Texture Enhanced Histogram Equalisation Using TV-L 1 Image Decomposition

Transcription

1 1 Texture Enhaned Histogram Equalisation Using TV-L 1 Image Deomposition Ovidiu Ghita, Dana E. Ilea, and Paul F. Whelan, Senior Member, IEEE Abstrat Histogram transformation defines a lass of image proessing operations that are widely applied in the implementation of data normalisation algorithms. In this paper we present a new variational approah for image enhanement that has been onstruted to alleviate the intensity saturation effets that are introdued by standard ontrast enhanement methods based on histogram equalisation. In our work we initially apply total variation (TV) minimisation with a L 1 fidelity term to deompose the input image with respet to artoon and texture omponents. Contrary to previous works that rely solely on the information enompassed in the distribution of the intensity information, in our approah the texture information is also employed to emphasize the ontribution of the loal textural features in the ontrast enhanement proess. This is ahieved by implementing a non-linear histogram warping ontrast enhanement strategy that is able to maximise the information ontent in the transformed image. Our experimental study addressed the ontrast enhanement of a wide variety of image data and omparative evaluations are provided to illustrate that our method produes better results than onventional ontrast enhanement strategies. Index Terms Contrast enhanement, TV-L 1, image deomposition, histogram warping, entropy maximisation. T I. INTRODUCTION HE optial and sensing limitations that are assoiated with standard digital image aquisition systems prompted the demand for flexible image proessing strategies that are able to maximise the visual transitions between objets that are present in the image data. Thus, among many low-level image proessing tasks, the development of automati ontrast enhanement (ACE) strategies forms one distint diretion of researh. The main reason behind this onsiderable interest is motivated by the fat that ACE tehniques are often used as preursors to higher level image analysis tasks suh as image segmentation, feature extration and pattern reognition and Ovidiu Ghita is with the Centre for Image Proessing and Analysis (CIPA), Shool of Eletroni Engineering, Dublin City University, Dublin 9, Ireland (phone: ; fax: ; ghitao@ eeng.du.ie). Dana E. Ilea, Paul F. Whelan are with the Centre for Image Proessing and Analysis (CIPA), Shool of Eletroni Engineering, Dublin City University, Dublin 9, Ireland ( danailea@eeng.du.ie, paul.whelan@du.ie). their appliation substantially enhanes the performane of the overall omputer vision systems. The main priniple behind ACE is to aentuate the intensity transitions between the objets aptured in the image data and this proess usually involves a range of linear or non-linear histogram transformations. More speifially, to maximise the image information, the histogram transformation needs to redistribute the probability of ourrene for eah intensity level i M (M being the range of intensity values) in the output image to attain a uniform distribution (when the probability for eah intensity level in the ontrast enhaned image is the same) [1]. Based on this approah, several ontrast enhanement (CE) algorithms have been proposed either in the Fourier/wavelet [2,3,4] domain or in the image (spatial) domain [5,6,7,8,9,1,11]. Among these tehniques the latter proved more popular when applied to onsumer images aptured by standard digital ameras, and the major objetive resides in the identifiation of a intensity mapping funtion g(.) that allows the maximisation of the image ontrast: O = g(i ), where O is the output (ontrast enhaned image), I is the input image and (i,j) denotes the pixel position in the image grid. To ahieve ontrast enhanement and maintain the appearane of the enhaned image similar to that of the original image, the intensity mapping funtion g(.) has to be monotonially inreasing over the domain K that overs the range of intensity values in the input image I, K M, where M is the omplete range of intensity values (for monohrome images M=[,255]). In this way, a wide spetrum of linear and non-linear funtions an be theoretially employed for ontrast enhanement where the most simplisti formulations are those that implement ontrast strething and gamma orretion. It is useful to note that these basi ontrast enhanement approahes return aeptable results only in situations where the intensity domain K is stritly a sub-domain of M, i.e. K M. To answer this limitation several related approahes based on random spatial sampling or loal statistis have been atively explored [7,12,13,14], but the improvement in ontrast enhanement has been obtained at the expense of inserting undesirable intensity artefats (suh as stairase effets) due to abrupt disontinuities in the loal image ontent. To irumvent the ompliations assoiated with the implementation of subjetive spatially onstrained strategies and the ourrene of stairase effets, the ontrast enhanement has been approahed as a global histogram warping proess, and among many potential implementations

