Technical Section

High-order wavelet reconstruction for multi-scale edge awaretone mapping$

Alessandro Artusi a,n, Zhuo Su b,c, Zongwei Zhang b,c, Dimitris Drikakis d, Xiaonan Luo b,c

a Graphics and Imaging Laboratory, Department of Informatics and Applied Mathematics, University of Girona, Spainb National Engineering Research Center of Digital Life, Guangzhou, Chinac School of Information Science and Technology, Sun Yat-Sen University, Guangzhou, Chinad Fluid Mechanics and Computational Science, Cranfield University, United Kingdom

a r t i c l e i n f o

Article history:Received 17 February 2014Received in revised form1 August 2014Accepted 3 August 2014Available online 21 September 2014

Keywords:High dynamic range imagingTone mappingEdge aware techniqueMultiscaleWaveletHigh order interpolation

a b s t r a c t

This paper presents a High Order Reconstruction (HOR) method for improved multi-scale edge awaretone mapping. The study aims to contribute to the improvement of edge-aware techniques forsmoothing an input image, while keeping its edges intact. The proposed HOR methods circumventlimitations of the existing state of the art methods, e.g., altering the image structure due to changes incontrast; remove artefacts around edges; as well as reducing computational complexity in terms ofimplementation and associated computational costs. In particular, the proposed method aims atreducing the changes in the image structure by intrinsically enclosing an edge-stop mechanism whosecomputational cost is comparable to the state-of-the-art multi-scale edge aware techniques.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

High Order Reconstruction (HOR) methods, introduced by Hartenet al. [1], have been used extensively for solving the hyperbolicconservation laws and the Hamilton-Jacobi equations [2]. Additionally,these methods have been applied to image processing (imagecompression), denoising [3] and segmentation [4]. Due to their abilityto reduce oscillations around function discontinuities, these methodscan be potentially used as an edge aware interpolation tool. Edge-aware techniques such as anisotropic diffusion [5], bilateral filtering[6,7] and neighborhood filtering rely on sophisticated type of spatiallyvarying kernels. Often, they tend to either generate artificially staircas-ing effects or ringing effects around sharp edges [8]. These artifacts canbe reduced using a post-processing step at the price of increasing thecomputational cost and the number of parameters used [9].To have better control of the details over the spatial scale, one canapply edge-aware techniques in a multi-scale fashion. However, thebilateral filtering is inappropriate for multi-scale detailed decomposi-tion [10]. Other edge-aware techniques that support the multi-scaleapproach [10,11,9] also encompass some flaws, e.g., they are not able

to achieve a plausible reproduction of all important image features[12] and may change the image structure.

Therefore, there is a need to develop methods that are reducingas much as possible any change into the image structure withoutincreasing the complexity or computational cost.

In this paper, we link the edge-aware problem to the typicalproblem of interpolation. In particular, we propose a novel waveletscheme that uses a robust predictor operator, based on the HORmethod, which intrinsically encloses an edge-stop mechanism toavoid influence of pixels from both sides of an edge. To have abetter control of details over the spatial scale, we employ the HORmethod in conjunction with a multi-scale scheme.

We demonstrate the usability of the proposed method to solvea typical problem in the context of High Dynamic Range (HDR)imaging, called tone mapping as defined in Banterle et al. [13].

The approach is formulated as follows; we decompose an inputHDR image, making use of wavelet decomposition and through theuse of HOR methods separate its coarse and fine features (details).The coarse and fine features are then manipulated to achieve thedesired tone and details levels. Finally, the output image isreconstructed. The advantage of the above approach is that it doesnot require the introduction of any edge-stopping function thatlimits possible image-structure changes.

To understand this concept, Fig. 1 shows the distortion mapas output of the Dynamic Range Independent metric (DRIM)

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Computers & Graphics

http://dx.doi.org/10.1016/j.cag.2014.08.0020097-8493/& 2014 Elsevier Ltd. All rights reserved.

☆This article was recommended for publication by Luis Santos.n Corresponding author.E-mail address: artusialessandro4@gmail.com (A. Artusi).

Computers & Graphics 45 (2014) 51–63

introduced by Aydin et al. [12] for [10,11,9] and the techniqueproposed in this paper. The original HDR image is used asreference, and the output of the tone mapping operator is comparedto it. A certain amount of lost of contrast (green) is clearly visible, andthis may change the overall image structure [12]. The map showsthat using the present HOR reduces the number of pixels affected byloss of contrast by more than 50%.

Moreover, the intensity profile may change as shown in Fig. 2. TheFattal method [11] may have an undesirable increase of the intensityprofile to the maximum output value 1 (1st zoomed area). Thestructure of the original profile may be undesirably modified (greenline) as shown for the method [9] (2nd enlarged area). Thesemethods may result in prohibitive computational costs (see Pariset al. [9]). An efficient implementation [14] of the method presentedby Paris et al. [9] is also discussed in Section 6.

