Patent Application: US-9439205-A

Abstract:
a method for graphical hairstyle generation presents an interactive technique that produces static hairstyles by generating individual hair strands of the desired shape and color , subject to the presence of gravity and collisions . a variety of hairstyles can be generated by adjusting the wisp parameters while the deformation is solved efficiently accounting for the effects of gravity and collisions . wisps are generated employing statistical approaches . as for hair deformation , a method is used based on physical simulation concepts but is simplified to efficiently solve the static shape of hair . on top of the statistical wisp model and the deformation solver , a constraint - based styler models artificial features that oppose the natural flow of hair under gravity and hair elasticity , such as a hairpin . our technique spans a wider range of human hairstyles than previously proposed methods , and the styles generated by this technique are pretty realistic .

Description:
it is readily observable that hair exhibits clustering behavior ; a group of hair strands that are spatially close and geometrically similar is called a wisp . we view the task of synthesizing a hairstyle as generating a collection of wisps . this section describes how we generate the hair strands and wisps by controlling the statistical properties of hair . our wisp model consists of a master strand 10 and numerous member strands 20 , and the overall shape of the wisp is a generalized cylinder . this section describes the wisp generation algorithm under the assumption that the geometry of the master strand 10 is given ; the method for generating the master strand 10 is presented in section 2 . 2 . we represent a strand as a catmull - rom spline , which is defined by and passes through a sequence of control points { p 1 , p 2 , . . . , p n }. the spline p ( s ) is parameterized within the range [ 0 , 1 ], so that p ( 0 )= p 1 and p ( 1 )= p n correspond to the root and tip of the strand respectively . within a wisp , the degree of similarity among the strands is controlled by the length distribution , the deviation radius function , and the strand - shape fuzziness value . * length distribution l ( u ) is a probability density function that gives the probability that a member strand 20 will have length u , relative to the length of the master strand 10 as shown in fig1 and 2 . * deviation radius function r ( s ) specifies an upper limit on the positional offset of the member strands 20 from the master strand 10 at the parameter value s ; it controls the shape of the generalized cylinder representing the wisp as shown in fig3 and 4 . * strand - shape fuzziness value a controls the variation of the strands within a wisp . fig5 and 6 show the appearance of wisps at σ = 0 . 1 ( low variance ) and σ = 1 ( high variance ). now , the member strands 20 are formed from the master strand 10 by applying l ( u ), r ( s ) and σ . the k - th control point p k of a member strand 20 is computed by adding a displacement d k to the k - th control point p k m of the master strand 10 : p k = p k m + d k . assuming that the displacement at the root d 1 = p 1 − p 1 m is known , d k is computed iteratively using the equation : d k = r k r k - 1 ⁢ d k - 1 + e , ( 1 ) where r k − 1 and r k are the deviation radii at p k − 1 and p k respectively , e is a three - dimensional noise vector . the vector e cannot have an arbitrary value because d k must lie within the sphere of radius r k according to the definition of the deviation radius function . let the noise vector be expressed as e = y { overscore ( x )}, where { overscore ( x )} is a unit direction vector and y is a scalar value . to determine the noise vector e : 2 . y is randomly chosen while | d k |& lt ; r k is satisfied . to find y , a ray is cast originating from p k o = p k m + r k / r k − 1 d k − 1 in the direction of { overscore ( x )} until the ray intersects the sphere at a point p k s . let l be the distance from p k o to p k s . a value for y is then chosen from the uniform distribution of the range [ 0 , min ( l , σ · r k )] as illustrated in fig7 . therefore , d k remains within the sphere , and the control points of the member strands 20 are not concentrated at the boundary of the sphere . the above procedure is repeated until the member strand 20 attains the desired length $ u $ relative to the master strand 10 . the wisp modeling algorithm assumes that the master strands 10 have been already given . this section presents how we actually model the geometry of a master strand 10 . it is generally accepted that hair strands from a single person resemble each other , which forms a unique characteristic of the person &# 39 ; s hairstyle . in our wisp model , an effective way of ensuring that the entire hair possesses a common pattern is to enforce that all the master strands 10 have the same pattern . then the member strand 20 generation algorithm ( section 2 . 