2 2 the ontrast enhanement tehniques based on global histogram equalization (GHE) proved most ommon [6,7,15]. While the appliation of GHE methods maximises the information ontent (entropy) in the transformed image, it is important to mention that this global transformation introdues saturation effets and over-enhanement. Alleviation of these problems has attrated substantial researh interest and several approahes based on multi-objetive optimisation [7], ontrast limited adaptive histogram equalisation [16,17,18] and on the adaptive ombination of global and loal ontrast enhanement [19,2] have been intensively explored. In this paper we propose to address the side effets introdued by GHE using a multi-stage TV-L 1 ontrast enhanement approah and we will experimentally demonstrate that our approah, as opposed to more onventional GHE tehniques, substantially redues the over-enhanement and the intensity saturation effets. Additional pratial advantages that are assoiated with the proposed ontrast enhanement algorithm are also demonstrated in the ontext of edge detetion and artoon rendering. This paper is organised as follows. Setion II briefly review the theoretial and pratial issues related to the seletion of the histogram transformation for ontrast enhanement. In Setion III the proposed variational approah for ontrast enhanement is introdued, whih is followed in Setion IV by a detailed analysis of the experimental results. Setion V onludes the paper with a summary of the main findings resulting from our study. information ontent in the output image. In this regard, if we replae w(s)=1 in (2) we obtain the standard minimummaximum intensity streth operation g ( s) = s, s [, N] and L N if we replae w(s) with the histogram of the input image h(s), s [, N] we end up with the standard histogram L s equalisation proess g( s) = h( r) dr (where ard( Ω) ard (Ω) denotes the ardinality of the image domain Ω where the input image I is defined), whih is the most ommon approah applied for global ontrast enhanement. Although many formulations for w(s) an be devised, the use of histogram equalisation in the ontext of ontrast enhanement is motivated by two main reasons. Firstly, the histogram equalisation proess redistributes the intensity information in suh a way that the inter-histogram bar spaing is minimised. Seondly, this histogram transformation approah is able to (theoretially) maximise the information ontent in the ontrast enhaned image. While these properties are opportune for ontrast enhanement, it is important to highlight that histogram equalisation exhibits several limitations suh as the inidene of intensity saturation and over-enhanement. II. HISTOGRAM TRANSFORMATION-BASED CONTRAST ENHANCEMENT The vast majority of ontrast enhanement proedures are based on various histogram transformations that attempt to maximise the information ontent in the input image via intensity mapping. If we assume that the input image is defined 2 as a disrete signal I :Ω K, where Ω Z denotes the disrete image domain and K M is the domain overed by the intensity values of the pixels in the input image, then the ontrast enhanement resides in the identifiation of the funtion g(.) that implements the intensity mapping as follows: q = g( s), s K, q M (1) (a) (b) where s = I ( i, j) Ω represents the intensity values of the pixels in the input image and q is the orresponding intensity value in the ontrast enhaned image. As indiated in [8] this problem an be formulated as a general variational minimisation model and the intensity mapping funtion g(s) an be implemented using a family of funtions w(s) as follows: L s g( s) = N w( r) dr (2) w( s) ds where N and L are the right bounds of the intensity domains K and M, respetively (i.e. N = sup(k), L = sup(m)). Hene, the ontrast enhanement an be reformulated as the problem of finding the funtion w(s) that is able to maximise the () Fig. 1. Global histogram equalisation ontrast enhanement. (a) Input image. (b) GHE ontrast enhaned image. Close-up details taken from the input () and ontrast-enhaned (d) images to illustrate the intensity saturation and over-enhanement artefats that are introdued by the appliation of the histogram equalisation proess. These problems are learly illustrated in Fig. 1 where the histogram equalisation proess has been applied to the standard Cameraman test image. For visualization purposes, lose-up details are provided to highlight the intensity artefats that are introdued during the ontrast enhanement proess. To redress these undesirable ontrast enhanement problems that are introdued by histogram equalisation, in this work we (d)

