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Contents : the patch. A further improvement reported in Geo04 is Following the example of Electrodynamics and Quantum ble for inertial effects gravitation electromagnetic and other remove the scratch from the shadow area in Figure 5 using a fourth order "bi-Poisson" equation which matches both Mechanics we will replace conventional derivatives with cointeractions. Introduced by Einstein Grossmann and Weyl only source material from the illuminated area. pixel values and gradients at the boundary. mans do not perceive lumiFollowing Following Following Following the the example the the example example example of of Electrodynamics of Electrodynamics of Electrodynamics Electrodynamics and and and Quantum and Quantum Quantum Quantum variant derivatives . They are closely related to the measure EG96 Wey23 they define the so-called "minimal" interacFigure 8: Scratch removed by covariant cloning from the This technology was first implemented in Photoshop 7.0 This technology was first implemented in Photoshop 7.0 Sec00 for a general survey ment process and in Theoretical Physics they are responsition. Using covariant derivatives in the above sense is new to Mechanics Mechanics Mechanics Mechanics we we will we will we replace will will replace replace replace conventional conventional conventional conventional derivatives derivatives derivatives derivatives with with with cowith cococosame illuminated area as in been Figure 7. successful Method described in This simple approach has very described + A ( x y ) (4) 2 Ado02 and first described in the Poisson Image Editing section 4. in the media as "redefining the way retouching is done in ble for inertial effects gravitation electromagnetic and other Ado02 and first described in the Poisson Image Editing the field of computer vision. y y examples of illusions.) The variant variant variant variant derivatives derivatives derivatives derivatives . They . They . They . are They are closely are are closely closely closely related related related related to the to to the measureto the the measuremeasuremeasurephotography". An Internet search on Healing Brush reveals paper PGB03 . The algorithm is based on solving the Poisinteractions. Introduced by Einstein Grossmann and Weyl its popularity. paper PGB03 . The algorithm is based on solving the PoisCovariant derivatives in our approach describe adaptation son equation with right hand side (source term) taken from and surrounded by a variable EG96 Wey23 they define the so-called "minimal" interacment ment ment process ment process process process and and in and and Theoretical in in Theoretical in Theoretical Theoretical Physics Physics Physics Physics they they they are they are responsiare are responsiresponsiresponsiand A are the x and y components of the vector Here A1 pixel values at the boundary of the patch but the cloned 2 of the visual system inof the following way. As 4). suggested in the image in some area texture (see Figure If the pebbles are still easy to spot. There is too much variation tion. Using covariant derivatives in the(source above sense is new to son equation with right hand side term) taken from Fig ual system's adaptation the Geo05 a perceptually correct gradient is written based on ble ble for ble for ble inertial for inertial for inertial inertial effects effects effects effects gravitation gravitation gravitation gravitation electromagnetic electromagnetic electromagnetic electromagnetic and and other and and other other other too high contrast or dynamic range in the "healed" area function A (x y)of image which used to describe adaptation of 2. Problems with Poisson cloning grayscale is f is ( x y ) and the sample area the image is the field computer vision. min of the image. This problem is inherent in the nature of the image in some area of texture (see Figure 4). If the the following simple recipe: Each derivative is replaced with g ( x y ) Poisson cloning is solving the Poisson equation ess (perceived brightness) in Our current paper describes an improvement to both Poisson the Poisson equation (1) which transfers variations of g interactions. interactions. interactions. interactions. Introduced Introduced Introduced Introduced by by Einstein by Einstein by Einstein Einstein Grossmann Grossmann Grossmann Grossmann and and and Weyl and Weyl Weyl Weyl the visual system. It represents the additional freedom which Covariant derivatives in our approach describe adaptation a "derivative + function" expression: Poisson Cloning Covariant Cloning cloning the Healing Brush. Poisson cloning between without and modifying their amplitude even if new brightness grayscale image is f ( x y ) and the sample area image is Following terminology from areas can be a problem values of aredifferent modified lighting to matchconditions the surroundings. of the system in the following way. As suggested inand EG96 Wey23 EG96 Wey23 EG96 Wey23 EG96 Wey23 visual they they they define they define define define the the so-called the the so-called so-called so-called "minimal" "minimal" "minimal" "minimal" interacinteracinteracinteracFigure 10: The central rectangle has constant pixel values. redefines our perception of gradients based on adaptation Figure 5: Original image of pebbles a scratch. without this improvement. This often is the case with face g ( x y ) Poisson cloning is solving the Poisson equation Geo05 a perceptually correct gradient is written based on f ( x y ) g ( x y ) (1) retouching to remove wrinkles when unwrinkled skin is avariant change in lightness. tion. tion. tion. Using tion. Using Using Using covariant covariant covariant covariant derivatives derivatives derivatives derivatives in the in in the above in the the above above above sense sense sense is sense new is is new is new to new to to to and will be specified later in equations (8) (9) and (10). The derivative is replaced with the following simple recipe: Each only available in areas of different lighting. ( x y ) (3) + A 1constraining the new Figure 2: Detail from Vision Figure 1. with Dirichlet boundary condition Adaption of Human Poisson Equation x expression: x a "derivative +Figure function" the the field the the field field of field computer of of computer of computer computer vision. vision. vision. vision. gradient visible in 10 is due to covariant derivative f ( x y ) to match the original image at the boundary. The simultaneous contrast illusion Figure 10 is an To provide a clean example of the problem let's try to Figure 10: The central rectangle has constant pixel values. 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 Covariant Covariant Covariant derivatives derivatives derivatives derivatives in our in in our in approach our our approach approach approach describe describe describe describe adaptation adaptation adaptation adaptation only source material from the illuminated area. f ( x y ) g ( x y ) (1) umans through a given visual Figure 2 shows detail in the same picture. Film grain + A (x y) nance directly. (See Gaz00 Sec00 for a general survey (4) Everywhere in this paper ( x y ) (3) + A 2 1 noise and a scratch are visible. The goal is to remove the Fig y y x x of the of of the of visual the the visual visual visual system system system system in the in in the in following the the following following following way. way. way. As way. As suggested As As suggested suggested suggested in in in in on lightness perception and examples of illusions.) The ation. The state of adaptation sam scratch in a seamless way. In Figure 3 (left) we see the result from with boundary condition constraining the new The Figure simultaneous contrast illusion Figure by 10 is an Dirichlet figure shows an 1. uniform gray band surrounded a variable and A are the x and y components of the vector Here A sect 1 2 Geo05 Geo05 Geo05 Geo05 a perceptually a a perceptually perceptually a perceptually correct correct correct correct gradient gradient gradient gradient is written is is written is written written based based based on based on on on It is well known that the Laplace equation f 0 with judgement of brightness and of inpainting. The method does a good job at interpolating 2 2 example which Due shows do not perceive lumibackground. to that our humans visual system's adaptation the functionthe A(x original y) which is to describe the adaptation of used Orginal image ofused pebbles and a scratch. f ( x y ) to match image at the boundary. Figure 9: Areas for Poisson cloning in Figure 7 and colors in the inpainted area but suffers aesthetically. It lacks + . (2) nance directly. (See Gaz00 Sec00 for a general survey the following the the following following simple simple simple simple recipe: recipe: recipe: Each Each Each derivative Each derivative derivative derivative is replaced is is replaced is replaced replaced with with with with Source area used for Poisson cloing and covariant Figure 6: simplest Scratch removed by simple inpainting. band appears to vary in lightness (perceived brightness) infollowing 2+ 2 Dirichlet boundary conditions is the way to recondo not reflect this adaptation the visual system. Itrecipe: represents the additional freedom which A ( x y ) (4) covariant reconstruction Figure 8. 4 Submission ID 1033 / the EG Photoshop Healing x y 2 pixe the look and feel of real texture. It is too smooth. Adding reconstruction. y y onopposition lightness to perception and examples of terminology illusions.) The its surroundings. Following from peb redefines our perception of gradients based on adaptation a "derivative a a "derivative "derivative a "derivative + function" + + function" function" + function" expression: expression: expression: expression: adaptation of human solution vision. Also our general approach is 4. Main Equations struct (or inpaint) a defective area in an image. Let's write hat are acceptable to that vinoise is the simplest often used with inpainting Also g ( x y ) is the texture we want to transfer to the too figure shows an uniform gray band surrounded by a variable Physics we will call this contravariant change in lightness. Figure 6 shows the result of inpainting. Again it is too he same picture. Film grain and A are the x and y components of the vector Here A and will be specified later in equations (8) (9) and (10). The 1 2 close to Geo05 . of t techniques. Following the example of Electrodynamics and Quantum inpainted region. Texture is assumed translated to the smooth. 