1 ) extends this pattern to the entire hair . we obtain the master strands 10 of a common pattern by synthesizing new strands that resemble a given prototype strand 30 . to synthesize a new similar strand from the prototype 30 , we adopt the example - based markov chain models [ 6 , 10 , 28 ], but formulate it as a gibbs distribution to mathematically control the degree of similarity between the prototype 30 and the synthesized strand 40 . instead of directly manipulating the control points { p 1 , p 2 , . . . , p n } of a master strand 10 , we decompose them into an outline component { c 1 , c 2 , . . . , c n } 12 and a details component { δ 1 , δ 2 , . . . , δ n } such that p i = c i + δ i ( i = 1 , 2 , . . . , n ). in fig8 b , the straight line segments at the center correspond to the outline component 12 of the strand ; they are connected with 3 - dof rotary joints 14 , and c i represents the position of the joint center 14 . we use the segments of the equal length between the joints 14 to simplify the calculation of deformations in section 3 . the displacements from the joint centers 14 to the control points correspond to the details component . the purpose of the decomposition is to separate the intrinsic geometry of the strand from the deformations applied to it by styling procedures . the subsequent deformations of the strand modify only the joint angles of the outline component 12 , but do not modify the details component . as a consequence , the geometrical characteristics — which are encoded by the details component — can be preserved . the outline component 12 of the strand consists of straight line segments , thus is determined straightforwardly once the segment length l s is given . this section describes how the details component is generated from a prototype strand 30 . the control points of a new strand must be established so that the strand resembles the given prototype 30 . assuming that a hair strand can be modeled as a markov chain , we can construct the details component by sequentially determining δ i ( i = 1 , 2 , . . . , n ) until the curve has reached the desired length while observing only the neighborhood of δ i . let { δ 1 *, δ 2 *, . . . , δ n *} be the details component of the prototype strand 30 . to determine δ i , we examine the prototype strand 30 and find the best match δ k * while restricting the comparison only to the neighborhood of δ i as illustrated in fig9 . we adopt a similar approach used in “ curve analogy ” [ 10 ], which can be referenced for the detailed description of the curve synthesis algorithms using markov chains . in fig9 , it is shown synthesizing a new strand from a prototype 30 . to find δ i , the window ω ( δ i ) is compared with every possible window in the prototype 30 , then δ k * is selected from the prototype 30 based on eq . ( 4 ). the neighborhood is modeled as a window — a set of control points — around the current control point . intuitively , the size of the window should be on the scale of the largest regular structure of the curve ; otherwise , the structure may be lost . we also let the neighborhood be causal : we construct the neighborhood containing only the points preceding the current point . therefore , in the sequence of the details component { δ 1 , δ 2 , . . . , δ n }, the neighborhood of δ i is defined by ω ( δ i )={ δ i − h , δ i − h + 1 , . . . , δ i − 1 }, where h is the window size . let ε ( ω i , ω j ) be the euclidean distance between the two h - sized windows ω ( δ i ) and ω ( δ j ): ɛ ⁡ ( ω i , ω j ) = ∑ t = 1 h ⁢  δ i - t - δ j - t  2 . ( 2 ) now , the problem of determining δ i so that ω ( δ i ) resembles the prototype pattern can be reduced to finding a portion ω ( δ k *)={ δ k − h *, δ k − h + 1 , . . . , δ k − 1 } of the prototype strand 30 such that min k ⁢ { ɛ ⁡ ( ω i , ω k * ) } . ( 3 ) then we can deterministically specify δ i to be δ i = δ k *. however , the diversity of synthesized strands 40 can be increased by stochastically selecting the details component instead of the optimal choice given by eq . ( 3 ). we introduce the gibbs distribution as a selection probability p k for each possible window of the prototype strand 30 : p k = 1 z ⁢ exp ⁡ ( - ɛ ⁡ ( ω i , ω k * ) t ) , ( 4 ) where t is the temperature of the system , which can be controlled by the user , and z is the partition function defined by z = σ k exp (− ε ( ω i , ω k *)/ t ). the strand diversity increases as t increases , which is demonstrated in fig1 . at the topmost level , the user sets the number of wisps n w and the total number of hair strands n s . the user then interactively constructs the density , length and protrusion maps which define the corresponding hair properties for every point in the scalp region . the effects of the density and length maps are illustrated in fig1 . density and length maps and their effects : ( a ) a density map ( top )— with bright area 54 representing high density and dark area 52 representing low density — and its effect on hair ( bottom ), ( b ) a length map ( top )— with bright area representing long hair and dark area 52 representing short hair — and its effect on hair . once n w is given , the root positions of the master strands 10 are determined by evenly distributing the number of points over the scalp region . voronoi diagrams can then be generated to set the boundary of each wisp . finally , using the density map and n s , the roots of the member strands 20 are randomly located within each voronoi region . note that the length and protrusion maps are used to determine the length and protrusion direction of the master strands 10 . for the member strands 20 , these properties are determined by the algorithm presented in section 2 . 