3 3 propose a multi-stage variational-based ontrast enhanement algorithm that will be detailed in the next setion of the paper. III. PROPOSED CONTRAST ENHANCEMENT METHOD The proposed ontrast enhanement approah involves a multi-step algorithm that initially applies the TV-L 1 model to attain the artoon-texture deomposition of the input image. After the extration of the artoon-texture image omponents, ontrast enhanement is ahieved by applying a non-linear histogram warping proess that emphasises the ontribution of the texture information in the intensity distribution of the ontrast enhaned image. An overview that details the proposed ontrast enhanement algorithm is depited in Fig. 2 and full details about eah omputational omponent will be provided in the remainder of this setion. Fig. 2. Overview of the proposed ontrast enhanement algorithm. A. TV-L 1 Cartoon-Texture Deomposition An important omponent of the proposed ontrast enhanement strategy is represented by the proess relating to the artoon-texture deomposition. The main objetive of this proess (in the ontext of the proposed appliation) is to extrat the texture omponent, as this information emphasises the meaningful patterns ontained in the input image and rejets the undesirable intensity transitions that are aused by variations in illumination onditions. The problem of artoontexture deomposition an be effiiently solved as a global variational model [21,22,23,24]. Total variation (TV) models have been widely employed for data denoising [25,26,27] and reently have found other interesting appliations in the fields of fae reognition [2], texture enhanement [28], inpainting [29,3] and blind deonvolution [31]. Using this variational approah, the input image an be deomposed as follows: I = t, where and t denote the artoon and texture omponents, respetively. The artoon image, whih ontains the detextured objets (or non-osillatory omponents) from the input image I, an be determined by minimising the following expression: 2 Ω min λ I dω (3) where Ω Z denotes the image domain and the symbol. L defines the L 1 norm. The TV-L 1 model depited in (3) 1 onsists of two distint terms that are alulated over the image domain Ω. The first term in (3) implements a PDEbased de-texturing proess (i.e. the total variation of the artoon omponent ), while the seond term defines a fidelity term that fores the intensity values in to remain lose to L 1 those in the original image I. The TV-L 1 variational model is ontrolled by the Lagrange multiplier λ R, whih is inversely proportional to the strength of the data smoothing proess. Using the alulus of variations (see Appendix A) we an derive the Euler-Lagrange equation of (3) and the steady state solution an be iteratively obtained by artifiial time disretization as follows: I t = λ, (t=) = I (4) I where t is the partial derivative of with respet to the time variable t, is the divergene and denotes the gradient operator. The implementation of (4) in the disrete image domain requires the approximation of the partial derivatives with entral differenes where the solution is found by the steepest gradient desent as indiated in (5). n 1 = tλ n t x x y n I 2 n ( I ), ε x n n n y y n β β s. t. β, ε where (i,j) denotes the position of the pixel in the image, and (5) are the forward and bakward disrete gradients, respetively, is the magnitude of the gradient, t is the time step, n is the iteration index, and x and y are the disrete spatial distanes of the image grid. The expression in (5) implements the mean urvature with the fidelity I (onstraint) term λ and is onvergent if the CFL I t ondition is upheld ( α, x y α R [31]). As mentioned in [22], the TV-L 1 variational model exels when applied to image deomposition and in this proess the optimal seletion of the parameter λ R plays an important role. While this parameter is usually seleted in onjuntion with the level of noise present in the image or based on speifi geometrial onstrains assoiated with the image deomposition proess (for more details refer to [22,31]), in the proposed implementation the optimisation of this parameter will be onduted to maximise the image ontent (entropy) in the ontrast enhaned image. This will be explained later in the paper. In the proposed ontrast enhanement algorithm, the TV-L 1 model is applied to perform artoon-texture deomposition and the artoon image λ is obtained by applying (5) to the input image. If we assume that the input image is noise-free, the texture omponent an be determined by simply subtrating the artoon omponent λ from the input image I as follows:

4 4 tλ = I λ, λ D (6) where D is the interval of variation for the parameter λ, λ and tλ are the artoon and texture images, respetively, that are obtained for a value λ D. Fig. 3 illustrates the results obtained when the artoon-texture deomposition is applied to the Cameraman image shown in Fig. 1. λ=.1 λ=.5 the information ontained in the texture omponent. In this regard, the first step is to identify the pixels that are assoiated with strong textures after the appliation of the artoon-texture deomposition. The binary texture map is obtained by applying (7). b 1 if I > ρ t =, ρ R (7) Otherwise where t b is the binary texture map and ρ is a small positive value. (The parameter ρ ontrols the strength of the texture information that will be used in the ontrast enhanement proess and in our implementation ρ has been set to 1.). One the identifiation of the texture pixels that are assoiated with the input image is finalised, the next step implies the evaluation of the neighbourhood Г around eah texture pixel. Sine the neighbourhood Г enompasses the loal texture, whih is usually defined by a heterogeneous loal distribution, our aim is to identify the extreme intensity values within the neighbourhood Г that will be used to alter the intensity distribution H that is alulated from the artoon omponent λ. The aim of this histogram transformation is to overemphasise the ontribution of the loal texture in the alulation of the histogram H and the result of this proess is illustrated in Fig. 4. As illustrated in Fig. 4, it an be observed that the appliation of the texture-enhaned histogram transformation generates a more uniform distribution than the distribution alulated from the original (input) image, and this not only allows the preservation of the textural features during the ontrast enhanement proess, but also avoids the introdution of saturation effets. All operations assoiated with the proposed texture-enhaned histogram transformation are shown in equation (8) and in the pseudoode sequene depited in (9). H = U h ( p), h ( p) = δ ( ( i, j) Ω p M 1 if where δ (, p) = if, p) dω, = p p (8) λ=.9 Fig. 3. Cartoon-texture deomposition ( t =.1) for different values of the parameter λ. Left: Cartoon images. Right: Texture images. Note that the strength of the texture dereases with the inrease in the value of λ. B. Texture-Enhaned Histogram Equalisation After artoon-texture deomposition, the next omponent of the proposed algorithm addresses the alulation of the histogram transformation that is applied for ontrast enhanement. As indiated in Setion II our aim is to onstrut a ontinuous monotoni inreasing transformation that is able to avoid the intensity and over-saturation effets that are introdued by the standard histogram equalisation proess. The main idea behind this approah is to implement a loal intensity mapping proess that alters the shape of the histogram alulated from the artoon image with respet to Texture enhaned histogram transformation: for end (i, j) Ω if ( t b end ) == 1 onstrut Γ for( p [ l, h]) end i, j and alulate l = inf ( ), H ( p) = H ( p) 1 Γ h = sup( ) where H is the intensity distribution (histogram) alulated from the artoon image λ, Ω is the image domain, λ Γ λ (9)

5 5 M [,255] defines the intensity domain, b t is a texture pixel at loation (i,j) in the image, Г is the 3 3 neighbourhood b around t, inf(.) and sup(.) are the infimum and supremum operators, l and h are the lowest and highest intensity values within Г in the artoon image λ,. Frequeny Histogram transformation Original histogram Transformed histogram Intensity value Fig. 4. Texture-enhaned histogram transformation. Note the more uniform distribution of the intensity information in the transformed histogram. Blueirled line histogram alulated from the Cameraman artoon image (λ=.1) - see Fig. 3. Red-barbed line transformed histogram. After the alulation of the texture-enhaned distribution H (using 8 and 9), the next step involves the onstrution of the umulative distribution g(s) that will be used as a look-uptable in the histogram equalisation proess. s g( s) = κ H ( r) dr, O = g( I ) ( i, j) Ω (1) L where κ =, L = sup( M ) L H ( r) dr is a saling