The proposed approach retains the same advantages introduced bythe traditional edge aware approaches such as Paris et al. [9], andFattal [11], namely with respect to obtaining local properties andproviding robust smoothing, hence avoiding the use of pixels from

both sides of the edge. The main contributions of this work can besummarized as follows:

1. Establish a link between the robust smoothing concept to thereconstruction problem of a non-smoothed function.

2. Achieve a complex solution of the edge-aware problem,through a simple and flexible point-wise manipulation by usingHOR method.

3. Propose an edge-aware filter that produces halo free results;reduces the changes in the image structure as defined by theDRIM metric and its computational cost is increasing linearlywith respect to the number of the input pixels N.

2. Related work

Edge aware filters: Edge aware techniques are used to smoothan image while keeping its edges intact, preventing pixels located

Fig. 1. Comparison of the state-of-the-art multiscale edge aware based tone mapping operators and the present HOR: 1st row: output of the various techniques. 2nd row:distortion map of the DRIM metric [12]. This map is showing the pixels that shows a distorsion with 95% of probability to been seen by the Human Visual System (HVS). Bluepixels are areas where invisible contrast is introduced; red pixels are areas where reversal of visible contrast is noticeable and green pixels shows areas of lost of contrast. Themap is showing of a reduction of more than 50% of the pixels affected by loss of contrast when the HOR method is used. Parameters used - (a) Farbmann et al. [10] multiscaleapproach balanced - (b) Fattal's [11] α¼0.9, β¼0.16 and γ¼0.8 - (c) Paris et al. [9] σr ¼ log ð2:5Þ, α¼0.5 and β¼0.0 (for conveying the local effect) - (d) the present HOR β¼0.7,γ¼0.9. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 2. Intensity profile for the tone mapping operators on an HDR mage for line 300: The 1st zoomed area, clearly shows how Fattal's [11] method (undesirably) increasesthe intensity profile to the maximum value of 1. In the 2nd zoomed area (Paris et al. [9] green line), the intensity profile is modified. (For interpretation of the references tocolor in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–6352

on one side of a strong edge from influencing pixels on the otherside. This concept can be used to separate high frequencyinformation from low frequency information such as texture anddetails. Once this separation is peformed the high and lowfrequencies information can be independently manipulated andre-composed.

In the past, techniques able to preserve edges [6,8,5] have beenapplied to image manipulation [15–17,11]. These techniques produceacceptable results, but often introduce visible ringing effects arisingfrom the Poisson equation [15] and filtering, as discussed in [10,8].Moreover, they need several parameters, that are image dependent,making their set-up difficult for practical applications [17]. Ourapproach offers a solution, that produces results at least as good asthe above techniques, runs linearly in time with respect to thenumber of the input pixels and is not dependent on a large numberof parameters.

Multi-scale edge aware filters. Recently, several edge-awaretechniques that can be used in the multi-scale framework, have beenpresented. Typically, these methods exploit the multi-scale approachby making use of pyramid mechanisms such as Laplacian [18],Gaussian [19], and Wavelets [20].

The Laplacian approach, in the context of edge-aware, has beenrecently revised by Paris et al. [9] through the use of localtransformation which makes the Laplacian approach suitable foredge-aware operations. Farbman et al. [10] employed the weightedleast square to build an alternative edge preserving operator andextend it to multi-scales as well. Fattal et al. [15] used the GaussianPyramid to compress the high dynamic range of the input image,followed by the full image reconstruction through the use of thePoisson solver.

The aforementioned techniques share with our approach themulti-scale ‘philosophy’, but are using different methods such asthe Laplacian [10,9] and Gaussian [15] pyramids. Moreover, they arebased on the solution of a linear system [10], a Poisson solver [15],or bilateral filtering all of which generate artifacts around edges [8].Li et al. [21] proposed a multi-scale approach based on waveletswhere each sub-band signal is modified using a gain map thatcontrols the local contrast. Fattal [11] presented an edge avoidingtechnique based on a second generation wavelet. Our approachintegrates within the wavelet mechanism a HOR technique that doesnot require any edge-stop function for computing a large set ofweights in the interpolation step as in [11]. Consequently, using thepresent approach there is no need for any particular precautionagainst the strong edges and distortions of the image structure arereduced.

3. Background

Fixed stencil approximation techniques, such as piecewiselinear and cubic interpolation, are often used to reconstruct themissing points of a function. These methods are working well inthe case where the function is smooth; however, if the function isonly piecewise smooth the fixed stencil approximation may not beadequate near discontinuities. In fact, oscillations at the functiondiscontinuities are visible.