1 . in this section , we address the problem of determining the shape of wisps under the influence of gravity and collisions . the purpose of the hair deformation solver is to find the joint angles of the master strands 10 after these factors are taken into account . then the shape of the member strands 20 are routinely determined from the shape of the master strand 10 ( section 2 . 1 ). our hair deformation solver is based on two ideas . firstly , it incorporates the physical properties of hair , but avoids dynamics simulation . styling operations such as braiding assemble hair into a complex structure that is beyond the realm of the simple newtonian mechanics . instead of simulating those situations in a pure physically - based way , we deal with gravity and the current styling operation as a unified force field , and solve the deformation employing pseudophysical approaches . secondly , the deformation solver models hair as a continuum , as proposed by hadap and magnenat - thalmann [ 9 ], and interprets density as a measure of collision . when the density of a certain region is above a threshold , it is regarded as a collision and the hair is forced to occupy a larger volume . we implement the two ideas within a single framework . the deformation solver works quickly , and can be applied to a wide range of hairstyles . the styling force field φ ( x ), defined in 3d space , quantitatively combines the effects of gravity and styling operations to represent the desired flow direction and intensity at each 3d point x . the force fields rotate the joint 14 so that the outline segment 12 ( fig8 ) attached to the joint 14 is oriented along φ ( x ). however , hair has an elastic property that resists such deformation in proportion to the bending amount and stiffness . therefore , we determine the orientation of a segment by finding the joint angle that maximizes the equation where e φ represents the degree to which the strand is aligned with the force field , and e b represents the degree to which the strand is aligned with its rest position . let q be the unit direction vector of the segment . e φ is calculated by the function e φ = φ ( x )· q and its value is greatest when q is aligned with φ ( x ). e b represents the ( inversely proportional ) amount of bending by e b = q 0 · q , where q 0 is the direction of the segment when the joint angle is zero . finally , κ is the scalar value that models the bending stiffness of hair . fig1 a and 11 b show the resulting deformation for two different values of κ . since the optimization function in eq . ( 5 ) is linear , it can be easily solved to find the unit vector q * that maximizes it . the joint angle corresponding to q * can then be calculated straightforwardly . the above procedure determines the angle for a single joint 14 . to determine the shape of an entire master strand 10 , the procedure is repeated along the strand , from the root to the tip . we now account for the hair - to - head and hair - to - hair collisions . detecting collisions between every pair of strands would be computationally intractable . fortunately , when we observe a hairstyle , collisions between individual strands are not noticeable . when wisps cross each other , however , it produces unmistakable artifacts . therefore , our collision resolution method is developed to handle collisions at the wisp level . we interpret that hair produces a density field . even though the geometries of the strands are defined after the wisps are generated ( section 2 ), the strands are regarded not existing yet . when the hair deformation solver starts processing strands , the density over the entire space is zero . once the solver calculates a segment , the density corresponding to the segment is created . when the procedure in section 3 . 1 would locate a wisp segment in a position x , it is first checked whether the following condition can be met : where ρ ( x ) is the existing density value at x , ρ δ ( x ) is the increased density that would be added due to the positioning of the wisp segment , and τ is the density threshold . when the sum on the left hand side of the equation ρ ( x )+ ρ δ ( x )& gt ; τ is greater than τ , this is treated as a collision and the segment is reoriented based on the gradient of the density field . when a small value is used for τ , the overall hair volume becomes larger , as demonstrated in fig1 c . the description of the previous two sections was based on continuous fields . to implement the algorithms in a discretized space , a three - dimensional grid is constructed around the head and shoulder that includes all regions where hair may potentially be placed during the styling process . the values of the styling force and density are then stored only for points on the grid . the styling force and density at an arbitrary point in the 3d space is obtained by performing a tri - linear interpolation of the values at the eight nearest grid points . the hair deformation solver works by repeating the following two steps : ( 1 ) deform the strands based on the fields , and ( 2 ) update the density field based on the deformation . one question remains : whether the steps should run in the breadth - first order or depth - first order . we choose the breadth - first order with the rationale that this handles hair - to - hair collisions better than the