fator that maps the intensity transformation within the interval M and O ( i, j) Ω is the ontrast enhaned image. C. Parameter Seletion As indiated in Setion III.A, the TV-L 1 -based image deomposition is ontrolled by the parameter λ R, whih is inversely proportional to the level of smoothing in the artoon image λ. Sine the aurate deomposition of the input image into artoon and texture omponents plays the entral role in the proposed ontrast enhanement strategy, the optimisation of the parameter λ should be onduted with the aim of maximising the information ontent in the ontrast enhaned image. Sine entropy [32] is a measure of the average information ontent present in the image, our objetive is to maximise the expression of the entropy E(.) shown in (11), when the value of the parameter λ is varied within the range [,1]. E( H n ) = H n ( s) log 2 ( H n ( s)), H n ( s) = 1 (11) s M O opt H n λ [,1] s M λ = arg max[ E( )] (12) where E is the entropy measure, H n is the normalised version H ( s) of the distribution H, i.e. H n ( s) =, λ opt is the r H ( r) M value of λ for whih the entropy measure is maximised and O H n denotes the normalised histogram of the ontrast enhaned image O ( i, j) Ω (see equation 1). IV. EXPERIMENTAL RESULTS In this experimental study we analyse the performane of the proposed algorithm when ompared to performanes offered by other relevant histogram speifiation/equalisation ontrast enhanement (CE) tehniques. In this regard, four representative ontrast enhanement algorithms were seleted for omparative purposes: onventional global histogram equalisation (GHE) [1], brightness preserving bi-histogram equalisation (BP-BHE) [17], minimum mean brightness error bi-histogram equalisation (MMBE-BHE) [15] and dynami histogram equalisation (DHE) [18]. Table I. Numerial results obtained by the proposed TV-L 1 TE-HE and other related histogram equalisation (HE)-based ontrast enhanement strategies. Image Method Entropy EPI α Cameraman image Berkeley image Couple image Kodak image Aerial image Lighthouse image GHE BP-BHE MMBE-BHE DHE TV-L 1 TE-HE GHE BP-BHE MMBE-BHE DHE TV-L 1 TE-HE GHE BP-BHE MMBE-BHE DHE TV-L 1 TE-HE GHE BP-BHE MMBE-BHE DHE TV-L 1 TE-HE GHE BP-BHE MMBE-BHE DHE TV-L 1 TE-HE GHE BP-BHE MMBE-BHE DHE TV-L 1 TE-HE The BP-BHE, MMBE-BHE and DHE ontrast enhanement algorithms were designed to irumvent the saturation and over-enhanement effets that are introdued by GHE by partitioning the histogram of the input image into two [15,17] and multiple omponents [18] prior to the appliation of the

6 6 histogram equalisation proess. Sine the objetives of the BP- BHE, MMBE-BHE and DHE algorithms are similar to those assoiated with the proposed variational approah, these ontrast enhanement shemes are partiularly suitable to be inluded in this experimental study. As indiated in Setion II, the main objetive of this paper is to propose a new CE algorithm based on TV-L 1 textureenhaned histogram equalisation (TV-L 1 TE-HE). To allow for a diret omparison between the performanes obtained by our algorithm and GHE, BP-BHE, MMBE-BHE and DHE ontrast enhanement algorithms, the first experiments are onduted on benhmark images ( Cameraman, Couple, Lighthouse, Aerial and other standard images inluded in the Kodak and Berkeley databases) and the algorithm performane will be sampled by metris suh as entropy and edge enhanement. The experimental results depited in Figs. 5 to 8 augment the numerial data reported in Table I (in this table additional results are also reported for images depited in Fig. 9) and they illustrate that the proposed algorithm is able to outperform the other CE tehniques investigated in this study with respet to the avoidane of intensity saturation and overenhanement. As indiated earlier, edge enhanement is another metri that is ommonly used in the assessment of CE tehniques and the results reported in Figs. 1 and 11 show that the proposed algorithm outperforms the GHE, BP-BHE, MMBE-BHE and DHE with respet to both edge loalisation and edge enhanement (the edge attenuation generated by MMBE-BHE and the notieable edge distortions introdued by GHE and BP-BHE an be learly observed in Fig. 11). (a) (b) () (d) (e) (f) Fig. 5. Contrast enhanement results Cameraman image. (a) Input image. (b) GHE. () BP-BHE. (d) MMBE-BHE. (e) DHE. (f) Proposed TV-L 1 TE-HE. Fig. 6. Close-up details from images depited in Fig. 5. (Left-right): input image, GHE, BP-BHE, MMBE-BHE, DHE, TV-L 1 TE-HE. Note the avoidane of the intensity saturation effets and over-enhanement that is ahieved by the proposed TV-L 1 TE-HE when ompared to GHE and other HE-based ontrast enhanement strategies.