Essential non-oscillatory scheme: Essential non-oscillatoryschemes (ENO) have been introduced by Harten et al. [1] to solvethis problem. The ENO scheme makes use of adaptive stencils, thusthe use of discontinuity cells is avoided. Let us consider a signalfunction f(x) with given grid of points of evaluated values such asv½i� ¼ f ½xi�.

The ENO scheme reconstructs f from the point values v

assuming that f is piecewise polynomial. This means that for eachcell Ii � ½xi�1; xiþ1� a polynomial interpolant pi(x) is defined usingthe set of points defined in the stencil Si. The idea is to find

Fig. 3. Example of the HOR scheme mechanism. (Top row) The original staircasesignal. (2nd row) The uniform grid points: (circle red) input points, (square blue)points to be interpolated. (3rd row) The stencil points used by the HOR scheme.(4th and 5th rows): Two separated stencils used to define the two interpolants bythe HOR scheme. (For interpretation of the references to color in this figure caption,the reader is referred to the web version of this article.)

Fig. 4. Overview of the present approach. Firstly, a pyramid representation of theinput HDR image is produced using a forward wavelet lifting scheme withintegrated the HOR interpolation method presented in this paper. Secondly, thecoarse level of the pyramid structure (blue continue arrow) and the details levels(blue dashed arrows) are manipulated. Thirdly, the modified pyramid is collapsedto reconstruct the output tone mapped image. This is done, using the backwardwavelet lifting scheme with integrated the HOR interpolation model. (For inter-pretation of the references to color in this figure caption, the reader is referred tothe web version of this article.)

Fig. 5. Pyramid image representation, after having applied the forward waveletlifting WENO scheme. The coarse level is the image at the upper left corner (withred frame). The other images are representing the details, at different levels, for thehorizontal, diagonal and vertical directions. (For interpretation of the references tocolor in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–63 53

a stencil of kþ1 consecutive points, including xi�1 and xiþ1, wherethe signal f(x) is the smoothest in this stencil when comparing itwith the other possible stencils. To evaluate the smoothness of f(x)we can use the Newton divide differences of f:

f ½x0� � f ðx0Þ;

f ½x0; x1� �f ½x0�

ðx0�x1Þþ f ½x1�ðx1�x0Þ

;

……: ð1ÞIn general, the j-th degree divided difference of f(x) is equiva-

lent to

f ½xi�1; :; xiþ j�1� �f ½xi; :; xiþ j�1�� f ½xi�1; :; xiþ j�2�

xiþ j�1�xi�1: ð2Þ

Starting from a two points stencil

S2ðiÞ ¼ xi�1; xiþ1; ð3Þthe linear interpolation of the stencil S2 in a Newton form is

p1ðxÞ ¼ f ½xi�1�þ f ½xi�1; xiþ1�ðx�xi�1Þ: ð4ÞTo expand the stencil we have two possibilities, either add the leftneighbor xi�2 or the right one xiþ2. In both cases this will be aquadratic interpolation polynomial. This will differ from the linearpolynomial of Eq. (4), by the same function multiplied by twodifferent constants. These constants are the two 2-nd degrees ofdivided differences of f(x) in two different stencils defined by theleft and right neighbors. This procedure is continued until the kþ1points in the stencil are reached.

High order interpolation scheme (HOR): The typical problem ofthe ENO scheme is that it can exhibit oscillatory behavior and isalso fairly expensive in its implementation [22]. As an alternative,

the weighted ENO (WENO) variant has been proposed.1 WENOuses a convex combination of all the corresponding interpolatingpolynomials on the stencil to compute an approximated polyno-mial for each cell (Fig. 3). A convex combination is a linearcombination where the coefficients (weights) are all positive andtheir sum is equal to 1. The key points of the reconstructionscheme are (at 3rd order accuracy):

1. Stencils definition: Taking a cell defined in the interval½xi�1=2; xiþ1=2� (see Fig. 3), the stencils are defined as [22]

S1 ¼ ðxi�3=2; xi�1=2; xiþ1=2Þ;S2 ¼ ðxi�1=2; xiþ1=2; xiþ3=2Þ ð5Þ

2. Interpolation polynomials: For each stencil the linear interpo-lation polynomial is computed as

p1 ¼ f ½xi�þf ½xi�� f ½xi�1�

Δxðx�xiÞ;

p2 ¼ f ½xi�þf ½xiþ1�� f ½xi�

Δxðx�xiÞ ð6Þ

where the f ½x� elements are the available data points of thefunction to be reconstructed (red points in Fig. 3).

