Geometric clustering for line drawing simplification

Transcription

1 Eurographics Symposium on Rendering (2005) Kavita Baa, Phiip Dutré (Editors) Geometric custering for ine drawing simpification P. Bara, J.Thoot and F. X. Siion ε ARTIS GRAVIR/IMAG INRIA Figure 1: The two stages of our method. Lines of the initia drawing (eft) are first automaticay custered into groups that can be merged at a scae ε (each group is assigned a unique coor). A new ine is then generated for each group in an appicationdependent stye (at right, ine thickness indicates the mean thickness of the underying custer). Abstract We present a new approach to the simpification of ine drawings, in which a smaer set of ines is created to represent the geometry of the origina ines. An important feature of our method is that it maintains the morphoogica structure of the origina drawing whie aowing user-defined decisions about the appearance of ines. The technique works by anayzing the structure of the drawing at a certain scae and identifying custers of ines that can be merged given a specific error threshod. These custers are then processed to create new ines, in a separate stage where different behaviors can be favored based on the appication. Successfu resuts are presented for a variety of drawings incuding scanned and vectorized artwork, origina vector drawings, drawings created from 3d modes, and hatching marks. The custering technique is shown to be effective in a these situations. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation Picture/Image Generation 1. Introduction Line drawing is an important aspect of modern graphics; it aows an intuitive depiction of compex scenes with a remarkabe economy of means. This is probaby due to the abiity of the human visua system to perceive shape from intensity discontinuities. Artists have since ong earned to use this perceptua property to provide stunningy expressive pictures, by cevery tuning ine density across their drawings. However, most of the time in computer graphics, ines do not come with an appropriate density. Simpy scaing a drawing, for instance for dispaying on ow-resoution devices, creates a need for density adjustment; moreover, density reduction in 3d is not yet mature and most non-photoreaistic rendering (NPR) systems extract far too many ines. Thus there is a need to adapt ARTIS is a research team of the GRAVIR/IMAG aboratory,a joint unit of CNRS, INPG, INRIA and UJF. the number of ines in a drawing, otherwise the effectiveness of such a representation may be compromised. There is not a singe way to simpify a set of ines, depending on the envisioned appication. In the context of density reduction, we may want to adjust the ine density of a drawing where too many ines project in a given region of the image. This is needed when scaing a ine drawing, as we as when rendering from a 3d scene. In this context ony the most significant ines shoud be drawn. Leve-of-detai (LOD) representations for ine-based rendering (contours and hatching), where the number of ines must vary with scae, constitute another simpification approach. Finay, in the context of progressive editing (sometimes caed oversketching), the user refines a curve by successive sketches. This can be viewed as an iterative simpification of the set of ine sketches provided by the user. In this paper, we anayse the properties shared by these three appications and propose a common soution.

2 1.1. Probem statement We consider a drawing to be a digita image composed of a number of vectorized 2d ines. Such images can be obtained in various ways: by scanning and extracting ines from a hand-made drawing; by direct digita creation using appropriate input devices (mouse, tabet, etc); by detecting contours in an image [ZT98]; or by rendering a 3d scene in a ine stye [SS02,GG01,GTDS04]. We thus imit our approach to static 2d drawings. We are focusing on simpification of such static 2d drawings, i.e., the creation of another set of ines containing fewer ines than the origina set. We propose a generic approach for this type of probem, where simpification is controed by a singe distance-based scae parameter ε. Of course, this rather restricted view (in which spatia proximity is used as the main discriminating criterion) impicity assumes that a ines beong to a coherent set, over which simpification can be carried out using a very ow eve semantic description. In particuar this approach is not taiored to reguar structures or other higher order arrangements. However, nothing prevents the user from preprocessing the data to organize ines in different categories and to appy our method to simpify independenty each category Reated work We now review reevant work invoving ine drawing simpification. Two research fieds dea with ine drawing treatments but are beyond the scope of this paper: beautification of a drawing that essentiay tries to satisfy some geometric constraints to correct technica diagrams; and simpification of a singe curve that deas with maintaining the goba shape of a curve whie decreasing its resoution. We do not consider these two fieds but rather concentrate on techniques that simpify a set of curves without imposing a predefined mode. Progressive drawing toos [IMKT97, Bau94] are usefu in the context of sketch-based