7 7 (a) (b) () (d) (e) (f) Fig. 7. Contrast enhanement results Couple image. (a) Input image. (b) GHE. () BP-BHE. (d) MMBE-BHE. (e) DHE. (f) Proposed TV-L 1 TE-HE. Fig. 8. Close-up details from images depited in Fig. 5. (Left-right): input image, GHE, BP-BHE, MMBE-BHE, DHE, TV-L 1 TE-HE. Note the avoidane of the intensity saturation effets and over-enhanement that is ahieved by the proposed TV-L 1 TE-HE when ompared to GHE and other HE-based ontrast enhanement strategies. (a) (b) () Fig. 9. Additional standard test images used in the experimental evaluation. (a) Kodak database. (b) Aerial image. () Lighthouse image.

8 8 (a) (b) () (d) (e) (f) Fig. 1. Contrast enhanement results Berkeley database [33] image. (a) Input image. (b) GHE. () BP-BHE. (d) MMBE-BHE. (e) DHE. (f) Proposed TV-L 1 TE-HE. Note the risper edge enhanement obtained by the proposed TV-L 1 TE-HE when ompared to GHE and other HE-based ontrast enhanement strategies (for additional details please refer to Fig. 8).

9 9 Pixel intensity Input image GHE BP-BHE MMBE-BHE DHE TV-L1 TE-HE Pixel position on the seleted line Fig. 11. Edge enhanement Berkeley database image. Pixel intensities plotted for the highlighted line depited in image Fig. 1(a). Another useful aspet assoiated with the proposed TV-L 1 TE-HE ontrast enhanement strategy is its potential to be applied to image data orrupted by noise, as the artoontexture deomposition has the ability to rejet the noisy signal from the input image data. This useful property an be obtained by simply replaing the input image I in (1) with the artoon omponent λ that maximises the expression shown in (12). To quantify the improved performane assoiated with the proposed algorithm we have applied all ontrast enhanement shemes that are investigated in this study to the Cameraman image that has been orrupted with Gaussian noise (zero mean, standard deviation 1 grey-levels, N(,1)). Experimental results are presented in Fig. 12 and the effiieny of the ontrast enhanement proess is validated in the ontext of edge extration (see Fig. 13). To omplement the visual results presented in Fig. 13, numerial data are presented in the last olumn of Table I, where the auray of the ontrast enhanement is evaluated with respet to edge preservation. In this regard, the ontrast enhanement algorithms were applied to image data that has been orrupted with Gaussian noise (zero mean, standard deviation 1 grey-levels, N(,1)) and the auray of the edge preservation is measured with respet to the strength of the edge information in the noiseless image I using the orrelation index α that has been suggested in [34]. In (13) is the Laplaian operator, I is the original image, O = g( I n ) is the output image that is obtained after the appliation of the ontrast enhanement to the noisy image I n =I N(,1), I, O are the mean values alulated after the appliation of the Laplaian operator to images I and O, respetively, and the funtion Λ(.) is defined as follows, Λ( I, O) = I( i, j) O( i, j) (14) ( i, j) Ω The edge preservation index α takes values in the range [,1] and the higher its value the more aurate the edge preservation. The edge preservation results reported in Table I indiate that our TV-L 1 TE-HE ontrast enhanement algorithm produes better numerial results than the GHE, BP- BHE, MMBE-BHE and DHE ontrast enhanement strategies when applied to data orrupted by noise, and they further demonstrate the appropriateness of the artoon-texture deomposition in the ontext of ontrast enhanement. α = Λ( I I, O O) (13) Λ( I I, I I) Λ( O O, O O)

10 1 (a) (b) () (d) (e) (f) Fig. 12. Contrast enhanement results - Cameraman image orrupted with Gaussian noise, N(,1). (a) Input image. (b) GHE. () BP-BHE. (d) MMBE-BHE. (e) DHE. (f) TV-L 1 TE-HE. Fig. 13. Edge information extrated using the Canny edge detetor orresponding to the images (b-f) shown in Fig. 9. The last set of results is reported in the ontext of artoon rendering of olor portrait images. In this regard, we have applied the artoon rendering algorithm detailed in [35] to the olor ontrast enhaned images that are obtained using the

