Total Variation Denoising and Enhan ement ofColor Images Based on the CB and HSV ColorModelsTony F. Chan, Sung Ha Kang and Jianhong Shen �Abstra tMost denoising and enhan ement methods for olor images have beenformulated on linear olor models, namely, the hannel-by- hannel modeland ve torial model. In this paper, we study the total variation (TV)restoration based on the two nonlinear (or non- at) olor models: theChromati ity-Brightness (CB) model and Hue-Saturation-Value (HSV)model. These models are known to be loser to human per eption. Re entworks on the variational/PDE method for non- at features by several au-thors enable us to denoise the hromati ity and hue omponents dire tly.We present both the mathemati al theory and digital implementation forthe TV method. Comparison to the traditional TV restorations based onlinear olor models is made through various experiments.1 Introdu tionAny tool that attempts to denoise and enhan e digital olor images must rely ontwo orrelated ingredients | the representation of olors (i.e. the olor model)and the restoration methodology applied to the representation. Therefore, the urrent paper starts with a brief introdu tion to these two omponents.1.1 RGB, CB, and HSV olor modelsIn image pro essing, olor has been represented or modeled in various ways [7℄.In this paper we shall fo us on the RGB model and HSV model.In the RGB representation of olor images, at ea h pixel p = (x; y), theve torial value I(p) = (u1(p); u2(p); u3(p)) represents the intensity of the three�Resear h was supported by NSF Grant DMS-9626755 and by ONR under N00014-96-1-0277. The authors are with the Department of Mathemati s, UCLA, Los Angeles, CA90095-1555, USA, (f han, skangg�math.u la.edu), and S hool of Mathemati s, Universityof Minnesota, Minneapolis, MN 55455, USA, (jhshen�math.umn.edu). Journal ofVisual Communi ation and Image Representation, in press, as ofMay, 2001. 1

primary olors separately. Ea h mono hromati omponent ui is alled a han-nel. The RGB model has led to the CB model, whi h de omposes an RGB pixelvalue I(p) into two omponents | the brightness omponent u(p) = kI(p)k(Eu lidean length), and the hromati ity omponent f(p) = I(p)=u(p). The hromat ity omponent lives on the unit sphere S2. Features like this that liveon nonlinear manifolds are said to be non- at [6, 16℄. It has been shown by sev-eral authors lately that the CB model is well suited for denoising, edge dete tionand enhan ement, and segmentation [6, 15, 16, 17, 18℄.HSV is another olor system that is believed to be more natural than theRGB systerm for human per eption. The three variables are: hue H , saturationS, and value V . The S and V are linear features and take values in the interval[0; 1℄. S en odes the \purity" of olor: larger S orresponds to purer olor. Thevalue V stores the intensity information, so that larger V value means brighter olor. The hue omponent H , though also taking values in [0; 1℄, is neverthelessa ir ular or periodi feature: as H in reases from 0 to 1, the olor spe trarevolve from red, yellow, green, yan, blue, to magenta, and eventually ba kto red. Therefore, the hue lives on the unit ir le S1 in some sense, and isanother example of non- at features [8℄. The ir ularization is easily made bythe exponential mapping H ! exp(i2�H).1

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Figure 1: An iso-V surfa e and an iso-S line on it in the RGB spa e. SeeEq. (1,2) for the de�nition. Noti e that an iso-V & S line is a 3-D \hexagon,"as shown separately on the right panel. From a; d; � � � to f; a, the hue value Hin reases from 0 to 1.The pre ise transition from the [r; g; b℄ variables to [H;S; V ℄ is realized by:V = r _ g _ b (1)S = 1� r ^ g ^ br _ g _ b ; (2)2

