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the patch. A further improvement reported in [Geo04] is Following the example gravitation, electromagnetic and other ble for inertial effects, of Electrodynamics and Quantum remove the scratch from the shadow area in Figure 5 using a fourth order "bi-Poisson" equation, which matches both Mechanics, we Introduced by only source material from the illuminated area. will replace conventionalGrossmann with coderivatives and Weyl interactions. Einstein, mans do not perceive lumiFollowing variantexamplethey defineclosely related"minimal" interac- and Quantum and gradients at the boundary. Following the the example Electrodynamics and Quantumpixel values Following theexample of the so-called to the measure-Quantum Followingderivatives. 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Method described in with 2 Poisson Image Editing in the media as "redefining the way retouching is done in [Ado02], and computer vision.in electromagnetic and other ble for inertial effects, gravitation, the [Ado02], derivatives. They yare closely related tomeasure- section 4. andfield of They areare are closely relatedthe the measurethe first described y algorithm iscloselysolving the Pois- to measure-photography". An Internet search on Healing Brush reveals examples of illusions.) The variant derivatives. IntroducedThey basedGrossmann to the the measurevariant paper derivatives. by closely on related to variant derivatives. They Einstein, related and Weyl variant [PGB03]. The interactions. paper [PGB03]. andandTheoretical Physics solvingare responsi-popularity. TheTheoreticalapproachterm) takenthey responsiCovariant algorithm is based on adaptation the in our (source son equationin derivativesTheoretical describethey are arePois- its with rightTheoretical Physics interachandthe so-called "minimal" are and surrounded by a variable [EG96,Wey23], in define sidey components from the vector pixel values at the boundary of the patch, but the cloned ment process, the 2 aresystem x the following way. As suggested in responsiment 1 process, andthe in and Physics they they ment and and Physics responsiA in Here ment process,somein in of texture (see Figure 4). If of A process, visual theyarea theof Using covariant derivatives in the(source term) to image the tion. son equation with perceptually correct gradient is sense is based taken from pebbles are still easy to spot. There is too much variation, Fig right hand side above written new on ual system's adaptation, the [Geo05], a bleble for inertialof whichf gravitation, electromagnetic andand other contrast, Poisson cloning in the "healed" area min forble A(x,effects, gravitation, electromagnetic and other too high bleinertial y),effects, isgravitation, electromagnetic and of forgrayscale image is (x, y) and the describe the adaptation other inertial computer vision. to sample area image is effects, used function for inertial effects, gravitation, electromagnetic other2. Problems with or dynamic range, the field theinteractions. IntroducedsolvingEinstein,equation with andand Weylimage. This problem is inherent invariations of of image the followingcloning recipe:texture Grossmann and andthe Ourthe paper describeswhich transfers tothe nature g in Poisson simple isofbyEach Einstein,replaced 4). Weyl of Poisson equation (1), an improvement both Poisson derivative g(x,Introduced by by Einstein, Grossmann y), some area Einstein, (see Figure the Poisson is GrossmannIf Weylthe current ess (perceived brightness) in interactions. aCovariant It + function"our by additional freedom which interactions. Introduced in expression: interactions. derivatives Grossmann Weyl the visual system. Introduced approach describe adaptation Covariant Cloning and the Healing Brush. Poisson cloning between represents the "derivative Poisson Cloning without grayscale ofimage isthey in y) followingso-called "minimal" interac- ofmodifying lighting conditions can new abrightness ftheythe and the so-calledarea interac-is cloningaredifferent their amplitude even if be problem (x, define so-called "minimal" interac-areas modified to match the surroundings. the sample "minimal" interacimage FollowingThe central rectanglefrom values the they [EG96,Wey23],visual system defineso-calledbased