3. The covariant approach background. Due topixel our visual system's adaptation the the derivatives explicitly: function A( x y) which iswith used to describe the adaptation of Everywhere in this paper ept of modified or covariant ngle angle e ctangle has has constant has has constant constant constant pixel pixel values. pixel values. values. values. gradient visible in Figure 10 is due to covariant derivative the Covariant Derivative Mechanics we will replace conventional derivatives coe.band The goal is to remove the reconstruction area. In order to solve this problem we borrow from the Retinex In Figure 7 we see the resultwhich of Poisson cloning from appears to vary in lightness (perceived brightness) in with variant derivatives . adapted They are closely related to the measurethe visual system. It represents the additional freedom to the surroundings. Lan77 Hor74 and the von Kries vK02 theories of the illuminated area into the shadow area. It correctly matches e a useful tool for making the Lightness is perceived by humans through a given visual valu ment process and in Theoretical Physics they are responsiopposition to its surroundings. Following terminology from Figure 5: Original image of pebbles and a scratch. igure 3 (left) we see the result redefines our perception of gradients based on adaptation Figure 3 (right) shows the scratch removed by Poisson ble for inertial effects gravitation electromagnetic and other system in a given state of adaptation. The state of adaptation Physics we will call this contravariant change in lightness. submitted to Eurographics Symposium on Rendering (2005 ) and will be specified later in equations (8) (9) and (10). The with the adaptation of the viDirichlet boundary conditions for the Poisson equation cloning. The source and target areas for the Poisson cloning interactions. Introduced by Einstein Grossmann and Weyl ( x ( y x ) ( x y ( ) x y ) y ) (3) (3) (3) (3) + A + + A + A A s a good job at interpolating 1 1 1 1 It is well known that the Laplace equation f 0 with is critical to our fundamental judgement of brightness and they 2 2 EG96 Wey23 define the so-called "minimal" interacgradient visible in Figure 10 is due to covariant derivative make Poison cloning seamlessly match the boundary of are shown in Figure 4. x x x x x x x x tion. Using covariant derivatives in the above sense is new to is the simplest way to reconDirichlet boundary conditions color. If the equations we use do not reflect this adaptation f + f 0 (5) t suffers aesthetically. It lacks adapted to the surroundings. + . (2) the field of computer vision. Lightness is perceived by humans through given struct (or inpaint) a defective area in an image. Let's write 2 2 can not produce results that are to visual that vintrast st illusion astthey illusion illusion illusion Figure Figure Figure Figure 10 10 10 is 10 is an isacceptable an is ana an x x y y x y submitted to Eurographics Symposium on Rendering (2005) Covariant derivatives our approach explicitly: describe adaptation re. It system. isa too smooth. Adding system in given state adaptation. The state of or adaptation the in derivatives sual We findof the concept of modified covariant K02 adaptation to grayscale of the visual system in the following way. As suggested in mans humans at humans humans do do not do not do perceive not not perceive perceive perceive lumilumilumilumiSubmission ID 1033 / EG Photoshop Healing 5 It is well known that the Laplace equation f 0 with is critical to our fundamental judgement of brightness and Submission ID 1033 / EG Photoshop Healing 5 derivative used in Physics to be a useful tool for making the After performing the above substitution (3) (4) the Laplace Geo05 a perceptually correct gradient is written based on color. often used with inpainting Also g ( x y ) is the texture we want to transfer to the Dirichlet boundary the following recipe: Each derivative conditions is replaced withis the simplest way to reconIffor the change equations we use do with not reflect this adaptation sensors in the retina L survey M adaptation equations the of the vi- simple ec00 z00 f 0 Sec00 Sec00 Sec00 for a for general for a"covariantly" ageneral a general general survey survey survey + A + + A + A A ( x ( y x ) ( x y ( ) x y ) y ) (4) (4) (4) (4) equation (5) is converted into the covariant Laplace equa2 2 2 2 5. Implementation and experimental results and it is playing the same role as the vector potential in a "derivative + function" expression: 5. Implementation and experimental results and it is playing the same role as the vector potential in inpainted region. Texture is assumed translated to the struct (or inpaint) a defective area in an image. Let's write they can not produce results that are acceptable to that visual system. f + f 0 (5) y y y y y y y Figure perception. 