depth - first order does , especially because the master strands 10 have the segments of the equal length ( section 2 ). we summarize the procedure for the hair deformation solver in algorithm 1 . algorithm 1 hair deformation solver initializedensityfield ( ); // 1 inside and 0 outside the head for joint = 1 to j /* root - to - tip order */ do for wisp = 1 to w do deformbystylingforcefield ( ); /* eq . ( 5 ) */ for grid points occupied by the wisp segment do while checkcollision ( ) /* eq . ( 6 ) */ do bend the joint by δθ along the density gradient ; end while end for updatedensityfield ( ); // add the density of this segment end for end for the hair deformation solver is quite powerful in producing the styles that are based on the natural flow of hair under the gravity field . however , the method is not easily applicable to producing artificial hairstyles such as braids . to produce such types of hairstyles , we take a new approach — styling constraints . a constraint causes a constraint force field ψ ( x ) to be generated over a portion of the 3d space , as shown in fig1 . when and only when the hair deformation solver processes the portion being constrained , instead of the original styling force field φ ( x ), it uses a modified styling force field φ ′( x ) given by where the control parameter w is the weight of the constraint force field ψ ( x ) relative to the original styling force field φ ( x ). other than the selective superimposition of the constraint force field over the original styling force field , the hair deformation solver works in the same way . based on the results of our experiments , we have concluded that three types of styling constraints illustrated in fig1 can be used to create the most of artificial hairstyles . * point constraint : a point constraint 61 is specified by an attraction point and a tolerance radius . this constraint produces vector fields toward the attraction point , as shown in fig1 a . the constraint force vector is activated only until the wisp comes within the sphere . * trajectory constraint : a trajectory constraint 62 is specified by a trajectory and a tolerance radius . the constraint generates unit vectors ( 1 ) toward the nearest point of the trajectory for grid points lying outside the tolerance radius , or ( 2 ) in the tangential direction of the trajectory for grid points lying inside the tolerance radius , as shown in fig1 b . * direction constraint : a direction constraint is specified by a trajectory and an influence radius . for grid points lying within the influence radius , the constraint generates unit vectors in the tangential direction of the trajectory , as shown in fig1 c . fig1 shows examples of using the constraint - based styler : ( a ) applying a single [ point + trajectory ] constraint queue to a group of hair , ( b ) applying three [ point + trajectory ] constraint queues to three groups of hair respectively . fig1 c and 18 d show the rendered images of these hairstyles . the functioning of the constraint - based styler can be summarized as follows : 1 . select the portion of hair , consisting of a set of wisps , to apply the constraints ; 2 . build a constraint queue , a sequence of constraints , to apply to the selected portion . when the hair deformation solver processes those selected wisps , the constraint force field ψ ( x ) is superimposed with the styling force field φ ( x ) produced by the constraints . the constraint - based styler activates the first constraint in the queue and processes the selected wisps until they all meet the constraint . it then processes each subsequent constraint , one at a time , in the queue . when all of the constraints have been processed , the hair deformation solver resumes processing the wisps in the usual way . fig1 demonstrates the use of two different constraint queues . the geometric models of hair would not look natural unless they are rendered properly . rendering quality is more important for hair than other parts of the body . this section addresses two problems : how to determine the colors of individual strands , and how to render the hair geometries . fig1 shows the hair colors corresponding to different probability distributions : ( a ) constant color , ( b ) uniform distribution , ( c ) gaussian distribution , ( d ) variation of colors in the wisp level . the colors of strands taken from the same person might appear to be similar . however , closer examination reveals that no two hair strands have exactly the same color . to modulate the change of hair colors , a stochastic approach is employed . we adopt hsv color space , and model hue , saturation and value ( brightness ) channels as three independent probability density functions , using either uniform or gaussian distributions . the different effects produced by adopting these two distributions are shown in fig1 b and 15 c . we can also control the color variations at the wisp level , as shown in fig1 d . in order to explicitly render individual hair strands , the catmull - rom splines , representing the strands ( section 2 ), are converted to thin ribbons using the “ ricurves ” primitive in renderman ™ [ 2 ]. then the hair shading model proposed by kajiya and kay [ 11 ] is used for shading each strand . finally , the deep shadow map [ 18 ] is used to represent the self - shadowing among hair strands , which is essential for creating the volumetric appearance of hair . we implemented the presented techniques on a pc with an intel pentium ( 4 ) 2 . 