modeing, or within vector graphics packages such as Adobe Iustrator TM. These dedicated toos assist the user in adjusting the shape of a ine: they are essentiay semi-automatic, work iterativey and are not designed to edit more than one ine at a time. Thus they are not easiy adapted to other, non-progressive appications. Severa agorithms have been proposed to contro the density of ines in 3d renderings. Deussen et a. [DS00] present a simpification technique dedicated to trees and vegetation, which reies on their intrinsic hierarchy, and works in object space. Preim et a. [PS95] and Wison et a. [WM04] measure density in image space in order to imit the number of ines drawn for compex objects. Simiary, Grabi et a. [GDS04] introduce density measures in image space, used to seect the most significant ines. The use of information extracted from the 3d scene (sihouettes, creases, etc.) aows them to evauate this significance and order the ines by decreasing priority. The simpification process is then carried out by deeting the east significant ines. Simiar approaches have been proposed in the context of resoutiondependent dispay and printing, since they are reated to haftoning [SALS96, ZISS04]. Here again the authors add a notion of priority to ensure that the most important ines are drawn first for any tone eve and then offer a seection mechanism of some ines among the origina ones. In the fied of iustration, Winkenbach et a. [WS94] introduced the notion of indication: compex textures are ony fuy rendered in certain paces of the drawing at an appropriate density, to suggest the compexity of a pattern (such as a brick wa). Severa papers about non-photoreaistic rendering dea with eve-of-detai rendering for animated scenes. Praun et a. [PHWF01] present an image-based method to hande LOD in hatching. Their tona art maps (TAMs) are mip-mapped textures aowing rea-time dispay of hatching styes. Other LOD creation systems have been proposed, such as the WYSIWYG NPR system [KMM 02] that ets the user specify the appearance of the object for severa view points. Reevant LODs are then bended for a given view. The methods presented above ony deete the ess significant ines and do not consider any perceptua aspect of ine simpification. On the other hand, perceptua grouping approaches provide effective ways of consistenty grouping ines, even if they do not adress the probem of simpification directy. Most of the work in this area deas with the extraction of cosed paths in drawings [Sau03, EZ96], focusing on grouping criteria such as good continuation and cosure. However, other criteria are more reevant for simpification purposes, e.g. proximity and paraeism. Unfortunatey, even if each criterion has been studied in isoation [Ros94], their reative infuence is yet to be determined Contributions Our two main contributions reside in an attempt to mode the common properties of target appications of ine drawing simpification whie aowing various simpification behaviors: contrary to previous methods, we construct a partition of the origina set into consistent groups that can be repaced by an entirey new ine. To this end, we draw inspiration from perceptua grouping. We decompose the process into two main stages (see Fig. 1): a custering stage in which we group the origina ines and a geometric stage where a new ine is created for each group. Whie the former is entirey automatic and common to a appications, the atter is oriented toward the specific needs of each of the envisioned appications. Our approach is genera in the sense that it considers a set of minima and ow eve goas shared by those appications. Therefore, it is not abe to dea with higher-eve structures ike Winkenbach and Saesin [WS94].

3 We begin by describing our methodoogy in Section 2. The common, automatic custering stage is presented in depth in Sections 3 and 4. Our contribution here is the definition of a moduar agorithm that custers any kind of ine. Section 5 shows how to adapt our custering agorithm to different needs, giving simpification resuts in the contexts of density reduction, LOD and progressive drawing; for ines coming from different sources: scanned drawings or nonphotoreaistic renderings. Finay we discuss imitations of our method in Section Methodoogy We now present the principes of our simpification method, incuding forma definitions that wi hep to carify our approach Input ines We define a ine-drawing as a set of 2d ines hoding a set of attributes (coor, thickness, stye parameters, etc.), without any assumption on their nature. Thus a ine is ony defined by two end points and a continuous path between them. : [0,1] R 2 A where A is the space of attributes. In the foowing, we wi denote by [a,b] the part of restricted to [a,b] with 0 a,b 1. We are not interested in an exact parameterization of ines, but ony on their geometric properties. Therefore in the rest of the paper, we wi refer to a ine indiscriminatey to refer to the set of geometric points that constitutes it Objectives As aready stated, our main goa is to create a set of ines containing fewer ines than the origina one. For that, we first need to contro the amount of simpification accompished by our method. Our target appications a have a singe common parameter: the simpification scae. This