oninadaptation without this improvement. This often is the case with face [EG96,Wey23], define the the way. "minimal" [EG96,Wey23], Figure 10: terminology has constant pixel [EG96,Wey23], they define gradients As image of pebbles and a scratch. values. redefines our perception of the 5: Original suggested Figure g(x, y), Poisson a perceptually correctg(x, y) is Poisson equation [Geo05], cloning fis solving the written based on derivatives inabove above is new is new to to remove wrinkles when unwrinkled skin is (x,derivatives in the sense sense new to y) = gradient the above sense isis to toretouching (1) avariant change in lightness. tion.tion. Using covariantrecipe: Each derivative (8), (9) and (10).new only available in areas of different lighting. tion. Usingfollowing simplederivatives in is replaced with tion. Using covariant covariant later, and Usingthespecifiedderivatives in the the above sense The will be covariant in equations (x, y) + A1constraining the new(3) Figure 2: Detail from Vision 1. withofof computer expression: Adaption of Human Figure Poisson Equation aof Dirichlet Figure vision. "derivative + boundary 10x due to covariant derivative function" condition thethe the fieldcomputer original image at the boundary. fieldfield y) to in vision. the off (x, computer x field computer vision. gradient visiblematch the vision. is The simultaneous contrast illusion, Figure values. To provide a clean example of the problem, let's try to Figure 10: The central rectangle has constant pixel10, is an adapted remove the scratch from the shadow area in Figure 5 using example which shows that humans do not perceive lumi- to the surroundings. Covariant derivatives ininour approach describe adaptation Covariant derivatives in approach (x, approach describe adaptation (1) umans through a [Gaz00, Sec00]picture. generalCovariant derivativesfinourourg(x, y) describe adaptationonly source material from the illuminated area. Figure directly. (See given same for a Film grain, Covariant derivativesy) = our approach describe adaptation 2 shows detail in the visual nance survey (4) Everywhere in this paper + +1A2 (x, y) (3) A (x, y) noise and a scratch are visible. The goal isof illusions.)ofof the visual systemthe y following way.suggested in in in to remove thethe visual system the in the followingAs As As suggested Fig x following way. As suggested of theThe visual systemxin following way. way. suggested in of visual system in iny the the on lightnessstate of adaptation perception and examples ation.in a seamless way. In Figure 3 (left) we see the result The sam scratch Figure 1. contrast band surrounded by with Dirichlet boundary condition constraining the new from simultaneous The is an figure shows an uniform gray illusion, Figure 10,a variable sect [Geo05],wellHere perceptually correct gradient is is writtenwith onon [Geo05],perceptually arethe xLaplace equation written0based [Geo05], known that the 2 and y2components writtenfbased on on [Geo05], a 1perceptually correct gradient =based a a a A and A2 correct gradient is of the vector perceptually correct gradient is written based It is judgementThe method does a good job atperceive lumiof inpainting. of brightness and interpolatingthe example which Due to that humans system's adaptation, y) to match the original used describe the adaptation of background. shows our visual do not function A(x, y), which is image at the boundary. to Orginal image of pebbles Poisson cloning fsurvey (x, Figure 9: Areas used for and a scratch. in Figure 7 and colors inappears (See [Gaz00, Sec00](perceived brightness) the following simple= Each Figurederivativereplaced withwith area used for Poisson cloing and covariant the inpainted area, but suffers aesthetically. It the infollowing simple recipe: Each derivative is replaced with 2 + the Scratch removed way recipe: Each derivative (2) to recon- Source . nance 4directly. to thisin lightness Submission ID 1033 /lacks following simple recipe: +is2derivative is by simple inpainting. withcovariant reconstruction, Figure 8. for a generalDirichlet boundary conditionsEachy) simplest isis replaced the EGthe Healing simple recipe: x Ay(x, 6: following visual system. It represents the2 additional freedom which replaced band reflect vary adaptation, do