10: The central rectangle has constant pixel values. or color Adaptand xamples and examples examples examples of of illusions.) of illusions.) of illusions.) illusions.) The The The The Electrodynamics. Electrodynamics. x x y y the derivatives explicitly: sual system. We find the concept of modified or covariant tion : It would be rather difficult to try implement a direct iteraIt would be rather difficult to try implement a direct iterareconstruction area. Submission ID 1033 / EG Photoshop Healing 5 Fig Scratch removed by simple inpainting. In the von Kries approach vK02 adaptation to grayscale Figure 6: Scratch removed by simple inpainting. derivative used in Physics to be a useful tool for making the cation of the (L M by S )by vector y ) (3) and +A A1 (x nd y ray band band surrounded band surrounded surrounded surrounded by by a variable a a variable variable a variable cov After performing the above substitution (3) (4) the Laplace tive solver for equation (11). In general the problem we face and and and A A are are A the are are the x the and the x x and and x y and components y y components components y components of of the of the of vector the the vector vector vector Here Here Here A Here A A A tive solver for equation (11). In general the problem we face x x 1 1 1 1 2 2 2 2 is generalized to three types of sensors in the retina L M equations change "covariantly" with the adaptation Submission ID in 1033 / EG Photoshop Healing and 5 The simultaneous contrast illusion Figure 10 is an of the viequation (5) is converted into the covariant Laplace equawith such equations is not only complexity and performance Here is how we define A ( x y ) in the case of our improvewith such equations is not only complexity and performance 5. Implementation experimental results and it is playing the same role as vector potential Here is how we define A ( x y ) in the case of our improvespace. Local effects of adapvisual sual lsual visual system's system's system's system's adaptation adaptation adaptation adaptation the the the the scratch removed by Poisson and S which are responsible for color perception. Adaptafunction function function A ( x A A ( y x ) ( A x y which ( ) x y ) which y ) which which is used is is used is used to used describe to to describe to describe describe the the adaptation the the adaptation adaptation adaptation of of of of example which shows that humans do not perceive lumi- function system. Figure 6propri shows the result of inpainting. Again it isiterative too tion : but the fact that a it is not clear if a given f + f 0 (5) ment of Poisson cloning. Following Geo05 we assume the but the fact that a propri it is not clear if a given iterative of Electrodynamics. ment Poisson cloning. Following Geo05 we assume the nance directly. (See Gaz00 Sec00 for a general survey smooth. tion is described by a multiplication of the ( L M S ) vector 3. T x x y y + A2 (x y)the (4) 5. Implementation and experimental results and it is playing same role as the vector potential in It would be rather difficult to try implement a direct iteraave been used in Geo05 to ss htness ightness ness (perceived (perceived (perceived (perceived brightness) brightness) brightness) brightness) in in in in Dirichlet conditions for the Poisson equation areas for the Poisson cloning y boundary y scheme for a given equation will converge and what are the visual system is completely adapted to the area of texture the the visual the the visual visual visual system. system. system. system. It represents It It represents represents It represents the the additional the the additional additional additional freedom freedom freedom freedom which which which which on lightness perception and examples of illusions.) The scheme for a given equation will converge and what are the visual system is completely adapted to the area of texture In the von Kries approach vK02 adaptation to grayscale by afigure matrix diagonal LMS space. Local effects of adapIn Figure 7 (11). we see the result of the Poisson cloning we fromface In o Electrodynamics. tive solver for equation In general problem shows an uniform in gray band surrounded by a variable + A + A + A + A )( ) f + ( )( ) f 0 (6) ( After performing the above substitution (3) (4) the Laplace are the x and yto components of the vector Here A1 and A2i.e. conditions for convergence. adapted g ( x y ) . In other words adaptation is such that 1 1 2 2 conditions for convergence. i.e. adapted to g ( x y ) . In other words adaptation is such that It would be rather difficult implement a direct iteraon of the visual system. La illuminated area into to the try shadow area. It correctly matches Following s. gs. ings. Following Following terminology terminology terminology terminology from from from from make Poison cloning seamlessly match the boundary of isFollowing generalized to three types of sensors in the retina L M tation of the von type system's have been used the inredefines Geo05 to background. Due Kries to our visual adaptation redefines redefines redefines our perception our our perception