54 ghz cpu and an nvidia geforce fx 5600 gpu . to check the intermediate results during styling processes , we also implemented a renderer on a programmable graphics hardware [ 17 ]. using our hair modeling system , both ordinary users and professional hairstylists were asked to either reproduce hairstyles from beauty magazines or to create novel hairstyles . to help the readers understand how the whole system works , we summarize the steps taken for the hairstyling . the user starts the styling work on a given 3d polygon model of head and shoulder . in a preprocessing step , the scalp surface is specified as a polygonal surface , which defines the region where hair follicles exist . then , the following steps are taken : 2 . tune the global parameters — the number of wisps n w and the number of strands n s . 3 . edit the geometry of the prototype strand 30 : the user models the prototype strand 30 with a 3d spline curve . we also provided pre - modeled samples representing the various types of curly and wavy hair so that he / she could simply select one from the samples . 4 . tune the wisp parameters — the length distribution l ( u ), deviation radius function r ( s ), and fuzziness value and σ . 5 . setup the styling force field ( see section 6 . 1 for the details ). 6 . tune the deformation parameters — the bending stiffness κ and the density threshold τ . 7 . specify the hair color : when the user uses the gaussian color model , he or she inputs the mean and variance of three hsv channels . fig1 shows some hairstyles created by taking the above modeling steps . approximately 80 , 000 to 100 , 000 hair strands were used for generating each hairstyle . the computation time greatly depended on the number of control points in a strand as well as n w and n s . in the initial setup , most time is spent on user interactions for preparing input to the system . however , once the input is provided , usually the result could be seen in less than 10 seconds on the hardware renderer which rendered about 20 , 000 strands . the accompanying video shows step by step how the above modeling procedure is done . when a desired hairstyle is formed after tuning the parameters , it finally went through a software renderer , which produced the results shown in fig1 . the software rendering took about 30 minutes to render a hairstyle . fig1 shows real hairstyles and the corresponding synthetic imitations modeled after the real ones : ( a ) a real hairstyle and ( b ) a corresponding synthetic one generated by applying the statistical wisp model and hair deformation solver ; ( c ) a real updo hairstyle and ( d ) a synthetic imitation of it obtained by applying the constraint - based styler . currently , we construct the styling force field using a procedural approach . for convenience , we provide the users predefined force fields — e . g ., gravity field , pulled - back hair field . to allow general users to fully exploit our system , a tool for editing vector fields , such as the one described in [ 8 ], should be implemented in the future . it needs to be noted that our use of vector field is somewhat different from that in the previous work [ 8 , 31 ] where almost all the aspects of hairstyles are directly controlled by the vector field . for instance , the vector field should be carefully constructed to prevent the penetration of hair into head . the stand details are also controlled by the vector field : e . g ., curly hair is generated by applying perturbations to the vector field . on the other hand , the construction of the force field in our system is much less cumbersome since the field is used only for the rough - level control of a hairstyle . the detailed properties of hair depend on separate controls . preventing the penetration of hair into the head is done by the collision resolution procedure ( section 3 . 2 ). geometric details of hair strands are generated inside the wisp model ( section 2 ). therefore , even though the current procedural construction is not an ultimate interface for specifying the force fields in general , it already provides enough flexibility to generate the variety of hairstyles in fig1 . as shown in fig1 , a system 80 for interactive hairstyle generation using statistical wisp model and pseudo - physical approaches includes a hair strand generating module 81 , a hair deformation solving module 82 , a hair styling module 83 , a hair rendering module 84 , a styling force field constructing module 85 , and a parameter controller 86 . the styling force field constructing module 85 includes a vector field editor 87 . the hair strand generating module 81 is for generating hair strands and wisps by controlling the statistical properties of hair using a statistical wisp model . the hair deformation solving module 82 is for solving the hair deformation due to gravity and collisions between the hair by a pseudo - physical approaches using a styling force field . the hair styling module 83 is for styling the hair by superimposing - a constraint force field with the styling force field and solving the hair deformation . the hair rendering module 84 is for rendering the hair strands using probability density functions to represent