not and feel of real texture. It is too smooth. Adding the (4) Photoshop pixe theopposition to its surroundings. Following terminologyThe look reconstruction. y y on lightness perception and examples of illusions.) from peb perception expression: a structaEquationsredefinesfunction" xexpression:anresulttheto theLet's itwrite "derivative + A +andfunction" and area6 shows the ofon of inpainting. Again, is too a a "derivative +our function"wey components image. "derivativefunction" the of expression: adaptation "derivative + defective gradientstobased expression:in transfer vector adaptation of human solution, often viAlso, our general 4. Main inpaint) specified later, Figure hatPhysics,acceptablegray band surrounded byin lightness.(orAlso, g(x, y) a the texture in equations (8), (9) and (10). The are picture.this contravariant approach inpainting noise is thewe will callvision.to that used with is a variable simplest is want too figure shows[Geo05]. an uniform Film grain,change he same to Here will be A2 are and 1 close of t techniques. Due to our visual system's adaptation, the the example of Electrodynamics and Quantumassumed translated to the Following inpainted region. Texture is smooth. 3. The covariant approach derivatives invisiblederivatives 10 is due to function A(x, y), paper ept has has is pixelcovariant ngleofconstant toor pixel values. theEverywhere conventional which iswith co-Derivative covariant derivative angle has constant values. ebackground.constant remove the ctanglegoal constant pixel values. Mechanics, we will replace explicitly: used to describe the adaptation of has modified pixel values. gradient this in Figure the e.band appears to vary in lightness (perceived brightness) inderivatives. They are closelyarea.It Covariant theFigure 7, we see the resultwhich cloning from In order to solve this problem we borrow from the Retinex with The reconstruction variant related represents the visual system. to the measure- In additional freedom of Poisson adapted to the surroundings. [Lan77, Hor74] and the von Kries [vK02] theories of the illuminated area into the shadow area. It correctly matches e opposition totoolshows making the terminologyment process, and in Theoretical Physics they are responsiaFigure 3 (right)perceived byscratch removed bygivenfrom useful its surroundings. humans through a Poisson for the Following Lightness is we see the result visual valu Figure 5: Original image of pebbles and a scratch. igure 3 (left) redefines our perception of gradients based on adaptation ble for inertial and later, other forEurographicsPoisson equation and Weyl the (9) and (10). The systemwe a given state of adaptation. The state of adaptation effects, gravitation, electromagnetic in equations (8), Symposium on Rendering (2005) in adaptation of the viPhysics, submitted and will be specified conditions +(x, (x, (x, y) with The to our at interpolating of brightness and IntroducedItbyisEinstein,known that+ A1+ +y) 1y) f = 0 with Dirichlet boundary to AA A y) cloning.the will call this contravariant the Poisson cloning source and target areas for change in lightness. interactions. Grossmann (3)(3) (3) (3) interac-Laplace 1 1 (x, s ashown in Figure fundamental judgement good job 4. well the equation is critical [EG96,Wey23],make Poison cloning seamlessly to2 the boundary of they define the so-called "minimal" 102is due gradient visible in xx x covariant derivative are aboveFigure new tox the simplest way to reconx x conditions ismatch x x tion. Using sense is Dirichlet in the boundary f + color. If the equations we useIt lacks do not reflect this adaptation, covariant derivativesthe surroundings. f = 0, (5) t suffers aesthetically. adapted to (2) a = 2 + in 2 . image. Let's write the field of computer vision. (or inpaint) struct ntrast can not perceivedresults 10,isisacceptable to visualvist they illusion, Figureis 10, an a given that illusion, Figure 10,humansan is an astLightness is produce by 10, are an illusion, Figure that through illusion, Figure x x defective areay an on Rendering (2005) y x y submitted to Eurographics Symposium Covariant re.sual system. We findof adaptation. of modified or covariant derivatives in our approach explicitly: It is given state the concept The state of adaptation system in a too smooth. 