perception of of gradients of gradients of gradients gradients based based based based on on adaptation on adaptation on adaptation adaptation function A(x your ) equation which isis used toperception describe the adaptation of x x y y with such equations is not only complexity and performance Here how we define A ( x y ) in the case of our improve(5) is converted into the covariant Laplace equag ( x y )) is covariantly constant the covariant derivatives of g g ( x y is covariantly constant the covariant derivatives of g solver for equation (11). In general the problem we face + A + A + A + A )( ) f + ( )( ) f 0 (6) tiveThe ( band appears to vary in lightness (perceived brightness) in the visual system. It represents the additional freedom which 1 1 2 2 and S which are responsible for color perception. Adaptaapproach we take in our case is based on the followderive a mathematical description of the visual system. The approach take in our case is based on the followbut the fact that propri it is not clear iterative submitteda towe Eurographics Symposium on Rendering (2005 )if a given x x y y ment of Poisson cloning. Following Geo05 we assume the variant contravariant ntravariant ravariant change change change change in lightness. in in lightness. in lightness. lightness. tion : opposition to its surroundings. Following terminology from and are zero. and will and and will be will will specified be be specified be specified specified later later later in later equations in in equations in equations equations (8) (8) (9) (8) (8) (9) and (9) and (9) (10). and and (10). (10). The (10). The The The redefines our perception of gradients based on adaptation are zero. with such equations isof not only complexity and performance Submission ID 1033 / EG Photoshop Healing 3 Here is how we define A ( x y ) in the case of our improvetion is described by a multiplication of the ( L M S ) vector ing unique property equation (11): ing unique of equation (11): after differentiation can written as Physics we will call this contravariant change in lightness. which scheme for a property given equation will converge and what are the visual system is completely adapted the area of texture and will be specified later in equations (8) (9) and (10). The be to mpleIn mathematical recipe that which after differentiation can be written as but the fact that a propri it is not clear if a given iterative this paper we provide a simple mathematical recipe that ment of Poisson cloning. Following Geo05 we assume the submitted to Eurographics Symposium on Rendering (2005 ) by a matrix diagonal in LMS space. Local effects of gradient adapgradient visible in Figure 10 is due to covariant derivative the patch. A further improvement reported in Geo04 is gradient gradient gradient visible visible visible visible in Figure in in Figure in Figure Figure 10 10 is 10 due is 10 is due is due to due covariant to to covariant to covariant covariant derivative derivative derivative derivative conditions for from convergence. i.e. adapted to g(x y). In other words adaptation is such that Let's start Let's start fromequation will converge and what are the scheme a given describes effects of adaptation illustrated in Figure 10. In the adapted to the visual surroundings. a fourth to order "bi-Poisson" equation which matches bothfor system is completely adapted the area of texture llustrated inis Figure 10. In the tation ofLightness the von Kries type have been used in Geo05 to Covariant is covariantly perceived by humans through a given visual Laplace Equation g ( x y ) constant the covariant derivatives of g pixel values and gradients at the boundary. adapted adapted adapted adapted to the to to the surroundings. to the the surroundings. surroundings. surroundings. + A + A + A ) f + ( )( ) f 0 (6) ( +A f conditions for convergence. The approach we take in our case is based on the followusual equations we simply replace each derivative with a co- i.e. adapted to g(x y ) . In other words adaptation is such that 1 )( 1 2 2 f system in a given state of adaptation. The state of adaptation derive a mathematical description of the visual system. ( x y )) g ( x y ) 0 (8) + A ( 1 x x y y ( x y )) g ( x y ) 0 (8) + A ( 0 (12) are zero. ace each derivative with a co1 0 (12) y mans humans by humans humans through through through through a given a a given given a visual given visual visual visual It is well known that the Laplace equation f 0 with is critical to our fundamental judgement of brightness and x f + f div A + 2 A grad f + A A f 0 . (7) variant derivative. These covariant derivatives are specified ing unique property of equation (11): x g g(x y) is covariantly constant the covariant derivatives of g g The approach we take in our case is based on the followDirichlet boundary conditions is the simplest way to reconThis simple approach has been very successful described color. If the equations we use do not reflect this adaptation which after differentiation can be written asA A f 0. In that this paper we provide a simple mathematical recipe that so the covariant gradient is equal to