3. Convex combination: The interpolation polynomials are com-bined following a convex combination:

Pi ¼ai0

ai0þai1p1þ

ai1ai0þai1

p2 ð7Þ

Fig. 6. Comparisons with state-of-the-art method Fattal's method [11]. 1st row: Fattal [11] using wavelet red and black model with α¼0.8, β¼0.11 and γ¼0.68 – 2nd row: thepresent approach with β¼0.3 and γ¼0.7 – 2nd column: gradient of a zoomed area, it showing the degree of edge preservation: (a) TMOs output and (b) gradient. (Forinterpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

1 WENO schemes have been widely used in computational fluid dynamics; see,for example, Drikakis et al. [23–25] and references therein.

A. Artusi et al. / Computers & Graphics 45 (2014) 51–6354

where

ai0 ¼Ci0

ϵþðISÞ1� �2:0;

ai1 ¼Ci1

ϵþðISÞ2� �2:0 ð8Þ

IS are the smoothness indicators, which are calculated asðISÞ1 ¼ ðf ½xi�� f ½xi�1�Þ2:0 and ðISÞ2 ¼ ðf ½xiþ1�� f ½xi�Þ2:0. The gradi-ent magnitude is well known to be a good estimator of edgeinformation. Based on this observation, we have used theimage gradient to select the coefficients C as given by [22],allowing the interpolation step to be aware of edge informationin order to avoid an edge-stopping function.� ∂Eðf Þ=∂f 40 : Ci

0 ¼ 1=2 and Ci1 ¼ 1;

� ∂Eðf Þ=∂f o0 : Ci0 ¼ 1 and Ci

1 ¼ 1=2.

4. HOR wavelet scheme

In this paper we propose a robust smoothing through the use ofa polynomial interpolant that makes use of the smoothest stencils.It is integrated in a wavelet scheme (lifting scheme) to takeadvantage of the multi-scale representation such as the capabilityto retain image information at different scale. Fig. 4 shows theprinciple of the present approach. Firstly, a pyramid representa-tion of the original input image I is produced using a forwardwavelet lifting WENO scheme. Secondly, the coarse (blue contin-ued arrows) and fine levels (blue dashed arrow) are manipulated.Thirdly, the modified multiscale representation is collapsed to theoutput image using the backwards wavelet lifting WENO scheme.A multi-scale representation can be obtained by making use of anested series of decimation D and reconstruction R operators. As aD operator, we have used a simple splitting operation which

Fig. 7. Comparisons of different methods. 1st column: Fattal [11] using wavelet red and black model with parameters as per web project page [26] – 2nd column: Thepresent approach with β¼0.7 and γ¼0.9 – 2nd row: gradient of the zoomed area in the 3rd row. Distortions at the edges are visible: (a) Fattal [11] and (b) Present HOR. (Forinterpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–63 55

separates the pixels of the input at level k in two different gridsbased on the index number (odd and even). For the R operator theWENO scheme has been employed according to which the level kis reconstructed from kþ1 using the following equation:

~Ik½x; y� ¼w0½x; y�ðIkþ1½x�1; y�

þ Ikþ1½x�1; y�� Ikþ1½x�3; y�½x�1; y��½x�3; y� ð½x; y��½x�1; y�ÞÞ

þw1½x; y�ðIkþ1½x�1; y�

þ Ikþ1½xþ1; y�� Ikþ1½x�1; y�½xþ1; y��½x�1; y� ð½x; y��½xþ1; y�ÞÞ: ð9Þ

Eq. (9) is equivalent to Eq. (7) where w0 and w1 are its factorialterms. The difference in the indices between Eq. (9) and (7) is dueto the fact that we have inserted zero pixels at kþ1 level andwould like to retain integer numbers in the indexing of the grid.Fattal [11] presented a robust average operator, for both type ofwavelet approaches, red�black and weighted CDF, making use ofan edge stop function to compute the prediction weights. In ourcase, as described in Eq. (7), we present a convex combination ofpolynomial interpolants. However, these polynomial interpolantsare linear, thus we can consider the overall operator as acombination of linear interpolants.

At the boundaries of the input image, we have adopted astandard extrapolation approach to generate the missing values in

the stencil. The restored ~Iklevel is later used to obtain the details

of the kþ1 level dkþ1 ¼ Ik� ~Ik. To preserve the overall sum of the

coarse elements Ikþ1, and based on the fact that the operator R can

be seen as combination of two linear interpolants we have decidedto use a linear interpolator as update-operator U:

Uðdkþ1Þ½x; y� ¼ dkþ1½x�1; y�þdkþ1½xþ1; y�4

; ð10Þ

and the level kþ1 of coarse elements is updated using

Ikþ1 ¼ Ikþ1þUðdkþ1Þ.This process is repeated both for the rows and columns of the

input image.Discussion: An example of the behavior of the present HOR,

integrated in the wavelet scheme, is shown in Fig. 5. The coarse, c,and ‘details’ coefficients, d, (vertical, diagonal and horizontal) forthree levels are shown. Edges are detected by the wavelet schemeavoiding the influence of pixels on both sides at each scale. This isobtained without the introduction of an edge stop functionutilized for the computation of the set of weights used in theinterpolation step as proposed by Fattal [11].