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In general, the problem Here x1 performing the x x x components Laplace the solver Figure is generalized "covariantly" of sensors in the retina, viequations changeto three types with the adaptation of the L, M, 1033 5 The simultaneous contrast illusion, Figure 10, is an equation (5) we converted y) in theSubmission ID in with such Healing how we same role in the case Implementation and HereA(x, which isisintoas describeour improve- adaptationequationsand experimental results performance, ity),iswhich define A(x,tothe covariantpotentialthe is playingistheis usedused to to of Laplace equa- EGwithof of vector the adaptation such of Here isy), y),define A(x,is to case of our improve- / adaptation of is not only complexity and performance, how which used y) useddescribe the the 5. Photoshopequations is not only complexity and space.S,Localare adaptation, perceive thefunction A(x, visualsystem's responsiblePoissonperception. Adapta- A(x,A(x, y), which sual system.system's by offor thethe lumi- function tion: lsual example whichadaptation,do color the visualwhich effects system's shows adaptation, scratchsystem's that adaptation, removed humans adap and function of Poisson cloning. Following f[Geo05], describethe function describe (5) adaptationfact that, a 6propri, itresultnot inpainting. Again, it isiterative not Figure shows the of too but the fact that, a propri, it is not clear if a given iterative + = 0, ment of Poisson cloning. fFollowing [Geo05], we assume the but the is clear if a given ment + A (x, y) x x we assume nance directly. (See [Gaz00, Sec00] for a general surveyM, S) vector Electrodynamics. smooth. difficult to try implement a direct itera- 3. 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It correctly matchesface A1 )(x and + g(x, y). of+ vector + 2 (3), is A2 f A (adap-Poisonperforming1thefseamlesslyadaptation +the )that= 0,conditionsfor equation (11).to try implement a direct itera- [La After Laplace are the components In other words, Here A1 and A2i.e. adapted yto g(x, y). above substitution )( (4), is such that other ongeneralizedterminology sensors inused of Following terminology the Following visual our visual system'sbeen fromredefines y),i.e.ourcloningthe adaptationy) thegradientsbased on adaptationfor convergence. complexity and performance, s.isFollowingDue Kries type have from the [Geo05] redefinesis ourdescribe of intoofofcovariant our based adaptation be rather difficult gs.tationthethe to three system. adaptation, retina, redefinesHereperception A(x,yin thegradients based on conditions illuminated area into ings. background.von terminology from in maketo ouradaptedtoperceptionof gradients Laplace equa- adaptation is not only Following to types of from terminology match improve-gon It wouldof L,redefines our towe M, A(x, which is x of perception adaptation function used x g(x, y) is how isperceptionthe + A caseof+ A ) f = 0,of g withsolverequations (11). 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It represents the additional freedom whichthe covariant derivatives of (6) appears toresponsible for color perception. the 2 )( 2 and S, which changedescription of brightness) system. visual ment of x + A1 )( x +constant, y [Geo05], we assume the are vary in lightness. derive achange in in lightness. but The The submitted variant contravariant surroundings.in in lightness. and willour perceptionbegradientscloning. in equationsyof our(9)(8), andand the1033 / EG Photoshop Healingin iscomplexity andgiventhe followntravariantto change Following terminology from S) vectorand are beisspecified on Followingequations (8),andSubmissionTheapproachawe takeSymposium on Rendering (2005)based performance, ravariant its change lightness. M, and willwillspecified later, inin in equations (9) (9) (10). Thethat, to Eurographics only not clear if a on iterative tion: Poisson later, adaptationequations (8), (9) (10). 