the perceived gra f + f div A + 2 A grad f + (7) are zero. iant derivatives are specified ptation. adaptation. tion. aptation. The The The state The state state of state adaptation of of adaptation of adaptation adaptation struct (or inpaint) a defective area in an image. Let's write in the media as "redefining the way retouching is done they can not produce results that are acceptable to that viSubmission ID in 1033 /property EG Photoshop Healing 3 ing unique of equation (11): Let's start from and describes effects offind adaptation illustrated incovariant Figure 10. In the the derivatives Here the vector function A(x y) An Internet (A1 (x y) Aon y)) Brush reveals dient. In the example of Figure 10 constant pixel values in explicitly: photography". search Healing sual system. We the concept of modified or and perform perform the the differentiations. differentiations. The The result result is is 2 (x equal to the perceived graIt is It It well is It is well is well known well known known known that that that the that the Laplace the the Laplace Laplace Laplace equation equation equation equation f f 0 f with f 0 0 with with 0 with udgement ls al ntal judgement judgement judgement of of brightness of brightness of brightness brightness and and and and the patch. A further improvement reported in Geo04 is f its popularity. derivative used in Physics to be a useful tool for making the usual equations simply replace each derivative with a codescribes adaptation of the visual system. It is related to the band havewe nonzero covariant derivative and describe the Let's start from ( x y )) g ( x y ) 0 (8) + A ( 1 0 (12) a fourth order "bi-Poisson" equation which matches both equations change "covariantly" with the adaptation of the vi( x y )) g ( x y ) 0 (9) ( + A x 2 ( x y )) g ( x y ) 0 (9) ( + A g 2 f + f div A + 2 A grad f + A A f 0 . (7) variant derivative. These covariant derivatives are specified the "guidance field" in Poisson Image Editing PGB03 perceived gradient. Here the vector function A ( x y ) ( A ( x y ) A ( x y )) e 10 constant pixel values in Dirichlet Dirichlet Dirichlet boundary boundary boundary boundary conditions conditions conditions conditions is the is is the simplest is the the simplest simplest simplest way way way to way reconto to reconto reconrecon y pixel values and gradients at the boundary. sual system. o se use do not do not do reflect not not reflect reflect reflect this this adaptation this this adaptation adaptation adaptation Dirichlet f y 1 2 f+ f 0 (5) Scratch removed by Poisson cloning from the ( x y )) g ( x y ) 0 (8) + A ( grad f gradg g ( gradg ) ( gradg )) (12) f Figure 7: Scratch removed by Poisson cloning from the illu1 x x y y 0 grad f gradg g ( gradg ) ( gradg f so that the covariant gradient is equal to the perceived grax illuminated area. - 0 . - 2 + 2 2. Problems with Poisson cloning g 0 . - 2 - + 2 In the von Kries approach vK02 adaptation to grayscale minated area. 2 describes adaptation of the visual system. It is related to nt derivative and describe the struct struct struct (or struct (or inpaint) (or (or inpaint) inpaint) inpaint) a defective a a defective defective a defective area area area in area an in in an image. in an image. an image. image. Let's Let's Let's write Let's write write write and perform the differentiations. The result is at ults s that that are that are acceptable are are acceptable acceptable acceptable to that to to that to that vithat viviviSolving for A ( x y ) produces the specific form of the vector After performing the above substitution (3) (4) the Laplace 2 f f g g g Solving for A ( x y ) produces the specific form of the vector Here the vector function A ( x y ) ( A ( x y ) A ( x y )) dient. In the example of Figure 10 constant pixel values in This simple approach has been very successful described f f g g g submitted to Eurographics Symposium on Rendering (2005 ) 1 2 is generalized to three types of sensors in the retina L M Our current paper describes an improvement to both Poisson equation (5) is converted into the covariant Laplace equa(13) in the media as "redefining the way retouching is done in function that we are going to use: (13) function that we are going to use: and have S which are responsible for color perception. and Adaptadescribes adaptation of the visual system. It is related to the band nonzero covariant derivative describe the cloning and the Healing Brush. Poisson cloning between the "guidance field" in Poisson Image Editing PGB03 the the derivatives the derivatives derivatives derivatives explicitly: explicitly: explicitly: explicitly: tion : the ncept oncept pt concept of tion of modified of modified of modified modified or or covariant or covariant or covariant covariant and perform the differentiations. The result is ( x y )) g ( x y ) 0 (9) ( + A photography". An Internet search on Healing Brush reveals 2 is described by a multiplication of the (L M S) vector areas of different lighting conditions can be a problem y in Poisson the "guidance field" Image Editing PGB03 perceived gradient. its popularity. We see that (13) is