scae, which we denote by ε, is thus the ony parameter needed by our approach. Intuitivey, we ony simpify the existing information at a scae smaer than ε, keeping a the information present at a arger scae. In the appications we envision, we first need to ensure that the overa configuration of the origina drawing is respected in the simpified one. Regarding perceptua grouping, this means taking proximity and continuation effects into account. To this end, we impose a coverage property which consists of creating new ines ony in regions where initia ines can be found. However, we are not ony focusing on the ine positions, but aso on their shape. We want the new ines to respect the way the initia ines have been created. For instance, in Fig. 2-(a), the new ine fods onto itsef to cover the origina (a) (c) Figure 2: (a) The simpified ine (in pink) has a fod whie the initia group (in back) does not - (b) Two simpified ines are preferred to represent this group - (c) The simpified ine does not refects the initia orientation and shape of the group - (d) Using three simpified ines better maintains the shape of the hatching group. ines that form a fork (Y-shapes), athough no such fod was initiay present. We prefer a soution such as the one shown in Fig. 2-(b), using one more simpified ine, but with more fideity to the goba shape of the origina ines. The same probem can be found in other exampes, such as hatching groups used to shade regions (see Fig. 2-(c),(d)). In order to preserve the shape of the origina drawing, we thus need another perceptua property: we want the new ine and the custered ines to be parae at the scae ε. To do that, we impose a morphoogica property on simpified ines that prevents them from foding onto themseves. Finay, sti foowing perceptua grouping, we want to be abe to reject the simpification of a pair of ines if their attributes (e.g. coors) are too different. Foowing our objectives, we now give forma definitions reated to our simpification approach Definitions We begin by the definition of an ε-ine and use it to define a group that can be simpified by a singe ine at the scae ε. Foowing our morphoogica property, an ε-ine is a ine that does not fod onto itsef at the scae ε. It corresponds to the fact that, for each point of, there is no point aong the norma at a distance ess than ε that aso beongs to (see Fig. 3). We assume to be G 1 in order to ensure that its norma is uniquey defined at each point: Definition 1 Let be a ine. is an ε-ine if and ony if is G 1 and { q = p + σ n (p) p, q, σ = p q ε (b) (d) where n (p) is the norma vector of at point p. This definition is equivaent to saying that does not intersect either of its two offset curves +ε and ε (see Fig. 3). We now define an ε-group as a group of ines that can

4 -ε +ε p q (a) -ε +ε Figure 3: (a) q is aong the norma at p thus the ine has a fod and hence is not an ε-ine - (b) is an ε-ine. be simpified by a singe ε-ine, as stated in our coverage property (see Fig. 4): < ε/2 < ε/2 Figure 4: An ε-group is a group (in back) that can be covered by an ε-ine (in pink) at the scae ε. (b) The custering stage is the main contribution of this paper, thus it is presented in detai in the next two sections. We then give in Section 5 various exampes of the geometric stage and show that our approach can address specific appications without demanding too much effort on the user side. 3. Custering We use a greedy agorithm to partition the set of input ines. It is based on the iterative custering of pairs of ε-ines and maintains the ε-group property during the entire process: custering a pair of ε-ines that each represent an ε-group resuts in a new ε-ine that represents the merged group. The first step consists of converting the origina ines into ε-ines by spitting them at their points of intersection with their offset curves (see Section 4.1). Our custering agorithm then iterativey custers the pairs of ε-ines (Section 3.1) that have the minimum error (Section 3.3) unti no more custers can be created. At each step we store the hu of the custered pair in order to take into account the resut of previous custerings in the next steps (Section 3.2). Definition 2 A group of ines G is an ε-group if and ony if there exists an ε-ine such that : d SH (,G) < ε 2 where d SH is the symmetric Hausdorff distance defined between two sets of points by: d SH (P,Q) = max(h(p,q),h(q,p)) h(p, Q) = max (min p q ) p P q Q An ε-group is thus a group that meets the proximity, continuation and paraeisms requirements at the scae ε Our approach Foowing our definitions, our approach states that a simpified ine-drawing is a set of ε-ines that covers the origina drawing at a scae ε. We first need to custer the origina ines in a set of ε-groups before being abe to create any new ine. Therefore our simpification method is organized in two stages: 1. A custering stage first groups the ines of the origina drawing in ε-groups. No ine is created at this stage and the process is entirey automatic using a greedy agorithm to iterativey group the origina ines. 2. A geometric stage then creates a singe ine for each custer of the origina drawing. For each of the three target appications, we use the custers differenty and appy dedicated strategies. ε ensures that two ines of the same ε-group are