiment, we compare the CDH method [4] and

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Fig. 5. From left to right: noisy image “Parrot” corrupted by 30% random-valued impulse noise, restoration with ACWM filter, restoration with CHN method,and restoration with proposed method.

TABLE IVSNRS OF RESTORATIONS OF GAUSSIAN BLURRED IMAGES CORRUPTED

BY DIFFERENT LEVELS OF SALT-AND-PEPPER NOISE FOR “PARROT”.

Noise CDH method Our method20% 25.55 25.8940% 24.30 23.6560% 22.44 22.5080% 19.55 20.54

the method proposed in Section III-B for restoring Gaussianblurred images with salt-and-pepper noise. In the experiment,we consider Gaussian blur with window size 7×7 and standarddeviation 5. Fig. 6 shows the degraded images and the restoredimages obtained with the CDH method and our method. Inaddition, we list the SNRs of the restored images in Table IV.We see that both methods yield good restorations. However, asthe proposed denoising method, our denoising and deblurringmethod does not require any parameter adjustment.

V. CONCLUSION

We have introduced a total-variation based exact model forrestoring images corrupted by impulse noise. Since impulsenoise only partly corrupts images, we start with an exactconstrained minimization problem for which the CHN model[5] can be viewed as an `1 approximation of our model.By separating the noisy pixels and the noise-free ones, ourformulation yields an unconstrained problem with only thenoisy pixels as variables. This reduces the size of the problem,especially for low noise-levels. We also extend our methodto the simultaneous deblurring and denoising case. The mainadvantage of our method is that it is parameter-free, andwe have demonstrated numerically that our method providescompetitive results in less time when compared to the CHNmethod.

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Fig. 6. First row: Gaussian blurred image “Parrot” with salt-and-pepper noise with noise level 20%, 40%, 60%, and 80%. Second row: results obtained withthe CDH method. Third row: results obtained with our method.