where for a pair of real numbers a and b: a _ b := max(a; b), a ^ b := min(a; b).The formula for H appears to be more ompli ated, but essentially is a simplepie ewise linear fun tion along any iso-S & V line (See Fig. 1). In the �gure, oneobserves a typi al \folded" hexagonal iso-S & V line. As one goes from a; d; � � �to f; a, the H value is de�ned so that it in reases linearly from 0 to 1. Thus onea h one of the six segments, the net in rement is 1=6. With this in mind, Theformula H an be expressed easily. For examples, along the �rst segment [a; d℄(along whi h the r and b oordinates are �xed)H = H(g) = 16 �g � br � b� ;and along [b; e℄ (along whi h the r and g oordinates are �xed)H = H(b) = 26 + 16 � b� rg � r� :Many lassi al pro essing tools for olor images are based on the RGB model,mostly be ause linear spa es are easy to work with. As a result, despite theirsimilarity to human olor per eption, the CB and HSV olor models have beenless favored due to their non- atness. The urrent paper intends to onstru trestoration models based on su h non- at olor models.1.2 The methodology of variational restorationThe variational and PDE method has attra ted mu h attention in image pro- essing be ause of its exibility in modeling and numerous advantages in nu-meri al omputation. Appli ations an be widely found in image segmentation,denoising, deblurring, enhan ement, inpainting, and motion estimation (see thetwo monographs [9, 19℄, for examples).The general framework of the variational approa h for denoising and enhan -ing olor images based on the linear RGB olor models an be lassi�ed intotwo ategories | the hannel-by- hannel approa h and the ve torial approa h.In the hannel-by- hannel approa h, ea h hannel ui is assumed to be on-taminated by noise �i so that the observation be omes u0i (x; y) = ui(x; y) +�i(x; y). A typi al hannel-by- hannel denoising model arries the form ofminui Ri(ui) subje t to 1jj Z jui � u0i j2 = �2i ; (3)where is the image domain, jj its area, Ri the regularity fun tional, and �ithe noise level. For example, in the total variation approa h (Rudin, Osher andFatemi [11, 12℄), one takes Ri(ui) = Z jruijdxdy:The TV measure has been proven both mathemati ally and omputationally tobe able to re ognize and enhan e edges embedded in noise [2, 12℄.3

In the ve torial denoising approa h, the ost fun tional is on the ve torialfun tion I = (u1; u2; u3):minI R3D(I) subje t to 1jj Z kI � I0k2 = �2; (4)where R3D is a regularity fun tional for ve tor-valued fun tions. Typi ally inappli ations, suppose Ri is a suitable 1-dimensional regularity fun tional for thei-th hannel, then one simply takes R3D = pR21 +R22 +R23. In the ase whenR3D is onne ted to the total variation and anisotropi di�usions, we observethe work of Sapiro and Ringa h [13℄ and Blomgren and Chan [1, 3℄.In pra ti e, both of the onstrained optimization problems (3) and (4) arerepla ed by their un onstrained forms. For example, for the ve torial ase, wesolve instead minI �R3D(I) + �2 Z kI � I0k2� :Here � is an appropriate panelty weight, whi h depends on the noise level. Inpra ti e, it is often estimated or hosen a priori [5, 12℄.1.3 Variational restoration for non- at featuresRe ently, the variational and PDE method has been generalized to non- at fea-tures by So hen, Kimmel and Malladi [14℄, Perona [10℄, Tang, Sapiro and Cas-selles [15, 16℄, Chan and Shen [6℄, and most re ently by Kimmel and So hen [8℄.Tang et al. and Chan et al. generalize the total varitional model, while Kimmelet al.'s model pro�ts from their previous general framework of Polykov a tionand Beltrami operators [14℄. All these works intend to restore edges of non- atfeatures.The urrent paper is the extension and ompletion of the works of [6, 15, 16℄on the total variation approa h for enhan ing and denoising non- at imagefeatures. We study in details the TV model and its numeri al implementation forrestoring olor images based on the CB and HSV olor models. Previous workson general non- at features enable us to work with the hromati ity feature f(on the sphere S2) and hue feature H (along the ir le S1) dire tly. Detailed omparison to the hannel-by- hannel and ve torial approa hes are illustratedthrough numeri al examples. Our results show onvin ingly the advantages ofthe CB and HSV olor models over linear ones.The paper is organized as follows. Se tion 2 introdu es the mathemati almodels, and Se tion 3 details their numeri al implementations. Numeri al ex-periments and omparison are explained in Se tion 4.2 The TV Formulation on CB and HSV ModelsIn this se tion, we explain the total variation formulation for the two nonlinear olor models. 4