5. Tone mapping manipulation

In this subsection, we will show how to make use of theproposed technique in the classical tone manipulation problem.Tone manipulation allows to reduce the intensity of the luminancerange of HDR content. This objective is achieved through compres-sion of large-scale variation and keeping the fine level information.The filtering approach is applied to the natural logarithmic scale ofthe luminance, keeping the color ratio unaltered as in Paris et al. [9],using a gamma correction of 2.2.

Fig. 8. Results from the application of Fattal's et al. [11] technique making use of the new contractive concave mapping as specified in [26]: (a) Fattal [11] β¼0.7 - Weak and(b) Fattal [11] β¼0.5 – strong. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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Fig. 9. Images used in the experiments. The numbering in Tables 1 and 2 follows the order of the images from the top to the bottom and from the left to the right.

Table 1DRIM results over the set of images presented in Fig. 9. We show the percentage of pixels with probability of 95% that present the distortion of reverse (R), loss (G), oramplification (B) of contrast.

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To manipulate the tone and the details of the input HDR image, wehave followed a similar approach to the one used by Fattal [11]. Thetone is linearly manipulated modifying the coarse coefficient c of thecoarsest level n through a parameter β, as βcn. This allows us toachieve the compression of the vast dynamic range available in theinput HDR image. A second parameter γ is used to manipulate thedetails. This is obtained from the progressive decreasing of the ‘details’

coefficients dk, such as γkdk where k is the number of levels varyingbetween 1 to n. The β and γ parameters are in the range of ð0:0;1:0�.

Since our approach shares several aspects with the techniquepresented by Fattal [11], we first provide an analysis and compar-ison to show how the present technique performs with respect tothe preservation of edges, while at the same time adjusting thetone of the input image.

Table 2DRIM results over the set of images presented in Fig. 9. We show the percentage of pixels with probability of 75% that present the distortion of reverse (R), loss (G), oramplification (B) of contrast.

Fig. 10. Output and DRIM comparison with state-of-the-art edge aware approaches. 1st row – output of the edge aware technique; 2nd row – DRIM metric [12] withprobability of 75%; 3rd row – DRIM metric [12] with probability of 95%. Parameters used – (a) Farbmann et al. [10] multiscale approach balanced – (b) Fattal's [11] α ¼ 0.9,β¼0.19 and γ¼0.5 – (c) Paris et al. [9] σr ¼ log ð2:5Þ, α¼0.5 and β¼0.0 (for conveying the local effect) – (d) the present HOR β¼0.7, γ¼0.9. (For interpretation of the referencesto color in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–6358

The present technique produces results comparable to thisstate-of-the-art operator, while offering the advantage of not usingan extra edge-stop function. The technique of [11] is capable to

capture more details but at the cost of introducing some distor-tions at the edge level, as shown in Figs. 7(a) (zoomed lamp areaand its edge map) and 6(b) (edge map).

Fig. 11. Output and DRIM comparison with state-of-the-art edge aware approaches. 1st row – output of the edge aware technique; 2nd row – DRIM metric [12]with probability of 75%; 3rd row – DRIM metric [12] with probability of 95% (for both images). Parameters used – (a) Farbmann et al. [10] multiscale approach balanced –

(b) Fattal's [11] α¼0.8, β¼0.12 and γ¼0.9 - (c) Paris et al. [9] σr ¼ log ð2:5Þ, α¼0.5 and β¼0.0 (for conveying the local effect) – (d) the present HOR β¼0.7, γ¼0.9. (Forinterpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–63 59

One may reduce these distortions by making use of a newcompression technique, as suggested in [26] (Fig. 8). However,artifacts may appear as shown in Fig. 8(b) 2nd row.

6. Experimental results

The HOR approach has been implemented in Matlab and theexperiments have been performed on a Mackbook air with Intel i7-core CPU 1.8 GHz, 64-bit machine and 4 GB of RAM. We havecompared our technique with the latest edge aware state-of-the-artmulti-scale approaches, applied to the tone mapping problem, such as[9–11]. We have used the Matlab code as well as parameters providedby the authors.

We have chosen the set of images as shown in Fig. 9. This setconsists of 18 images with different dynamic range that span fromoutdoor to indoor and from light to dark illumination conditions.

6.1. Quality

To provide a fair comparison, we have selected the parametersof the different techniques to convey similar appearance in term ofcontrast, edges and details preservation to all the techniquespresented in this comparison.