at a distance 2 smaer than ε 3.1. Custering pairs of ε-ines Foowing our definitions, a pair of ε-ines ( 1, 2 ) can be custered if and ony if ( 1, 2 ) is an ε-group. This definition is not constructive: it ony says that there must exist an ε-ine that covers ( 1, 2 ). We now describe a way of buiding this new ε-ine from 1 and Possibe configurations for ( 1, 2 ) to be an ε-group Observation 1 The coverage property (Def. 2) impies that if ( 1, 2 ) is an ε-group then there exists a point p 1 (resp. p 2 ) of 1 (resp. 2 ) such that p 1 p 2 < ε. If not, it woud not be possibe to find a ine such that d SH (,( 1, 2 )) < ε/2. We ca overapping zones the portions where 1 and 2 are at a distance ess than ε, formay defined as: Definition 3 An overapping zone, Z, is a pair of ine portions ( 1 [a1,b 1], 2 [a2,b 2]) such that: d SH ( 1 [a1,b 1], 2 [a2,b 2]) < ε where {a 1,a 2 } and {b 1,b 2 } are the extremities of Z. Observation 2 The morphoogica property (Def. 1) impies that ( 1, 2 ) is an ε-group if it is not a fork. Indeed, if it were the case, then any ine representing ( 1, 2 ) woud have to fod onto itsef. This impies that the overapping zones must not fork, thus there must be at east one of the two ines that ends at each extremity of the zone. A simpe forking configuration is shown in Fig. 5-(a). Other forking configurations exist, but are not represented because we mainy direct our attention to vaid zones (Fig. 5(b) and (c)). We ca such an overapping zone a path and define it by:

5 (a) Fork (b) Four possibe configurations with one path (c) Two paths: cosed curve Figure 5: Possibe configurations of a pair of ε-ines. Ony (b) and (c) can form an ε-group. Definition 4 A path on a pair of ines ( 1, 2 ) is a maxima overapping zone, Z, such that there is at east one extremity of 1 or 2 at each extremity of Z. Knowing that each ine has 2 extremities, there are five combinations for the paths between two ines, iustrated in Fig 5-(b), (c). Thus, if a pair of ines does not correspond to one of these five combinations, it is not an ε-group. The ony configuration that corresponds to a cosed curve is when a pair of ines ( 1, 2 ) has two paths (see Fig 5-(c)). For the sake of brevity, we wi not detai this case in the foowing, as it is essentiay equivaent to the others. However Section 3.4 shows that our agorithm correcty handes it Buiding the new ε-ine Now that we have identified vaid configurations (paths), we buid the new ine, and make sure that it is an ε-ine, i.e. that it respects Definition 1. We create an ε-ine that passes from 1 to 2 and ies in the midde of the path. is obtained by concatenating the portions of ines outside the 2 1 path (in purpe) with a ine created inside the path by interpoating between 1 and 2 from one extremity to the other (in pink). Such a construction insures that is an ε-ine outside the path since 1 an 2 are themseves ε-ines. However for zones inside the path, some particuar cases when is not an ε-ine exist. Indeed, may fod onto itsef if the curvature of 1 or 2 is too cose to 1/ε. In these cases, ( 1, 2 ) is simpy not considered as an ε-group Buiding the hu of an ε-group The new ine we created ony ensures that ( 1, 2 ) is an ε- group; we cannot use it iterativey since it woud not take into account the error made by the custering of 1 and 2. Indeed, consider an ε-ine 3 ; determining if ( 1, 2, 3 ) is an ε-group is not directy equivaent to determining if (, 3 ) is an ε-group. In order to propagate the custering resut of ( 1, 2 ) to, we define the hu of an ε-group by assigning a varying thickness to the ε-ine that represents it. This thickness describes the resut of the custering of two or more ines and is used in subsequent custerings (see Fig. 6). < ε Figure 6: To decide if the four thin ines are an ε-group we use their two representatives (in pink and purpe) and compute the error measure (see Section 3.3) between the farthest ines, which wi be represented by the hu. For each point (x), the points of the hu + (x) and (x) are obtained by taking the extrema intersections aong the norma with ( 1, 2 ) (see Fig. 7). 1 + Figure 7: The hu (in orange) of a ine (in purpe) representing a pair of ines ( 1, 2 ) (in back) is defined by the farthest points of ( 1, 2 ) aong each norma of (dashed ine). A the definitions given in the previous section are easiy extended by considering the two hus instead of the two ε-ines. Indeed, to decide if a pair of ε-ines ( 1, 2 ) is an ε- group we ony need to compute distances between pairs of points. By considering 1 +, 1,+ 2, 2 for the distance computation, we can determine the overapping zones and then the paths between 1 and 2 whie taking into account the two ε-groups they aready represent. Therefore, whie computing overapping zones between two ε-ines, the distance between 1 (x 1 ) and 2 (x 2 ) wi be taken as: - 2 max { + 1 (x 1), + 2 (x 2), + 1 (x 1), 2 (x 2), 1 (x 1), + 2 (x 2), 1 (x 1), 2 (x 2) } Note that the hu of an ε-ine is not defined on points p where there is no intersection with 1 and 2 and the norma at p. For those points, we project the cosest points of the two hus as shown in Fig. 8. This wi ony have an impact on the error computed on the hu as expained in the next section and our choice favors pairs of aigned ines (i.e., those with good continuation).