the randomness of hair color . and , the styling force field constructing module 85 is for constructing a styling force field using procedural approach with a means for editing a vector field using the vector field editor 87 . since the five hairdressing operations — cutting , permanent waving , combing , tying and braiding — would produce most human hairstyles , showing that our modeling system is capable of implementing these operations can give an estimation of the applicable range of the proposed technique . * cutting : it can be implemented by directly editing the length map . * permanent waving : it can be achieved by giving a prototype strand 30 of the desired shape to the statistical wisp model . * combing : it can be implemented using a direction constraint . first , a portion of hair is selected for combing , then the desired trajectory is indicated . finally , we specify the weight w of eq . ( 7 ) that controls the influence of the constraint force field relative to the styling force field . * tying : the portion to tie is selected , and a point constraint 61 is imposed so that the selected portion is attracted to the point . a value of w = 1 is used so that the portion is entirely affected by the constraint force field . once the constraint is met , the selected portion is then affected by the normal styling force field . * braiding : it can be implemented by applying a point constraint 61 followed by a trajectory constraint 62 to three or more groups of hair . each group of strands is first gathered at the point specified by the point constraint 61 . then the trajectory constraint 62 is applied so that the strands form braids . fig1 shows the hairstyles produced by the constraint - based styler after applying ( a ) a single point constraint 61 , ( b ) two point constraints 61 , ( c ) a point constraint 61 followed by a trajectory constraint 62 , ( d ) three sets of a point constraint 61 followed by a trajectory constraint 62 . all hairstyles commonly involved cutting , permanent waving and combing . we additionally applied tying operations to produce the styles in fig1 a and 18 b , a variation of the tying operation followed by a trajectory constraint 62 for an updo style in fig1 c , and the braiding operation for fig1 d . in the production , the accessories such as a ribbon did not affect the styling , but was added just to enhance the visual realism . the accompanying video shows the user interface for hairdressing operations . we have presented a technique for interactively modeling human hair . a variety of hairstyles can be generated by modifying a small number of wisp parameters while the effect of gravity is automatically solved by the hair deformation solver . the framework is quite versatile to be capable of realizing familiar but non - trivial styling operations such as permanent waving and braiding in a straightforward manner . the technique provides improvements over previous methods in the reality of the result and the range of hairstyles it can produce . the speedup , realism and diversity are possible due to the combination of the statistical wisp model , the hair deformation solver , the constraint - based styler and the stochastic hair color model . firstly , the statistical wisp model has relieved us from modeling each hair strand manually or by employing parametric functions . we have introduced an example - based markov chain model to synthesize a master strand 10 from a given prototype curve . we duplicate a master strand 10 to neighboring member strands 20 by creating statistically - controlled variations . secondly , we have proposed a solver to compute the deformations due to the gravity and collisions . by tailoring the solver to static hairstyles , we could - make it work fast . thirdly , some hairstyles , such as braids or updo styles , require tedious hairdressing operations , and have been difficult to synthesize . the constraint - based styler provides us an effective way of producing such styles . even though there are only three types of constraints , the combinations of them can produce a wide range of styles , as shown in fig1 . finally , by using probability density functions to represent the randomness of hair color , we have obtained more realistic results even with the traditional kajiya and kay shading model . we plan to - explore using a more sophisticated shading model proposed by marschner et al . [ 19 ], combined with the stochastic color model . the paper by the inventors , byoungwon choe and hyeong - seok ko , a statistical wisp model and pseudophysical approaches for interactive hairstyle generation , ieee transactions on visualization and computer graphics , vol . 11 , no . 2 , 160 - 170 , march 2005 , is incorporated by reference into this disclosure as if fully set forth herein . also , all reference papers , list below , referred to by the paper by the inventors are incorporated by reference into this disclosure as if fully set forth herein . while the invention has been shown and described with reference to different embodiments thereof , it will be appreciated by those skilled in the art that variations in form , detail , compositions and operation may be made without departing from the spirit and scope of the invention as defined by the accompanying claims . 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