We may observe that the DRIM metric is measuring changes incontrast, in other words the overall appearance of the image, andit is not able to detect if small-scale details are not well preserved.On the other hand, edge-aware techniques are able to preservewell small-scale details. This is preserved intrinsically by themechanisms described in the previous sections as well as by theresults shown here that are comparable with the existing state-of-the-art edge aware technique [11].

Based on the fact that small-scale details are to certain extentwell reproduced by the edge-aware techniques, our objective wasto examine how these techniques are able to convey the overallappearance of the input HDR into the tone mapped result. In doingso, we have decided to use the DRIM metric as specified below.

Since the DRIM metric accepts cd/m2 values, the input imagesneed to be calibrated. In the case of the tone mapped input image,we need to linearize the input signal and then map it to thedynamic range of the display where the image will be visualized.In our case, the γ value used for the linearization step is 2.2, andthe dynamic range chosen is [0.5, 100] cd/m2 . In the case of theHDR input image, there was no need to linearize the signal, andthe dynamic range has been chosen as [0.015, 3000] cd/m2.

DRIM results and discussion: Tables 1 and 2 show the results ofthe DRIM metric applied to the test set images. The numbersrepresent the percentage of pixels with probability for the distor-tion to be perceived by the HVS. Tables 1 and 2 show the resultswith probability 95% and 75%, respectively. The colors used todepict the type of distortion are the same with those used todescribe the distortion – R (red) reversal, – G (green) lost and – B(blue) amplification of contrast. We have colored the methods thatshow the higher probability, as well as the ones that showsignificant percentage of pixels with the specified probability.In the case of probability 95%, the significant distortion introducedby the state-of-the-art edge aware methods, as well as by thepresent HOR is mostly due to the loss of contrast; neither reversalnor amplification of contrast are significant. The lost of contrast isattributed to the fact that the edge-aware methods are usingsimple linear scaling for compressing the large luminance dynamicrange. This may affect the overall preservation of local contrast.With probability 95% the state-of-the-art methods may presenthigh percentage of pixels affected by loss of contrast. This is thecase of the images 1, 3, 5, 11,13 and 14. In most of the other cases,

Fig. 12. DRIM comparison with state-of-the-art edge aware approaches and simple TMO. DRIM metric [12] with probability of 95%. Farbman et al. [10] DRIM output isomitted because of the similarities in the results with the one obtained with the present HOR: (a) Present HOR; (b) Fattal [11]; (c) Paris et al. [9]; (d) Reinhard et al. [27]; and(e) Drago et al. [29]. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–6360

this number is negligible. For the images 1, 13 and 14, the HORshows a slightly higher percentage value for the loss of contrast.However, this value is either comparable or lower than the valueprovided by the state-of-the-art edge-aware methods. We havealso tried to analyze the results of the DRIM metric at lowerprobability such as 75% and the results are shown in Table 2.As expected, the percentage of pixels is drastically increased andmore images are affected by a significant percentage value. In thiscase, reversal of contrast (red) and in some cases amplification ofcontrast (blue) may appear. In the case of loss of contrast the Fattal[11] and Paris et al. [9] results show that the majority of the images

are affected by this type of distortion. This type of distortion alsoaffects the present HOR, but when compared with the state-of-the-art edge-aware methods shows a lower percentage of pixelsaffected by this distortion.

Only in the case of image 18 the present HOR shows highervalue for the loss of contrast. However, this percentage value isquite small and it is not actually perceivable by the HVS. Theresults are also affected by the reversal of contrast. In particular,several results of Fattals [11] method are showing this distortion.The present HOR shows reversal of contrast higher than theother methods only for three images (11, 12 and 18). Finally, the

Fig. 13. Output and DRIM comparison with state-of-the-art edge aware approaches. 1st and 3rd rows – output of the edge aware techniques; 2nd and 4th rows - DRIMmetric [12] with probability of 95%. Parameters used – (a) Farbmann et al. [10] multiscale approach balanced – (b) Fattal's [11] α¼0.8, β¼0.19 and γ¼0.9; – (c) Paris et al. [9]σr ¼ log ð2:5Þ, α¼0.5 and β¼0.0 (for conveying the local effect) – (d) The Present HOR 1st row: β¼0.4, γ¼0.8; 3rd row: β¼0.6, γ¼0.8. (For interpretation of the references tocolor in this figure caption, the reader is referred to the web version of this article.)

A. Artusi et al. / Computers & Graphics 45 (2014) 51–63 61

amplification of contrast (blue) does almost not exist, and the onlyimage that is affected by using the proposed HOR is the image 15.

DRIM visual analysis: Figs. 10, 11, 13 and 14 show results withthe corresponding DRIM distortion maps. Fig. 10 and 11 comparesthe DRIM maps at probability 75% (2nd row) and at 95% (3rd row)for each output result.