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Local effects of gradientin visible due in in Figure patch.due to is such that adap-gradient 10 g(x, Figure words, adaptation gradient visible Figure in insubmitted to is 10 to further reported derivative [Geo04] is gradient i.e. adapted isto Figure other Eurographics gradient Figure the 10 covariant conditionsstart from derivativefrom y). In derivative Let's scheme for a describes effectsKries type have been used Figure 10. In the the visual of adaptation illustrated in in [Geo05] to to surroundings. adapted a fourth order "bi-Poisson" texture, both system is Covariant Laplace Equation of equation, which matchesLet's startgiven equation will converge, and what are the completely adapted to the area llustrated von Figure humans In the visual tation ofLightness is perceived by 10. through a given the in g(x, y)the A )( surroundings. )( and + A ) f at the boundary. isthe surroundings. covariantgradients = 0, of (6) covariantly A ) f + ( + values derivatives g 1 other adapted to the surroundings.(x, y))g(x,2y) = 0 2 is such that adapted toto to the (y).constant, words, adaptation adaptedadapted surroundings.pixel A adapted + to g(x, + In A the conditions for convergence. our case is based on the followThe approach we take in ff usual equations we simply replace state of derivative with a co- i.e. ( each adaptation 1 in a given state of adaptation. derive system derivative with of the visual system. a mathematical descriptionThe co+A x = (8) + f = 0 y))g(x, y 0 (12) = 0, are that the Laplace x (xdivA11 (x,y ace humans our fundamentalcovariantofderivatives are specified known zero. covariantlyf constant,with· grady)+ A derivatives of(8)(7) ing unique property of equation=(11): eachthroughgiven visual visual avisual and 0, (12) g ybyvariantcriticalthrough a given visual mans through aThesegiven brightness humansderivative. a judgement humans to through a given It is well g(x, y) is equation is f + x + 2A covariant · A f = 0. f the g g The approach we take in our case is based on the followDirichlet boundary conditions is the simplest way to recon-be written as color. If the equations we use do not reflect this adaptation, which+ f divA + 2A · grad f + A ·has been = 0. after differentiation can This simple approach A f very successful, described In The covariant adaptation so derivatives gradientare adaptation iant this canstate ofare simpleequal ptation.paper we provide athatof acceptabletotothe perceived gra-inpaint) azero. area in an image. Let's write the media as "redefining the way retouchingingLet's1033 /property ofHealing (11): adaptation. The statespecified that vi-recipe that are defective tion.that theThe state of adaptation aptation. not produce results adaptation The state of is mathematical struct (or f in is done(7) EG Photoshop equation they Submission unique ID instart from 3 and perform the differentiations. The result is describes effectsWe find theof Figure modified or covariant 10. In the in explicitly: Here the vector function A(x, y) An Internet search on(x, y)) Brush revealsperform the differentiations. The result is dient. Insystem.example conceptillustrated in Figure values photography". = (A1 (x, y), A2 Healing sual the of adaptation of 10, constant pixel the derivatives and known adaptation the the Laplace equation fLet's that that thethe y) system. It equation the y))g(x, Laplace is f to Laplace 0 with udgement ofwe simplytoreplace eachandand describeItIt Itwell known thatthat itsvisual =equation related =in0[Geo04]0is startwith lsusual bandto usednonzerobrightnessderivative with a co- wellis well knownA1ofLaplaceA further improvement(8) f = = with alequal have in Physics covariantand and the It istheisis well known+ (x,the popularity. 0 equationreported f=with0 from ntal equations ofperceived derivative and judgement brightness tool for making judgement ofof be a useful grajudgement brightness brightness patch. f derivative the describes ( the (12) = 0, a fourth y) = "bi-Poisson" equation, which matches both order 0 equations change "covariantly" with the adaptation of the vix + Ain(x,·y))g(x, y) A · Editing fconditionsy))g(x, + y)0 the(A [PGB03], Arecon- recon+ ((conditions thetheAsimplest (9)way (x, reconfy +conditions and gradients0. theway towayrecondivA +22(x, pixel values simplest (x,(9) (7) variantsual system. reflectcovariant derivatives areDirichlet boundary derivative. These adaptation, vector function gradisImage f simplest the f + f = field" A 2A (5) A(x, is Poisson perceived gradient. thisthis adaptation, specifiedDirichlet"guidance0,yconditionsf is = == simplest to to Here the boundary + A (x, y))g(x, y) = 0 y), 2 to fy)) grad f 7:gradg bygg bycloning from the(gradg) e use not not this adaptation, 10, do reflectpixel values in constant this adaptation, Dirichlet boundary Dirichlet boundary is the at 1 boundary. way o do reflect senot not reflect do f Scratch removed removed 0, Poisson cloning from the illu-(12) (8) Figure gradg Poisson 1 x x y y ( = (gradg) f - 2 grad f · Scratch - g + 2 (gradg) ·· (gradg) = 0. so that the covariant gradient is equal to the perceived grax illuminated area. - = 0. In the von acceptableadaptationvi- vi- describes Solving inpaint) defective area Poisson cloning is related - nt thatisacceptabletypes [vK02]10, thatretina,pixelstructin(or(or Here for A(x,ay)defectiveProblems=Symposiumof theimage. Let's performf the·area.g g g + The resultg22 derivativeKriesthreeofdescribein the vi- M, valuesstruct inpaint) a vectorproduces theThis simplewith inimage.image. write 2write differentiations. 2 and to ofthatto the to grayscale struct (orabove substitutiony)aproduces to2.A(x, y) area(x,hastheIt y)) Let'sffwriteminated g structadaptation offunctionvisual system.beenvector inpaint)defective area in an1 any),RenderingLet's describedwrite inpaint) athe the specific form in image. defective in ofon an(x, (2005) Let's f area (A and to is atdient. are are to approach toto viults that acceptable sensors constant L, s areIngeneralized that areexample Figure thatthat acceptable After performing the(or for A(x, (3), (4), the Laplace specific approach an A vector g Solving the the g submitted Eurographics form 2 very successful, Poisson Our equation (5) is converted into the we are going to use:current paper describes an improvement to both done in covariant Laplace equa- the media as "redefining the way retouching is (13) function that we are going use: system. It (13) function field" in of the in and have nonzero covariant derivative and thethe the derivatives explicitly: S, are responsible for color perception. Adapta- the thederivatives explicitly: to visual and the Healing Editing [PGB03], describesthat ( + A Poisson = 0 adaptation (x, cloning related to the band ofwhich modified covariant describe the derivatives explicitly:y))g(x, y)Image isBrush. Poisson cloning between "guidance derivatives explicitly: 2 photography". An Internet search on Healing Brushand perform the differentiations. The result is ncept tion ismodified or orof covariant tion: oncept described by or covariant M, S) vector pt of of of concept modified or covariant modified a multiplication the (L, (9) reveals areas Image Editing conditions can y the "guidance field" in Poisson of different lighting [PGB03], be a problemsee that (13) is same as (11). Using the fact that perceived gradient. in LMS space. Local effects of adapits popularity. by a matrix diagonal g (gradg) · (gradg) We see that (13) is same as (11). Using the fact that f face withouty) = 0 this improvement. This often is the case withWe grad f gradg o to a atation usefulKriesmaking thein [Geo05] to beuseful tool tool type making the the a bebe usefulvon tool have been used the useful tool for for making a of the for for making (9) ( + A2 (x, y))g(x, = from · A(x, y) produces the specific form of the vector unwrinkled skin- 2 (11) is equivalent to (6) - g "guiding field" extracted from retouching to remove wrinkles when is (11) is equivalent to (6) with "guiding field" extracted 0. with + 2 2 + A1 ) f for + ( + A1 )( Solving+ ( + A2 )( y A2 ) f = 0, (6) gradg f f g g derive a mathematical description of the visual system. gradg in areas on Renderinglighting. x x y y A(x, to= only submitted Eurographics Eurographics Symposium of different (10) (2005) sampling farea Scratch removed by (8), cloning from the to our submittedy)touse: available Symposium on Rendering (2005) Figure gradg defined+ 2 (gradg) ·we come =(13) thef sampling area and - g by Poisson (9), (gradg)illu- 0.our - withwithadaptation of thethe the viy" with the the adaptation vi- vily" the the adaptation ofof vintly" with adaptation of the the - 2 grad · 7: and defined by (8), (9), we come to A(x, y) - g function that we are going to =2. Problems with Poisson cloning (10) minated area. g which after differentiationfor be written as Solving can A(x, y) specific