2.1 TV for the CB modelIn the RGB representation, a olor image is a mappingI : ! R3+ = f(r; g; b) : r; g; b > 0g:I an be separated into the brightness omponent u = kIk, and the hromati ity omponent f = I=kIk = I=u. Mathemati ally speaking, this is the spheri al oordinates for the Eu lidean spa e. We shall not introdu e the two Eulerianangles (i.e., the latitude and longitude) for f , thanking to our previous work onnon- at features [6℄.The brightness u an be treated as a gray image, and thus any s alar denois-ing model an be applied. The hromati ity omponent f stores the major olorinformation, and is \non- at" sin e it takes values on the unit sphere S2. Wethus apply the general framework of non- at TV denoising models as studied byChan and Shen [6℄. In the hromati ity-brightness approa h, the �nal restored olor image I is assembled from the two restored omponents:I(p) = u(p)� f(p):We now detail the mathemati s.Given a noisy image I0, let u0 : ! R be its brightness omponent. Thenthe s alar TV restoration model applied to u0 isminu Z jruj dx dy + �2 Z(u� u0)2 dx dy;The asso iated Euler-Lagrange equation (in a formal level) of this ost fun tionalis �r � ( rujruj ) + �(u� u0) = 0: (5)To avoid possible singularity for jruj in the denominator, we ondition it tojruja = qjruj2 + a2 for some small a in numeri al implementation, whi h isequivalent to minimizing dire tlyZ jruja dx dy + �2 Z(u� u0)2 dx dy:The total variation as a ost fun tional legalizes the existen e of edges or sharpjumps in the brightness omponent, whi h usually orrespond to the real phys-i al boundaries of obje ts [12℄. (As well known in the literature, if a = 1 (orafter being linearly s aled), the �rst term measures the total area of the graphof z = u(x; y).)Now we dis uss how to restore the noisy hromati ity omponent f0. Letf : ! S2 be a general hromati ity feature. Assume that f is smooth. Then�xf(p) and �yf(p) are two tangent ve tors in the tangent spa e Tf(p)S2. Let5

k�k be the indu ed Riemannian norm of Tf(p)S2 in R3. Then the total variationfor hromati ity is de�ned to be [6℄ETV (f) = Z e(f ; p) dp = Zqk�xf(p)k2 + k�yf(p)k2dp; dp = dx dy:Here sin e the unit sphere S2 is embedded in R3 and f = (f1; f2; f3) 2 R3, wehavee(f ; p) = krfk =qkfx(p)k2 + kfy(p)k2 =pkrf1k2 + krf2k2 + krf3k2:Let d be any appropriate distan e fun tion on S2 [6℄. For example, one an hoose the embedded distan e or the ord distan ed(f; g) := kf � gkR3 =p(f � g)2; f; g 2 S2: (6)Another natural hoi e would be the geodesi distan e or the ar distan ed(f; g) = ar os hf; gi;where hf; gi denotes the inner produ t in R3. Then the TV restoration modelbe omes minf ETV (f) subje t to 1jj Z d2(f0; f)dp = �2:The Euler-Lagrange equation for its un onstrained version is given by:���x� �xfkrfk�� ��y � �yfkrfk�+ �2 gradf d2(f0; f) = 0;where gradfd2(f0; f) denotes the gradient ve tor of the s alar fun tion d2(f0; f)on S2 and ��x and ��y the ovariant derivatives a ting on ve tor �elds on thesphere. A ording to [6℄, when the hromati ity sphere is naturally endowedwith the restri ted metri of R3, the ovariant derivatives are simply the ordi-nary derivatives (in R3), followed by an orthogonal proje tion onto the sphere.Therefore, ��x � �xfkrfk�+ ��y � �yfkrfk� = r � � rfkrfk�+ krfkf;as worked out step by step in [6℄. Furthermore, if we hoose the embeddeddistan e (6) for d, then [6℄12gradfd2(f0; f) = �f (f � f0) = ��ff0:Here �f is the orthogonal proje tion from TfR3 onto the tangent plane TfS2:�f g = g � hg; fif; for any ve tor g 2 TfR3:6