The visual analysis of the results shows that in the case ofprobability 75% the state-of-the-art methods show a consistentnumber of distorted pixels localized in large areas, when com-pared with the present HOR. On the other hand, when theprobability increases to 95%, the size of these areas are eitherreduced or are almost not affected by any distortion. However, in

Fig. 14. Output and DRIM comparison with state-of-the-art edge aware approaches. 1st and 3rd rows – output of the edge aware techniques; 2nd and 4th rows – DRIMmetric [12] with probability of 95%. Parameters used – (a) Farbmann et al. [10] multiscale approach balanced – (b) Fattal's [11] α¼0.8, β¼0.19 and γ¼0.9; – (c) Paris et al. [9]σr ¼ log ð2:5Þ, α¼0.5 and β¼0.0 (for conveying the local effect) – (d) the present HOR 1st row: β¼0.6, γ¼0.9; 3rd row: β¼0.7, γ¼0.9. (For interpretation of the references tocolor in this figure caption, the reader is referred to the web version of this article.)

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some cases the state-of-the-art methods are still showing largeareas of lost of contrast (green) and reversal of contrast (red).

Figs. 13 and 14 are showing other results with the distortionmaps with probability at 95%, where the all methods are showingsimilar behavior.

Comparison with simpler TMO's: One can observe that the globaloperators are faster and convey an overall better appearance(Artusi et al. [28]). For this purpose, we have computed the DRIMmaps for a well known global version of two TMOs published byReinhard et al. [27] and Drago et al. [29]. The comparison is limitedto the global operator showing that the quality of the results is notcomparable with the state-of-the-art edge aware techniques.

The results are shown in Fig. 12 for the distortion maps atprobability of 95%.

The results reveal that the DRIM obtained for the Reinhardet al. [27] and the Drago et al. [29] operators often show largerareas of amplification of contrast; see the window area in Fig. 12,in comparison with the results obtained by the majority of theedge-aware techniques employed in this experiment.

Fig. 12 (2nd row) shows reversal of contrast, in large areas ofthe window, for both global operators. On the other hand, theedge-aware techniques have very tiny areas affected by reversal ofcontrast. Moreover, we emphasize in general that global operatorsare not designed for edge-awareness and do not encapsulatemechanisms for retaining the fine details at different spatial scale,as in the case of the present HOR and edge-aware techniques.

6.2. Computational analysis

Another aspect that needs to be taken into account is thecomputational cost associated with the different algorithms. Here,we have performed a computational cost analysis for the proposedtechnique versus other state-of-the-art techniques.

Our approach presents computational complexity and asso-ciated cost comparable to the one presented in [11,10] andoutperforming the method of [9].

Specifically, the method presented by Paris et al. [9] requires1738 s to process an image size of 800�525, 420 s for an imagesize of 400�262 and 190 s for an image size of 267�174. Whencompared with the computational cost of our method and theapproaches of [11,10], the computational cost is significantlyreduced: 14 s to process an image size of 800�525, 3 s for animage size of 400�262, and 1 s for an image size of 267�174.

Recently, Aubry et al. [14] presented a fast implementation ofParis et al. [9] technique that significantly improves its computa-tional performances (50 times faster). However, our comparison isdone on the Matlab implementation of the all techniques used inthe evaluation, as provided by the authors, without including anyoptimization. Even if we apply the 50-fold improvement in themeasured time of the Matlab implementation of Paris et al. [9], thepresent HOR delivers an excellent overall performance.

7. Concluding remarks

We have introduced a new edge preserving technique that makesuse of a HOR method, which is able to preserve edges withoutintroducing artifacts and reducing any changes in the image structurewhen compared to the state-of-the-art edge preserving operators. Thepresent method does not require an extra stop-edge function, thusoffering simplicity. Furthermore, its computational cost increaseslinearly in time. We have demonstrated the accuracy of the presenttechnique on a variety of images and parameter settings. The use ofthe HOR technique in other applications such as details enhancementand image colorization is also possible and will be part of future work.The proposed HOR technique will be further implemented in graphics

hardware with reference to video applications, allowing substantialimprovements in computational performance.

Acknowledgments

This work was partially supported by Ministry of Science andInnovation Subprogramme Ramon y Cajal RYC-2011-09372,TIN2013-47276-C6-1-R from Spanish government, 2014 SGR 1232from Catalan government, NSFC-Guangdong Joint Fund (U1135003),Natural Science Foundation of China (NSFC) (No. 61320106008).

References

[1] Harten A, Engquist B, Osher S, Chakravarthy S. Uniformly high order essen-tially non-oscillatory schemes. J Comput Phys 1987.