of the vector produces form the In this paper we provide a simple mathematical recipe that f f g g g2 covariant reconstruction algorithm as follows: Our current paperclean exampleimprovement to bothcovariant reconstruction algorithm as follows: describes effects of adaptation illustrated in Figure 10. In the Poisson that function that we equation+ + fTo provide a 0,=form of the problem, let's try to (5) (5) is same as (11). Using the fact (13) are going(7),fwe obtain = f f0, an 0, the to + + f the = = of f (7),use: obtainscratchfinal the shadow the in Figurebetween (5) (13) f we the the Healing form of area cloning We see that f describes0, Poisson (5) Substituting in equation Substituting in usual equations we simply replace each derivative with a cocloning and the final Brush. remove from 5 using x A A f x x(7)gradgy lighting xx areassourcey x only ydifferent f + covariant· Laplace ·equation:y Reconstruction the illuminated area. (11) is equivalent to (6) with "guiding field" extracted from f divA + 2A grad fCovariant Image y y y from conditions can be a problem + x = 0. variant derivative. These covariant derivatives are specified of material covariant Laplace equation: We see that the image by the (11). Using the fact that (13) is same (1) Dividearea and definedas sampling we come to our so that the covariant gradient is equal to the perceived grathe sampling by (8), (9), (texture) image. (10) A(x, y) = - without this improvement. This often is the case with(1) Divide the image by the sampling (texture) image. face K02] adaptation toFigureto to grayscale hch [vK02] adaptationgrayscale in [vK02] adaptation grayscale values [vK02]Inadaptation 10, grayscale to constant pixel equivalent (6) Figure covariant Here the vector function A(x, y) = (A1 (x, y), A2 (x, y)) g dient. the example of retouching to remove wrinkles (3),unwrinkled skin produces the8:intermediate image field" extracted from when (4),(11) isLaplaceto Scratch removed image I(x, y). is reconstruction with "guiding I(x, cloning from the This produces the intermediate byas follows: This y). After performingvisual system.A(x, y)aboveavailable in areas of (3), (10) thecovariant same illuminatedalgorithmFigure 7. Method describedto our Afteradaptation of the thethe therelated- gradg After performingabove substitution (3), (4),lighting. Laplace area and defined by (8), (9), we come in After performing above substitution different(4), the Laplace performing the is = only substitution the the sampling above substitution (3), (4), the Laplace area as in describes It to the band have nonzero covariant derivative and describe the pes sensorsgradient.the the retina,M, M, the "guidanceSubstituting inImage gradg(7), g obtain the final form of the s ofof sensors in retina, M, L, of sensors in retina, L, L, M, sensors in the the retina, L, we field" fin Poisson equation [PGB03], g (gradg) · (gradg) Editing perceived in is converted into into (gradg) · (gradg) Laplace equa-section 4. gradconverted thethe the covariant Laplace reconstruction algorithm as follows: ff converted into covariant Laplace equaequation (5) f(5)Laplace ·equation: into 2 covariant Laplace covariant equaequation (5)-(5)converted g + 2 the covariant0. equation - 2 grad · gradg - + equation isis is equa= - g To provide a clean example of= 0.problem, let's try to covariant 2 f le color perception. Adaptaible for color perception. Adaptarfor color perception. Adaptafor color perception. Adaptaff to Eurographics Symposiumg Renderingwe obtain the final form of the gon (7), (2005) g22 sampling (texture) image. submitted g Substituting inf equation the g the shadow area in Figure (1) Divide the image by the ff(x, y) (x, y) tion:tion: covariant Laplace equation: remove the scratch from tion: tion: 5 using I(x, y) = of the patch, (14) I(x, boundary (11) pixel values at the y) = image I(x, y). but the cloned (14) (11) area. This produces the intermediate g(x, y) only source material from the illuminated ation of thethe the M, S) vector lication of of (L, (L,vector vector ltiplication (L, M, S) M, S) plication of the (L, M, S) vector (1) Dividepebbles image easy to spot. There is too(texture) image. the are still by the g(x, y) sampling much variation, too highremoved by covariant cloningin the the fromarea