same as (11). Using the fact that by a matrix diagonal in LMS space. Local effects of adap We see that (13) is same as (11). Using the fact that grad f gradg g ( gradg ) ( gradg ) f without this improvement. This often is the case with face o be ato be useful a be a useful useful a useful tool tool tool for tool for making for making for making making thethe the the ( x y )) g ( x y ) 0 (9) ( + A 2 from 0 . - 2 to - +2 tation of the von Kries type have been used in Geo05 to (11) is equivalent (6) with "guiding field" extracted retouching to remove wrinkles when unwrinkled skin is (11) is equivalent to (6) with "guiding field" extracted from 2 y Solving for A ( x y ) produces form of the vector + A + A + A + A2 ) f 0 the )( ) f + ( )( (6) specific ( f f g g g 1 1 2 gradg derive a mathematical description of the visual system. gradg x x y y A submitted to Eurographics Symposium on (2005 ) grad f gradg g ( gradg ) ( gradg ) f only available in areas of Rendering different on lighting. Figure 7: Scratch removed by Poisson cloning from the illusubmitted to Eurographics Symposium Rendering (2005 ) the sampling area and defined by (8) (9) we come to our (10) ( x y ) - with y" ly" ntly" with with the with the adaptation the the adaptation adaptation adaptation of the of of the viof the the vivivithe sampling area and defined by (8) (9) we come to our (10) A ( x y ) - (13) function that we are going to use: 0 . - 2 - + 2 2. specific Problems with Poisson cloning g minated area. 2 g which after differentiation can be written as Solving for A( x y ) produces the form of the vector In this paper we provide a simple mathematical recipe that f f g g g covariant reconstruction algorithm as follows: covariant reconstruction algorithm as follows: describes effects of adaptation illustrated in Figure 10. In the Our current paper describes an improvement to both Poisson To provide a clean example of the problem let's try to (13) We see that (13) is same as (11). Using the fact that function that we are going to use: f + f + f + f + f f 0 f f 0 0 0 (5) (5) (5) (5) Substituting obtain the final of the Substituting in in equation equation (7) (7) we we obtain theHealing final form of the usual equations we simply replace each derivative with a cocloning andscratch the Brush. Poisson cloning remove the from form the shadow area in Figurebetween 5 using x x x x x x y y y y y y y (11) is equivalent to (6) with "guiding field" extracted from f + covariant f divA + 2A Laplace grad fCovariant +A equation: A fx 0 .Image (7) variant derivative. These covariant derivatives are specified areas of different lighting conditions can be a problem Reconstruction source material from the illuminated area. gradg covariant Laplace equation: only We see that the (13) is same as (11). Using the fact that (1) Divide image by the sampling (texture) image. (1) theand image by the sampling (texture) image. so that the covariant gradient is equal to the perceived grathe sampling area defined by (8) (9) we come to our (10) A ( x y ) - without this improvement. This often is the case with face Divide K02 h vK02 ch vK02 vK02 adaptation adaptation adaptation adaptation to grayscale toto grayscale to grayscale grayscale (11) isproduces equivalent to (6) with "guiding field" extracted from Figure 8: Scratch removed by covariant cloning from the g to remove wrinkles when unwrinkled Here the vector function A(x y) (A1 (x y) A2 (x y)) dient. In the example of Figure 10 constant pixel values in retouching skin is This the intermediate image I ( x y ) . This produces the intermediate image I ( x y ) . covariant reconstruction algorithm as follows: After After After performing After performing performing performing the the above the the above above above substitution substitution substitution substitution (3) (3) (4) (3) (3) (4) the (4) (4) the Laplace the the Laplace Laplace Laplace gradg same illuminated area as by in Figure 7. Method described in our describes adaptation of the visual system. It is related to available in areas of different(10) the band have nonzero covariant derivative and describe the only lighting. the sampling area and defined (8) (9) we come to A ( x y ) - pes s of sensors of sensors of sensors sensors ingradient. the inin the in retina the the retina retina retina L M L L M L M M the "guidanceSubstituting section 4. inImage equation we obtain the final form )of the g field" in Poisson Editing(7) PGB03 perceived f grad f gradg g ( gradg ) ( gradg equation equation equation equation (5) (5) is converted (5) is is converted is converted converted into into into the into the covariant the the covariant covariant covariant Laplace Laplace Laplace Laplace equaequaequaequacovariant reconstruction algorithm as follows: f(5) grad f gradg g ( gradg ) ( gradg ) 0 . - 2 equation: - + 2 0 . - 2 - + 2 covariant Laplace 2 To provide a clean example of the try Divide to le ible rfor color for for color color perception. color perception. perception. perception. AdaptaAdaptaAdaptaAdapta2 form of theproblem let's ff to Eurographics ff equation g g g (1) the image by the ff