6 1 Figure 8: In paces where there is no intersection with the norma, we project the cosest points on the pair of ines Error measure of an ε-group In order to use a greedy agorithm we now need to choose which pairs of ε-ines we want to custer at each step. To this end we define an error measure. Intuitivey we want to custer the cosest ines first. By cosest we mean not ony spatiay cose but aso with simiar attributes. We first show how to compute the spatia error, then we expain how to incorporate an attribute error in order to orient the simpification toward a given appication. When computing the spatia error of a pair ( 1, 2 ) of ines, we want to favor pairs of ines that coud be custered with the smaest possibe ε. We thus define the spatia error E s( 1, 2 ) of an ε-group reative to the ε-ine chosen to represent it by the maximum thickness of the hu associated with. This heuristic favors the custering of the thinest groups first. This error is normaized between 0 and 1 using a division by ε: 2 E s( 1, 2 ) = max x [0,1] + (x) (x) /ε The user can aso define an attribute error measure e a(p 1, p 2 ) (normaized between 0 and 1) for a particuar attribute space if he or she wants to take it into account in the custering process. For the singe attribute we used in our impementation (i.e., coor), we found that a mean was better than a max to give a good estimation of the tota error between two groups. This gives the foowing attribute error: 1 E a( 1, 2 ) = e a( + (x), (x))dx 0 The spatia and attribute error measures are then cassicay combined in a mutipicative way to give the error measure E( 1, 2 ): E( 1, 2 ) = 1 (1 E s( 1, 2 )) (1 E a( 1, 2 )) The attribute error is ony computed for ε-groups, that is groups of ines that can be spatiay custered. In order to forbid custering if the attributes of the ines of the group are too different, we add the constraint that for an ε-group ( 1, 2 ) to be custered, it must satisfy E( 1, 2 ) < Cosed curves Most of this method hods for cosed curves and the agorithm is very simiar. However, we need to impement some additiona processes. First, the ines cosed at the scae ε are detected. Those are the ines whose endpoints are at a distance ess than ε. Moreover, when identifying paths, if the path configuration found in Fig. 5-(c) arises, the resuting ε-ine becomes cosed. 4. Impementation detais We have impemented the greedy iterative custering by an edge coapse agorithm appied on a graph whose edges represent pairs of ε-ines which are ε-groups Preprocessing input ines The ines we take as input can be of any kind. The ony constraints are that we need to sampe them. In our impementation, we use reguary-samped Catmu-Rom spines. For a distance computations invoving such sampes, we use an acceeration grid of ce size ε, aowing us to quicky find candidate sampes. In order to initiaize the agorithm, we need to convert an initia ine into an ε-ine. To do that, we foow, progressivey creating its two offset curves, and we spit as soon as it crosses one of the aready created offset curves. Note that this spitting process gives different resuts depending on the extremity at which one starts. We have not found any remarkabe difference in the resuts; however, one may want to choose a more symmetric way of spitting. After this initiaization step, each ine is its own hu Buiding the graph Once the input ines have been converted into ε-ines, we buid a graph with a node for each input ε-ine, and whose edges represents pairs of ines that can be custered, i.e. that are ε-group. A pair of ε-ines can ony be custered if it corresponds to one of the configurations shown in Fig 5-(b),(c). Thus, for an ε-ine pair, if there are more than two extremities at a distance greater than ε from the other ε-ine, we can reject it directy, saving a ot of computation time. In practice, we compute a hu for each potentia custer and store it on the corresponding edge aong with its error Updating the graph Then, at each step of the agorithm, we coapse the edge with minimum error and update the graph edges ocay. Coapsing an edge is done by creating a new node that stores the edge s ε-ine and hu. The coapsed edge is deeted and by definition of a hu, we ony have to inspect the edges incident to the coapsed nodes. Those edges are removed from the graph and new edges are created between the new node and its neighbors. We aso compute the attributes of the new ε-ine by ineary interpoating the attributes of the two origina ε-ines.