Scratch8:contrast, or dynamic range, y). We see that the covariant Laplace equation is more Figure intermediate image I(x,from "healed" the Scratch This produces the image. Thisremoved by covariant cloningnature of We see that gradg g Laplace equation is more f grad f the covariant (gradg) · (gradg) MS space. Local effects ofof adapS space. Local effects adapLMS space. Local effects adappace. Local effects of of adapthe problem (2) Solveoftheilluminated equation Figure. Methodthe the Laplace equation = 0. -2 · actually very different, from the Laplace - very different, from the Laplace +2 same Laplace as in above is inherent (2) Solveilluminated area area as in Figure 7. in described in complicated, and actually g complicated, and g the Poisson equation (1),f which transfers variations of g f f g2 (x, y) section 4. f f gradg g describing (gradg) (gradg) I(x, y) = ypebeenbeen used in in [Geo05] to vehave used in [Geo05] to toto e have been used [Geo05] have been used in [Geo05] equation.gradincorporates terms interaction(11) It incorporates terms describing ·interaction with without modifying their amplitude even if new brightness(14) equation. · It with 0. f - f = 2 - +2 g(x, y) 2 values are modified to match the surroundings. 5: A A)()( A1 AA) 1 )g ( ( a )(+2Aequation(11)) ) A2g)( 2 )(+ Af+ A= )(+1A+field g. In+)way,(this is+Poisson equationpebbles2f fa) f0,0, f (6) ( ( + (++external)(++fgIn1A(++ +FigurePoisson)(image of+withA2== = 0, (6) ( AtheAexternal field ) + aafway,fthis isAa2+Original+ A2 ) withand0,scratch. (6) (6) I(x, y) = f (x, y) the g. 1 1We1see 1 2 (14) ption thethe the visual system. iption ofvisual system. cription of visual system. n of of the visual system. that x covariant "right y equationyy the on the y handy y However, Laplace is more 0, pixel values at I(x, y) =of the patch, but the cloned (15) the boundary y) x x x x x term y y xa modified x I(x, y) = 0, (15) g side". g(x,

Covariant derivatives in our approach describe adaptation of the visual system in the following way. been very successful, describedin As suggested This simple approach has in the media as "redefining the way retouching is done in [Geo05], a perceptually correct gradient issearch on Healingbased on written Brush reveals photography". An Internet its popularity. the following simple recipe: Each derivative is replaced with Further development of the mathematical expression: the Adobe® Photoshop® Healing Brush tools behind a "derivative Photoshop Healing 4 Submission ID 1033 / EG+ function" Figure 7: Scratch removed by Poisson cloning from the illu2. Problems with Poisson cloning minated area. gle adaptation of humanpixel values. general approach is has constant vision. Also, our Our current paper Todor Georgiev describes an improvement to both Poisson 4. Main Equations cloning and the Healing Brush. Poisson cloning between anclose to [Geo05]. Marco, Venice. Following the example of areas of different lighting conditions can be a problem Electrodynamics and Quantum Submission ID ID ID /1033EG Photoshop Healing replace conventional derivativesoften isco- case with face Submission 1033 EGEG 1033 Photoshop Healing Submission ID1033 / /Photoshop Photoshop Healing without this improvement. This with the SubmissionSubmission ID/ Photoshop Healing 1033 EG / EG Healing 4 Mechanics, in Figure retouching to remove Poisson cloning. is Figure 4: Areaswe will 2 used for wrinkles when unwrinkled skin(3) y) are+ availablerelated of different lighting. A1 (x, areas to the measurevariant derivatives. They only closely in Figure 1: Basilica San Marco, Venice.approach is x in Theoretical Physics they are responsix adaptation of human vision. Also, our general 4. Main Equations Submission ID 1033 / EG P ment process, Figure 4: Areas on. our[Geo05].general approach is is4. 4.4.Main Equations in Figure 2 used for Poisson cloning. problem, let's try to so, Also, our general approach .close to our Figure 10, is is is Also, general approach an Also, our general approach Main Equations and 4. Main Equations Main Equations To provide a clean example of the illusion,

a fourth order "bi-Poisson" equation, which matches both pixel values and gradients at the boundary.

Covariant Image Reconstruction

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