The proje tion operator appears due to the fa t that we are working ex lusivelyon the hromati ity sphere, and thus \blind" to any hanges o uring perpen-di ularly to the sphere. Eventually, the restoration equation for the TV modelsimply be omes:ft (or 0) = r � � rfkrfk�+ krfkf + ��ff0: (7)We shall explain in Se tion 3 the numeri al implementation or the digitalversion of the nonlinear restoration equations (5) and (7). In parti ular, ournumeri al s heme works dire tly on the steady equation, thanking to the dig-ital TV �lter re ently proposed by Chan, Osher and Shen [5℄. It avoids the omplexity of keeping the nonlinear ow (7) stri tly on the hromati ity sphere.Numeri al methods for the dire t ow are also dis ussed in [15, 16℄.2.2 TV for the HSV modelIn the HSV olor model, the hue feature H is non- at and lives on the unit ir le S1, while the other two are both linear ones. Therefore, for the saturationS and value V , we simply apply the s alar TV denoising model (5). For the ir ular feature H , one opies the equation for hromati ity (7) based on thestraightforward modi� ation from S2 to S1:H = (H1; H2) 2 S1 � R2 and krHk =qkHxk2 + kHyk2:Our experiment (Fig. 6) shows that denoising separately the hue omponentH and the saturation omponent S does not produ e satisfa tory visual results.A possible explanation for this de� ien y is that to human visual per eption,the two omponents are highly orrelated. Therefore, a better approa h is torestore the ombination of H and S, whi h in the lassi al literature of olorimage pro essing, is also alled hromati ity.From Fig. 1, in the RGB olor spa e, for a �xed value V , the other twovariables H and S span the iso-V surfa e whi h onsists of three 2-D squares.The surfa e is not smooth sin e there are a orner and three folding lines.Working with su h irregular surfa es is in onvenient. Thus, we apply the disktransform to \straighten" this folded surfa e as follows. For ea h point on theiso-V surfa e with the hue and saturation (H;S), de�ne a omplex numberZ = S � exp(i2�H): (8)Sin e both H and S take all values in [0; 1℄, the image of ea h iso-V surfa e isthe unit disk in the omplex plane (see Fig. 2). Under the disk transform, ea hiso-S line on an iso-V surfa e (i.e., the 3-D hexagon on the right panel of Fig. 1)is mapped onto a ir le entered at the origin (see Fig. 2).The unit disk is a smooth and onvex domain and thus mu h easier to workwith. To restore the ombination of H0 and S0 in the HSV olor model, is now7

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Figure 2: The disk transform of an iso-V surfa e: Z = S exp(i2�H).equivalent to restoring a omplex-valued s alar feature Z0. Pre isely, for anygiven noisy image under the HSV model(H0(x; y); S0(x; y); V 0(x; y)); (x; y) 2 ;it is suÆ ient to denoise two s alar omponents | the real-valued signal V 0(x; y),and the omplex-valued signal Z0(x; y). Sin e both are s alar fun tions insteadof ve torial or non- at ones, we an simply utilize the same s alar TV model (5).The numeri al results shall onvin e us the advantage of working with the om-bination of H and S, or their disk transform.3 Numeri al ImplementationTo digitally implement the nonlinear di�erential equations, we have applied theapproa h of digital TV �ltering as proposed in [5℄. The digital TV �lter an been onsidered as the �nite di�eren e realization of the di�erential equations, but issimpler and more self- ontained in the general framework of graph theory [5℄.Let be a dis rete digital domain or a graph. Pixels in are denoted by�; �; � � � . In the onventional re tangular setting, it is also denoted by � = (ij).Denote N� the neighbors of the pixel �. For instan e, in the re tangular setting,N� = N(ij) = f (i+ 1; j); (i� 1; j); (i; j � 1); (i; j + 1) g:Of ourse, there is mu h freedom in de�ning the neighbors. For example, onemay in lude (i� 1; j � 1) in N(ij). If � 2 N�, we also write � � �.Let u : ! R be the brightness omponent. We de�ne its lo al variationto be jr�uj = qP���(u� � u�)2, and its onditioned form to be jr�uja =pkr�uk2 + a2 for some small a. 8

By introdu ing the weights !��(u) = 1jr�uja + 1jr�uja , Chan, Osher, andShen [5℄ showed that the Euler-Lagrange equation (5) is repla ed by a systemof nonlinear algebrai equations:X���!��(u) (u� � u�) + �(u� u0) = 0; � 2 :The te hniques of linearization and iteration s heme lead to the so- alled digitalTV �lter [5℄. The digital TV �lter F is nonlinear data dependent �lter F : u! von e the noisy image u0 is given. At any pixel � 2 ,v� = F�(u) = X���h��(u)u� + h��(u)u0�; (9)where the low-pass �lter oeÆ ients areh��(u) = !��(u)�+P �� !� (u) ; h��(u) = ��+P �� !� (u) :The digital TV �lter is applied in an iterative fashion. To restore and de-noise the brightness omponent u0, one starts with a random guess u(0) (forexample, u(0) = u0, onveniently though unne essarily), and then generateu(n) = F(un�1) for n = 1; 2; � � � . u(n) onverges to the optimal restorationu. We now dis uss how to apply the digital TV �lter to the hromati ity om-ponent. Let f0 : ! S2 be the noisy hromati ity. Taking the embeddeddistan e d(f; g) =p(f � g)2 for any f; g 2 S2, we de�ne the lo al variation ata pixel � of a hromati ity feature f : ! S2 to be:e(f ;�) = 24X��� d2(f�; f�)35 12 :The digitized total variation plus the �tting onstraint be omes [5℄ETV (f; �) = X�2n e(f ;�) + � X�2n 12d2(fo; f);whi h provides the digital ost fun tion for optimization. The digital Euler-Lagrange equation is shown to be [6℄:0 = X����f�(f�)� 1e(f ;�) + 1e(f ;�)�+ ��f�(f0�); � 2 : (10)Here �f is the orthogonal proje tion de�ned in the previous se tion. By setting!��(f) = 1e(f ;�) + 1e(f ;�) ;9