7 Finay, the two coapsed nodes are removed from the graph. But instead of deeting these nodes, we keep them in a history of coapse sequences which is stored as a tree under the newy created node. This gives us access to the underying input ines and the coapsing scheme of each custer. The agorithm stops when no more custers can be created. P. Bara, J.Thoot & F. Siion / 5. Resuts In this section, we give some resuts to iustrate the overa simpification process, i.e. both custering and geometric stages for each of the target appications: density reduction, eve-of-detai and progressive drawing. The geometric stage is ceary a more speciaized operation since the choice of the new ine to be drawn is eft to the chosen strategy. We impemented two standard", pre-defined strategies: Average ine: the new ine interpoates a the origina ines in the custer (with appication-defined weights); Most significant ine: the new ine is one of the origina ines, chosen according to an appication-defined priority measure (base on ength, nature... ). The former supports a broad range of simpification behaviors, whie the atter gives a simpe seection/deetion scheme. In the worst case, the tota process increases quadraticay with the number of input ines at a fixed scae parameter ε. In practice, for our exampes, it ranges from severa seconds to a minute. For each exampe we give the number of input ines and resuting custers. The simpification scae is shown by a circe of diameter ε. Density reduction Fig. 9 shows a straightforward iustration of our approach. Lines have been extracted from a scanned ine drawing. The user chooses a simpification scae ε and the ines are simpified. We appied an average ine strategy without smoothing the resuts, so that simpified ines exhibit the ε-ine of each group. Fig. 11 shows a simiar scenario that takes the coor attribute into account. The attribute error is a L a b coor distance. Fig. 12 shows the use of categories to separate ines of different nature: externa contour on one side and interna and suggestive contours [DFRS03] on the other side. In exampes coming from 3d renderings ike this one, we make use of object IDs and ine nature to detect categories automaticay. This aows for the use of two different geometric strategies: for the externa contour an average ine is drawn, whereas the ongest ine of each custer is drawn for the interna and suggestive contours. Moreover, the externa contour A video showing an oversketching session and two exampe LODs (incuding the tree of Fig. 10) is avaiabe at (a) (c) Figure 9: Density reduction: (a) The origina scanned and vectorized drawing: 357 input ines - (b) The resuting simpification: 87 custers - (c,d) Zoom on the above images. The scae ε is indicated by the circe in the upper eft corner. is simpified at a arger scae than the other ines. Resuting ines are better organized and keep the most saient features of the mode. Leve-of-detai Fig. 10 shows an exampe of a LOD sequence produced with our approach. Progressivey scaing down a drawing is equivaent to choosing an increasing ε. Thus we appy a series of simpifications with an ε step, each time starting from the previous, finer eve. Here again, two different geometric strategies are used: the average ine for the contour and the ongest ine for the hatchings. To do that, we created five categories by hand: one for the contours, and one for each of the four orientations for hatchings. Note that athough no particuar treatment was appied to preserve tone across simpifications, this resut is quite convincing. Tone preservation coud be expicity incuded in the method at the geometric stage, by choosing appropriate ine attributes such as width and/or coor. Progressive drawing Fig. 13 shows a drawing sequence using our progressive drawing too. Here, the custering agorithm is appied iterativey: The user chooses a sensitivity ε and draws a sketched ine over an initia drawing; the ines are then simpified; and finay, the resuting ines constitute the initia drawing for the next step. This too requires (b) (d)

8 376 input ines 269 custers 134 custers 81 custers Figure 10: LOD: A series of LODs made by progressivey increasing ε. Using different categories and geometric strategies prevents undesired hatching ines from merging. Compare the sma resized images with (right) and without (eft) simpification. an additiona feature: we ony want the simpification to be done between initia ines and the new sketch. Thus the input ines are organized in two sets: the initia ines and the new sketched ine. During the custering, ony the edges between pairs of nodes that ie in different sets are buit. Finay we choose a priority-based strategy because we want the ast drawn ine to have a greater priority than initia ines. In practice, that consists of using an average ine strategy, giving greater weights to sampes beonging to the ast drawn ine. This is made possibe by the history tree stored at each custer. We found this too to be very intuitive, particuary for modifying ines coming from 3d renderings or extracted from images. 