11 11 proposed and the HE-based methods (generalisation to olor was ahieved by onverting the input image to the Lab olor spae and the CE algorithms were applied to the L omponent) and experimental results are reported in Fig. 14. These additional results learly indiate that the artoon rendering in onjuntion with the proposed TV-L 1 TE-HE algorithm returns better results (note the more oherent rendering that is ahieved in areas defined by skin, whih are onsistent with the ambient illumination, and the natural enhanement of the highly textured regions suh as the eyes and the sarf areas) and they further demonstrate that the inlusion of the proposed ontrast enhanement algorithm in the development of more omplex image proessing tasks leads to improved performane. V. CONCLUSIONS The major aim of this paper was to introdue a new variational approah for histogram equalisation whih involves the appliation of the TV-L 1 model to ahieve artoon-texture deomposition. To avoid the ourrene of undesired artefats suh as intensity saturation and over-enhanement that are harateristi for onventional histogram equalisation methods, our approah formulates the histogram transformation as a non-linear histogram warping whih has been designed to emphasise the texture features during the image ontrast enhanement proess. The reported experimental results demonstrate that the proposed TV-L 1 TE- HE strategy is an effetive approah for ontrast enhanement and they also reveal that the algorithm detailed in this paper offers a flexible formulation that is able to outperform other histogram equalisation-based methods when applied to image data orrupted by noise. Our future studies will fous on the extension of the proposed variational algorithm to attain interframe onsistent ontrast enhanement when applied to low SNR video data and to examine the potential of applying loal restoration models that are able to address the image enhanement of medial data that are orrupted by multimodal noise models. APPENDIX A As indiated in Setion III.A the artoon image (or de-textured omponent) of the input image I an be obtained by minimising the funtional Ψ( ) = Ω λ I dω, where λ R is the Lagrange multiplier, Ω is the image domain and the symbol. L 1 denotes the L 1 norm. The funtional Ψ() an be re-written in the differential form F ( x, y,, x, y) = λ I L 1 and its solution an be obtained by applying the Euler-Lagrange equation as illustrated in (A2). Ψ( ) = F( x, y,, x, y ) dxdy (A1) ( x, y) Ω F F x F y x y = L 1 (A2) where x, y are the partial derivatives of the artoon image (i.e. x =, y = ). If we alulate the partial derivatives x y in (A2) and we onsider that I 1 = I we obtain the following expression: I λ = (A3) I This equation approximates the mean urvature flow when λ R [36]. In (A3) the first term implements a fidelity term with respet to the intensity values of the input image I and the seond term defines the mean urvature of ( denotes the divergene operator). If we assume that the artoon image is a funtion of the time variable t, then the expression shown in (A3) is a first-order Hamilton-Jaobi equation that an be solved using steepest gradient desent, If we use in (A4) the notation L I = λ (A4) t I t = and we express the t mean urvature in terms of the partial derivatives, = x x ( ) ( ) / 1/ x y y x y t y, then (A3) beomes, = x y I x ( ) y ( ) λ 2 2 1/ / 2 I x y x y (A5) where the stationary solution for is obtained for t. The implementation of (A5) in the disrete domain is obtained by approximating the partial derivatives,, with entral x y t differenes as indiated in (5) (see Setion III.A). REFERENCES [1] R.C. Gonzalez, R.E. Woods, Digital Image Proessing, 3rd Edition, Prentie-Hall, 28. [2] A.F. Laine, S. Shuler, J. Fan, W. Huda, Mammographi feature enhanement by multisale analysis, IEEE Transations on Medial Imaging, 13(4), , [3] J.L. Stark, F. Murtagh, E.J. Candès, D.L. Donoho, Gray and olor image ontrast enhanement by the urvelet transform, IEEE Transations on Image Proessing, 12(6), pp , 23. [4] Y. Wan, D. Shi, Joint exat histogram speifiation and image enhanement through the wavelet transform, IEEE Transations on Image Proessing, 16(9), pp , 27. [5] G. Sapiro, V. Caselles, Histogram modifiation via partial differential equations, in Pro of International Conferene on Image Proessing (ICIP), vol. 3, pp , Washington, DC, USA, 1995.