we rewrite the restoration equation (10) as�f�(X���!��f� + �f0�) = 0; � 2 :Chan and Shen [6℄ showed that this equation on the unknown optimal restora-tion f : ! S2 an be similarly solved by the digital TV �lter F . Unlike thebrightness omponent, sin e the feature lives on the unit sphere, g = F(f) nowneeds an extra step of normalization:~g� =P��� h��f� + h��f0�;g� = ~g�=k~g�k:One starts the iterative �ltering pro ess with an initial guess f (0), then generatef (n) = F(f (n�1)). Chan and Shen [6℄ showed that the limit of f (n) indeed solvesthe restoration equation (10).For the TV restoration model based on the HSV olor system, we apply thes alar TV �lter (9) to both the real s alar fun tion V 0, and the disk transformZ0 of H0 and S0, whi h is a omplex s alar fun tion. Sin e both the unitinterval and unit disk are onvex domains, the maximum prin iple of the TV�lter [5℄ guarantees that the restored V takes values in [0; 1℄, and Z on the unitdisk. Thus, the restored H and S an be well re onstru ted from Z.4 Numeri al Experiments and ComparisonThis se tion summarizes the performan e of the TV models based on the CBand HSV olor models.4.1 TV restoration based on the CB modelIn Fig. 3, we demonstrate the result of TV restoration applied to the hromati ity-brightness representation. In the middle panel, we have plotted the image with hromati ity restored only, and at the bottom, the image with both hromati -ity and brightness restored. The result is quite su essful. The visible noisyred and green dots have been swept out. The eyes and dark lines resume theiroriginal bla k olor, and the nose and lips be ome smoothly red as they shouldbe.4.2 Comparison of TV's on the CB and linear modelsIn Fig. 4, we ompare the restortation results by the CB based TV and linearTV's. The two olumns on the right side show the details of the �rst olumn.Compared to the hannel-by- hannel TV and ve torial TV, the hromati ity-brightness TV restoration seems to give better olor ontrol. This example onvin es us the advantage of working with the hromati ity and brightness10

omponents, and also the fa t that the CB representation is loser to human olor per eption.Fig. 5 shows another example for the omparison between the linear TV'sand the CB based TV. The olumn on the right zooms into that on the left. Wesee that the author's stamp has been best restored by the CB based TV. Thelinear TV's have somehow blurred the stamp, mostly due to their ineÆ ien yin dealing with the hromati ity omponent.4.3 TV restoration dire tly based on the HSV modelIn Fig. 6, we show the restoration result from the dire t appli ation of theTV models on the three omponents of the HSV representation. The zoom-in image learly shows the unsatisfa tory behavior of su h an approa h of theapproa h. The underlying reason, we believe, is the high orrelation betweenthe hue omponent H and the saturation S for human per eption.4.4 TV restoration on HSV and its disk transformThe pre eding se tions and the previous numeri al example lead us to onsid-ering the ombination Z of H and S via the disk transform (8). In Fig. 7, wedisplay the restoration result from the s alar TV on both the real fun tion Vand the omplex fun tion Z. From the zoom-in image, it is lear that su h ombination is mu h loser to human per eption and thus yields better olorrestoration.4.5 TV on CB and HSV with the disk transformThe disk transform Z in the HSV representation en odes both the hue andsaturation, and thus is similar to the hromati ity information in the CB repre-sentation. Meanwhile, the \value" omponent V apparently plays the same roleas the brightness omponent in the CB model. As a result, to no one's surprise,the performan e of these two approa hes should be very lose, as learly seenfrom Fig. 8.A knowledgmentsThe authors are grateful for the numerous onstru tive suggestions from ourreferees. In addition, Jianhong (Ja kie) Shen also wishes to thank ProfessorStanley Osher for his onstant en ouragement and support.Referen es[1℄ P. V. Blomgren. Total variation methods for restoration of ve tor valuedimages, (Ph.D. thesis). Also Te hni al report, UCLA Dept. of Math., CAM98-30, June 1998. 11