sampling (texture) image. ((x y )) submitted Symposium on Rendering (2005 )obtain the final g g Substituting in (7) we g x y tion tion :tion tion : : covariant : remove the scratch from the shadow area in Figure 5 using (14) I ( x y ) (14) I ( x y ) (11) pixel values at the boundary of the patch but the cloned (11) This produces the intermediate image I ( x y ) . Laplace equation: g ( x y ) only source material from the illuminated area. g ( x y ) is too(texture) ation lication ltiplication plication of the of of the of (the L the ( M L ( L M ( S L M ) S vector M ) S ) vector Svector ) vector are still easy spot. There much variation (1) Dividepebbles the image by to the sampling image. too high contrast or dynamic range in the "healed" area Scratch removed by covariant cloning from the We see that the covariant Laplace equation is more Figure 8: Scratch removed by covariant cloning from the We see that the covariant Laplace equation is more This produces the intermediate image I ( x y ) . f grad f gradg g ( gradg ) ( gradg ) MS S LMS pace. space. space. space. Local Local Local Local effects effects effects effects of adapof of adapof adapadapof theilluminated image. This problem is inherent in the nature of illuminated area as in above Figure. (2) Solve the Laplace equation 0 . - 2 - + 2 same area as in Figure 7. Method described in (2) Solve the Laplace equation complicated and actually very different from the Laplace 2 complicated very different g from the Laplace the Poisson equation (1) f which transfers variations of g f f and actually g g ( x y ) section 4. f grad f gradg g ( gradg ) ( gradg ) (14) I ( x y ) ype ve e have have been have been been used been used used in used Geo05 inin Geo05 in Geo05 Geo05 to toto to equation. It incorporates terms describing interaction with without modifying their amplitude even if new brightness (11) equation. It incorporates terms describing interaction with f - 0 . 2 - + 2 g ( x y ) 2 values are modified to match the surroundings. f g g g f ( x y ) Figure 5: Original image of pebbles and a scratch. the external field g . In a way this is a Poisson equation with + A + + A + A A + A + + A + A A A + + A + A A + A + + A + A A )( )( )( )( ) f + ) f ) ( + f ) + f ( + ( ( )( )( )( )( ) f ) f ) 0 f ) f 0 0 (6) 0 (6) (6) (6) ( ( ( ( the external field g . In a way this is a Poisson equation with 1We1see 1 1 1 1 1 1 2 2 2 2 2 2 2 2 (14) Iat (x yI)( ption iption cription n of the of of the visual of the the visual visual visual system. system. system. system. (11) that the covariant Laplace equation is more x y ) 0 (15) pixel values the boundary of the patch but the cloned x x x x x x x y "right y y y side". y y However yy I (x y 0 (15) ax modified g term on the hand g)(x y ) Covariant derivatives in our approach describe adaptation of the visual system in the following way. As suggested in This simple approach has been very successful described in the media as "redefining the way retouching is done in Geo05 a perceptually correct gradient is written based on photography". An Internet search on Healing Brush reveals its popularity. the following simple recipe: Each derivative is replaced with Further development of the mathematical tools behind the Adobe Photoshop Healing Brush a "derivative function" 4 Submission ID 1033 / EG+ Photoshop Healing expression: Figure 7: Scratch removed by Poisson cloning from the illu2. Problems with Poisson cloning minated area. gle adaptation has constant Our current paper describes an improvement to both Poisson Todor Georgiev of humanpixel vision. values. Also our general approach is 4. Main Equations cloning and the Healing Brush. Poisson cloning between to Geo05 . of different lighting conditions can be a problem anclose Marco Venice. Following the example of areas Electrodynamics and Quantum without this improvement. This often is the case with face Submission Submission Submission Submission ID ID 1033 ID ID 1033 1033 /1033 EG / EG /Photoshop EG / 1033 Photoshop EG Photoshop Photoshop Healing Healing Healing Healing 4 Submission ID / EG Photoshop Healing Mechanics will replace conventional derivatives with cloning. coFigure 4: Areaswe in Figure 2 used for Poisson retouching to remove wrinkles when unwrinkled skin is ( x y ) (3) + A 1 only available in areas of different lighting. variant derivatives . They are closely related to the measureFigure 1: Basilica San Marco Venice. x x adaptation of human vision. Also our general approach is 4. Main Equations Submission ID 1033 / EG P ment process and in Theoretical Physics they are responsiFigure 4: Areas in Figure 2 used for Poisson cloning. on. so . Also Also our Also our general our our general general general approach approach approach approach is is is is 4. 4. Main 4. Main 4. Main Equations Main Equations Equations Equations close to Geo05 . To provide a clean example of the problem let's try to illusion Figure 10 is an a fourth order "bi-Poisson" equation which matches both pixel values and gradients at the boundary. Covariant Image Reconstruction

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