6. Discussion In this paper we opted to remain very genera, trying to find the common properties of some target simpification methods. However, it is cear that such a ow-eve method can sti be speciaized to adapt to other specific appications. In particuar, we beieve that the separation of the custering and geometric stages is crucia for a simpification methods. Other attributes than coor coud be used in the attribute error definition. However, we did not consider input ines exhibiting wigging patterns and impicity assumed that they come at an appropriate scae. The probem of extracting the so-caed natura scae of a ine has been previousy adressed (e.g. [Ros98]). Our method is invariant under rotation, scae, and transation, since it operates ony on eucidean distances between pairs of points. However it has two imitations: it is not transitive and prevents simpifying forks. The former means that simpifying a drawing at scae ε 1, then simpifying the resut at scae ε 2 > ε 1 is not guaranteed to provide the same resut as a direct simpification at scae ε 2. However, this is not a probem in the appications we envision, for instance generating a discrete set of LOD representations. The atter assumes that the probem of forks is rather separate from geometric custering (appearing at a higher eve of processing and depending on the appication) and thus is eft as a post process. The choice of a greedy agorithm for custering impies that we ony reach a oca optimum in genera. This turns out to be sufficient in practice for the appications we have tested. Other optimization techniques coud be used if reaching a goba optimum is important. The evauation of a simpification method for ine drawings is not an easy task. Indeed, there is no simpe and obvious quaity measure for a simpified drawing. Visua evauation invoves a number of high-eve interpretation processes, which are difficut to mode and quantify. Our approach offers the convenience of a guaranteed geometric criterion: the resuting drawing is within a distance ε from the origina drawing. The direct evauation of the resut is the number of custers. However, in an attempt to provide finer evauation toos we identified two other criteria. First, the reduction in the number of ines composing the drawing; Second, the variation of the tota arc-ength in the drawing. Both are strongy reated to the geometric strategy chosen for an appication: keeping a ine per custer ceary decreases the tota number of ines, and the arc-ength may strongy vary depending on the new ines created. For instance, in Fig. 9 the number of ines was divided by 4 and the arc-ength reduced by 30%. In the exampes shown, we observe that a purey distancebased simpification shoud generay not be appied to a ines of the drawing at once, because there are categories of ines that shoud not be custered. one exampe is ines

9 P. Bara, J.Thoot & F. Siion / Figure 11: Top: input drawing. Midde: simpified drawing without taking coor error into account during the custering stage. Bottom: taking coor error into account better preserves the origina drawing (see the fence and the tree, the trunks and the eaves...). depicting different objects paced near each other. Segmenting the drawing and appying the simpification agorithm to each category is better for the scope of an automatic process. Naturay this raises the question of how to segment or define the categories automaticay, which is beyond the scope of this paper. Finay, we think that our approach coud be extended to animation. The idea woud be to guide the custering stage temporay, not ony to ensure tempora coherence, but aso to custer ines that go together across time. This coud be accompished by incorporating another perceptua grouping criterion: common fate, which states that the visua system tends to group eements with simiar veocity. 7. Future work Our approach can be extended in many ways. Forks, where a ine separates into two distinct ines, are not handed expicity in our technique. We pan to take them into account in the density reduction appication, aong with new geometric strategies. We pan to extend our LOD system to more eaborate transitions between eves: since we keep a the history of aggomerations, we have a the information needed to rec The Eurographics Association (a) (b) (c) (d) Figure 12: Simpification of a 3d rendering: (a) 3d mode and its ine rendering using sihouettes and suggestive contours (531 input ines) - (b) Simpification without any category (256 custers) - (c) Using two categories (externa and interna contours) each with a different scae and geometric strategy (294 custers) - (d) Same resut in a caigraphic stye. aize geometric morphs instead of bends. Moreover, we wi add new features to our oversketching too (such as aignment, anchoring, etc.). Another interesting issue ies in the couping of our method with contour detection in images, with appications in medica imagery and image-based rendering. Our approach is we adapted to these appications since it is abe to incorporate any kind of data associated with the extracted ines (gradient, coor, etc.) and to take them into account in the simpification process. Finay, a ong-term goa is to adapt our approach to the simpification of animated ine drawings, taking into account the observations made at the end of Section Concusions We have presented a new, generic approach to the simpification of ine drawings, that supports a arge variety of appications. The custering stage identifies groups of ines that capture the morphoogica structure of the origina drawing; the geometric stage buids the actua ines that wi represent this structure in the fina drawing.