12 12 (a) (b) () (d) (e) (f) (g) Fig. 14. Cartoon rendering results Kodak database. (a) Original image. (b) Appliation of the artoon rendering algorithm [35] to image (a). (-g) Appliation of the artoon rendering to the image that is ontrast enhaned using GHE (), BP-BHE (d), MMBE-BHE (e), DHE (f) and proposed TV-L 1 TE-HE (g).

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Larnder, Augmented pereption via artoon rendering: refletions on a real-time video-to-artoon system, ACM SIGGRAPH Computer Graphis, 4(3), pp. 8:1-8, 26. [36] G. Sapiro, Geometri Partial Differential Equations and Image Analysis, Cambridge University Press, ISBN: , 21. Ovidiu Ghita reeived the BE and ME degrees in Eletrial Engineering from Transilvania University, Brasov, Romania and the Ph.D. degree from Dublin City University, Ireland. From 1994 to 1996 he was an Assistant Leturer in the Department of Eletrial Engineering at Transilvania University. Sine then he has been a member of the Vision Systems Group (VSG) at Dublin City University (DCU) and urrently he holds a position of DCU-Researh Fellow. Dr. Ghita has authored and o-authored over 9 peerreviewed researh papers in areas of instrumentation, range aquisition, mahine vision, texture analysis and medial imaging. Dana E. Ilea reeived her B.Eng. degree (25) in Eletroni Engineering and Computers Siene from Transilvania University, Brasov, Romania and her Ph.D. degree (28) in Computer Vision from Dublin City University, Dublin, Ireland. Sine 28 she holds the position of Post Dotoral Researher within the Centre for Image Proessing & Analysis (CIPA), Dublin City University. Her main researh interests are in the areas of image proessing, texture and olour analysis and medial imaging. Paul F. Whelan (S 84 M 85 SM 1) reeived his B.Eng. (Hons) degree from NIHED, M.Eng. degree from the University of Limerik, and his Ph.D. (Computer Vision) from Cardiff University, UK. During the period he was employed by Industrial and Sientifi Imaging Ltd and later Westinghouse (WESL), where he was involved in the researh and development of high-speed omputer vision systems. He was appointed to the Shool of Eletroni Engineering, Dublin City University (DCU) in 199 and is urrently Professor of Computer Vision (Personal Chair). Prof. Whelan founded the Vision Systems Group (VSG) in 199 and the Centre for Image Proessing & Analysis (CIPA) in 26 and urrently serves as its diretor. As well as publishing over 15 peer reviewed papers, Prof. Whelan has oauthored 2 monographs and o-edited 3 books. His researh interests inlude image segmentation, and its assoiated quantitative analysis with appliations in omputer/mahine vision and medial imaging. He is a Senior Member of the IEEE, a Chartered Engineer and a fellow of the IET and IAPR. He served as a member of the governing board ( ) of the International Assoiation for Pattern Reognition (IAPR), a member of the International Federation of Classifiation Soieties (IFCS) ounil and President ( ) of the Irish Pattern Reognition and Classifiation Soiety (IPRCS). Prof. Whelan is a HEA-PRTLI (RINCE, NBIP) and Enterprise Ireland funded prinipal investigator.