[2] Shu CW. Essentially non-oscillatory and weighted essentially non-oscillatoryschemes for hyperbolic conservation laws. Technical Report, ICASE Report No.97-65, NASA—CR-97-206253; 1997.

[3] Chan TF, Zhou HM. Eno-wavelet transforms for piecewise smooth functions.SIAM J Numer Anal 2002;40:1369–404.

[4] Boucenna H, Halimi M. Image segmentation by the level set methods usingthird order WENO. In: 5th international conference: sciences of electronic,technologies of information and telecommunications; 2009.

[5] Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion.IEEE Trans Pattern Anal Mach Intell 1990;12:629–39.

[6] Tomasi C, Manduchi R. Bilateral filtering for gray and color images. In:Proceedings of the IEEE international conference on computer vision; 1998.p. 839–46.

[7] Su Z, Luo X, Alessandro A. A novel image decomposition approach and itsapplications. Vis. Comput. 2013;29(10):1011–23.

[8] Buades BCA, Morel JM. The staircasing effect in neighborhood filters and itssolution. IEEE J Trans Image Process 2006;15:1499–505.

[9] Silvan P, Hasinoff SW, Kautz J. Local laplacian filters: edge aware imageprocessing with a laplacian pyramid. In: ACM transactions on graphics(proceedings of ACM SIGGRAPH); 2011.

[10] Farbman Z, Fattal R, Lischinski D, Szeliski R. Edge-preserving decompositionsfor multi-scale tone and detail manipulation. In: ACM SIGGRAPH 2008 papers,SIGGRAPH '08. ACM; 2008. p. 67:1–10.

[11] Fattal R. Edge-avoiding wavelets and their applications. ACM Trans Gr 2009;28(3):1–10.

[12] Aydin TO, Mantiuk R, Myszkowski K, Seidel H-P. Dynamic range independentimage quality assessment. ACM Trans Gr 2008;27(3):69:1–10.

[13] Banterle F, Artusi A, Debattista K, Chalmers A. Advanced high dynamic rangeimaging: theory and practice. Natick, MA, USA: AK Peters (CRC Press); 2011.

[14] Aubry M, Paris S, Hasinoff S, Kautz J, Durand F. Fast and robust pyramid-basedimage processing. Technical Report; 2011.

[15] Fattal R, Lischinski D, Werman M. Gradient domain high dynamic rangecompression. ACM Trans Gr 2002;21(3):249–56.

[16] Chen J, Paris S, Durand F. Real-time edge-aware image processing with thebilateral grid. In: ACM transactions on graphics (Proceedings of ACM SIG-GRAPH); 2007.

[17] Tumblin J, Turk G. LCIS: a boundary hierarchy for detail-preserving contrastreduction. In: ACM transactions on graphics (Proceedings of ACM SIGGRAPH);1999.

[18] Burt PJ, Edward, Adelson EH. The laplacian pyramid as a compact image code.IEEE Trans Commun 1983;31:532–40.

[19] Burt PJ. Fast filter transform for image processing. Comput Gr Image Process1981;16(1):20–51.

[20] Akansu AN, Haddad PR. Multiresolution signal decomposition: transforms,subbands, and wavelets. 2nd ed. Inc. Orlando Fl, USA: Academic Press; 1992and 2000.

[21] Li Y, Sharan L, Adelson EH. Compressing and companding high dynamic rangeimages with subband architectures. ACM Trans Gr 2005;24:836–44.

[22] Liu X, Osher S, Chan T. Weighted essentially non-oscillatory schemes.J Comput Phys 1994;115:200–12.

[23] Mosedale A, Drikakis D. Assessment of very high-order of accuracy in LESmodels. ASME J Fluids Eng 2007;129:1497–503.

[24] Thornber B, Mosedale A, Drikakis D. On the implicit large eddy simulations ofhomogeneous decaying turbulence. J Comput Phys 2007;226:1902–29.

[25] Drikakis D, Hahn M, Mosedale A, B T. Large eddy simulation using highresolution and high order methods. Philos Trans R Soc A 2009;367:2985–97.

[26] Fattal R. [link]. URL ⟨http://www.cs.huji.ac.il/�raananf/projects/eaw/⟩.[27] Reinhard E, Stark M, Shirley P, Ferwerda J. Photographic tone reproduction for

digital images. ACM Trans Gr 2002;21(3):267–76.[28] Artusi A, Akyz O, Ahmet, Roch B, Michael D, Chrysanthou Y, et al. Selective

local tone mapping. In: Proceedings of IEEE international conference on imageprocessing (ICIP); 2013.

[29] Drago F, Myszkowski K, Annen T, Chiba N. Adaptive logarithmic mapping fordisplaying high contrast scenes. Comput Gr Forum 2003;22:419–26.

A. Artusi et al. / Computers & Graphics 45 (2014) 51–63 63