10 Figure 13: Oversketching: A 3d mode has been rendered in a ine drawing stye. The user adds new ines (in red) which are custered with the od ones; the scae ε (gray circe) can be changed at each step. The ow eve and generic nature of this process makes it a soid foundation for many appications. We have demonstrated three possibe scenarios: Density reduction of a set of ines where input ines, possiby cassified in categories, are repaced by a smaer set of ines; an automatic eve-ofdetai system for ine drawings with specific behaviors for sihouette ines and hatching groups; and a progressive drawing too, which provides an intuitive and fexibe interaction environment where the user creates or modifies existing ines with drawing gestures on the canvas. Acknowedgement We woud ike to thank Gies Debunne for his hep on the making of the video. This paper benefited greaty from suggestions given by the members of the ARTIS team and Lee Markosian. This work was supported in part by Region Rhone-Apes (DEREVE project and EURODOC grant). References [Bau94] BAUDEL T.: A mark-based interaction paradigm for free-hand drawing. In UIST 94: Proceedings of the 7th annua ACM symposium on User interface software and technoogy (New York, NY, USA, 1994), ACM Press, pp [DFRS03] DECARLO D., FINKELSTEIN A., RUSINKIEWICZ S., SANTELLA A.: Suggestive contours for conveying shape. ACM Transactions on Graphics 22, 3 (Juy 2003), [DS00] DEUSSEN O., STROTHOTTE T.: Pen-and-ink iustration of trees. Proceedings of SIGGRAPH (2000). 2 [EZ96] ELDER J. H., ZUCKER S. W.: Computing contour cosure. In ECCV (1) (1996), pp [GDS04] GRABLI S., DURAND F., SILLION F.: Density measure for ine-drawing simpification. In Proc. of Pacific Graphics (2004). 2 [GG01] GOOCH, GOOCH: Non-Photoreaistic Rendering. AK-Peters, [GTDS04] GRABLI S., TURQUIN E., DURAND F., SIL- LION F.: Programmabe stye for npr ine drawing. In Rendering Techniques 2004 (Eurographics Symposium on Rendering) (june 2004). 2 [IMKT97] IGARASHI T., MATSUOKA S., KAWACHIYA S., TANAKA H.: Interactive beautification: A technique for rapid geometric design. In UIST (ACM Annua Symposium on User Interface Software and Technoogy) (1997), pp [KMM 02] KALNINS R. D., MARKOSIAN L., MEIER B. J., KOWALSKI M. A., LEE J. C., DAVIDSON P. L., WEBB M., HUGHES J. F., FINKELSTEIN A.: WYSI- WYG NPR: drawing strokes directy on 3d modes. In SIGGRAPH 2002 (2002). 2 [PHWF01] PRAUN E., HOPPE H., WEBB M., FINKEL- STEIN A.: Rea-time hatching. In SIGGRAPH 2001, Computer Graphics Proceedings (2001), Fiume E., (Ed.), pp [PS95] PREIM B., STROTHOTTE T.: Tuning rendered ine-drawings. In WSCG 95 (February 1995), pp [Ros94] ROSIN P.: Grouping curved ines. In 5th British Machine Vision Conf (York, 1994), pp. pp [Ros98] ROSIN P. L.: Determining oca natura scaes of curves. Pattern Recognition Letters 19, 1 (1998), [SALS96] SALISBURY M., ANDERSON C., LISCHINSKI D., SALESIN D. H.: Scae-dependent reproduction of pen-and-ink iustrations. Computer Graphics 30 (1996), [Sau03] SAUND E.: Finding perceptuay cosed paths in sketches and drawings. IEEE Trans. Pattern Ana. Mach. Inte. 25, 4 (2003), [SS02] STROTHOTTE T., SCHLECHTWEG S.: Nonphotoreaistic computer graphics: modeing, rendering, and animation. Morgan Kaufmann, San Francisco, CA, USA, [WM04] WILSON B., MA K.-L.: Representing compexity in computer-generated pen-and-ink iustrations. In NPAR (2004). 2 [WS94] WINKENBACH G., SALESIN D.: Computergenerated pen-and-ink iustration. Proc. SIGGRAPH (1994). 2 [ZISS04] ZANDER J., ISENBERG T., SCHLECHTWEG S., STROTHOTTE T.: High quaity hatching. Computer Graphics Forum 23, 3 (2004), [ZT98] ZIOU D., TABBONE S.: Edge detection techniques - an overview. Internationa Journa of Pattern Recognition and Image Anaysis 8 (1998),