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Figure 3: Spheri al TV on the hromati ity omponent and s alar TV on thebrightness omponent. In (b), only the hromati ity is restored, while in ( ),both the hromati ity and brightness omponents are restored.[2℄ A. Chambolle and P. L. Lions. Image re overy via Total Variational mini-mization and related problems. Numer. Math., 76:167{188, 1997.[3℄ T. F. Chan and P. V. Blomgren. Color TV: Total variation methods forrestoration of ve tor valued images. Te hni al report, UCLA Dept. ofMath., CAM 96-5, 1996.[4℄ T. F. Chan and P. V. Blomgren. Modular solvers for onstrained imagerestoration problems. Te hni al report, UCLA Dept. of Math., CAM 97-52,1997.[5℄ T. F. Chan, S. Osher, and J. Shen. The digital TV �lter and nonlineardenoising. IEEE Trans. Image Pro essing, 10(2):231{241, 2001.[6℄ T. F. Chan and J. Shen. Variational restoration of non- at image fea-tures: Models and algorithms. SIAM Journal of Applied Mathemati s,61(4):1338{1361, 2000. 12

[7℄ R. Gonzalez and R. Wood. Digital Image Pro essing. Addison-Wesley,1992.[8℄ R. Kimmel and N. So hen. Orientation di�usion or how to omb a por u-pine ? J. Visual Comm. Image Representation, to appear, 2000.[9℄ J.-M. Morel and S. Solimini. Variational Methods in Image Segmentation,volume 14 of Progress in Nonlinear Di�erential Equations and Their Ap-pli ations. Birkh�auser, Boston, 1995.[10℄ P. Perona. Orientation di�usion. IEEE Trans. Image Pro ess., 7(3):457{467, 1998.[11℄ L. Rudin and S. Osher. Total variation based image restoration with freelo al onstraints. Pro . 1st IEEE ICIP, 1:31{35, 1994.[12℄ L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noiseremoval algorithm. Physi a D, 60:259{268, 1992.[13℄ G. Sapiro and D. Ringa h. Anisotropi di�usion of multivalued images withappli ations to olor �ltering. IEEE Trans. Image Pro essing, 5:1582{1586,1996.[14℄ N. So hen, R. Kimmel, and R. Malladi. A geometri al framework for lowlevel vision. IEEE Trans. Image Pro ess., 7(3):310{318, 1998.[15℄ B. Tang, G. Sapiro, and V. Caselles. Color image enhan ement via hro-mati ity di�usion. Te hni al report, ECE University of Minnesota, 1999.[16℄ B. Tang, G. Sapiro, and V. Caselles. Di�usion of general data on non- atmanifolds via harmoni maps theory: The dire tion di�usion ase. Int.Journal Computer Vision, 36(2):149{161, 2000.[17℄ P. E. Trahanias, D. Karako, and A. N. Venetsanopoulos. Dire tional pro- essing of olor images: theory and experimental results. IEEE Trans.Image Pro ess., 5(6):868{880, 1996.[18℄ P. E. Trahanias and A. N. Venetsanopoulos. Ve tor dire tional �lters |a new lass of multi hannel image pro essing �lters. IEEE Trans. ImagePro essing, 2(4):528{534, 1993.[19℄ J. Wei kert. Anisotropi Di�usion in Image Pro essing. Teubner-Verlag,Stuttgart, Germany, 1998.13

(a) Original Image

(b) Noisy Image : Channel by Channel noise

(c) Channel by Channel TV

(d) Vectorial TV

(e) TV on Chromaticity and Brightness

Figure 4: Comparison of the CB based TV and linear TV's (I): CB leads tobetter olor ontrol. 14

(a) Original Image

(b) Noisy Image : Channel by Channel noise

(c) Channel by Channel TV

(d) Vectorial TV

(e) TV on Chromaticity and Brightness

Figure 5: Comparison between the CB based TV and linear TV's (II): CB doesnot mix olors. 15

Noisy Image TV on H, S and V A Zoom−in

Figure 6: Cir ular TV on H , and s alar TV's on S and V . The unsatisfa toryperforman e shows the inappropriateness of treating the H and S omponentsseparately.Noisy Image Scalar TV on both V & Z A Zoom−in

Figure 7: S alar TV's on both the real fun tion V and the omplex fun tion Z,i.e., the disk transform of H and S.Noisy Image TV on V & Z (under HSV) TV on C & B

Figure 8: TV's based on the CB and HSV (under the disk transform) olormodels have the similar performan e. 16