Noname manuscript No. (will be inserted by the editor) Gershon Elber ¢ Elaine Cohen Probabilistic Silhouette Based Importance Toward Line-Art Non-Photorealistic Rendering Abstract When pictorial information is presented, de- The search for silhouette edges of a polygonal model tails of importance are typically emphasized. These in- can be exhaustive. It may require searching all the front- clude discontinuities in the geometry, highly curved re- facing polygons in a model for back-facing neighbors or gions, silhouettes, etc. vice versa. This test is a time-consuming task that is lin- This work analyzes the probability that certain smooth ear in the number of polygons n, whereas, in general, 1 surface regions or polygonal edges possess silhouettes. a polygonal model approximating a C continuousp ge- This probability analysis is then associated with the vi- ometry with n polygons will present only n silhouette sual importance of the local neighborhood, which is capa- edges [1]. In [1,17], output-sensitive algorithms that ap- ble of capturing discontinuities and highly curved regions. proach O(m) time complexity, where m represents the A non-photorealistic rendering technique is subseq- size of the output or the number of silhouette edges, were uently proposed to take advantage of the silhouette-based presented for orthographic and perspective views, respec- importance. Based on this importance analysis, we present tively. Markosian et al. [14] presented a probabilistic al- a completely automatic algorithm that creates line-art gorithm that randomly examines whether edges in the that captures visual features in the model in an appeal- model are silhouettes. Once a silhouette edge is detected, ing way. the algorithm serves as a seed for tracing the silhouette Key Words: Silhouettes, NPR rendering, Gaussian curve along the polygonal model. This algorithm does Sphere, Visibility Determination, Feature and Suggestive not guarantee that all silhouettes will be found but the Contours. longer the silhouette curve is, the greater the likelihood of its being detected. Image-based extraction of silhouette edges [18,20] is becoming popular due to the availability of graphics ren- 1 Introduction dering hardware (GPUs) that enables extraction of sil- houette curves in real time. Image processing is employed Silhouette curves play a major role in many non-photo- to detect locations at which front and back facing poly- realistic (NPR) line drawings. Since silhouette curves pro- gons share the same depth; that is, along the silhouettes. vide intuitive cues to the shape of an object, a large body Extraction of silhouettes from freeform surfaces is typ- of research has been devoted toward e±cient and accu- ically performed either by tessellating the freeform shape rate extraction of such curves. Algorithms have been pre- into polygons and using one of the methods discussed above or by fetching the silhouette edges directly from sented to extract silhouette curves from polygonal meshes 1 [1,12,14,18], parametric freeform surfaces [8], implicit free- the surface. Let S(u; v) be a regular C rational para- form surfaces [2,16] and even iso-surfaces of volumetric metric surface and let V be the viewing direction. The data sets [13,21]. silhouette curves can now be characterized via the fol- lowing rational constraint: ¿ À @S(u; v) @S(u; v) Gershon Elber £ ;V = 0: (1) Computer Science Department @u @v Technion, Israel Email: [email protected] In general, the solutions for Equation (1) can be found Elaine Cohen only numerically. Alternatively, divide-and-conquer sub- Computer Science Department division algorithms can be used [8]. Tracing the silhouette University of Utah, SLC, Utah, USA curves on the surfaces is common for implicit representa- Email: [email protected] tion solutions. 2 Gershon Elber, Elaine Cohen While the body of silhouette extraction methods is liable to produce great changes in the parabolic set. Re- large, the converse, as presented below, is rarely investi- cently, curvature-based feature lines were sought in the gated: context of non-photorealistic rendering [3]. Zeros of the normal curvature in the direction of a projection of the Query 1 What is the probability of a small sur- view direction onto the tangent plane, denoted as the ra- face region (a polygon edge) containing (being) a dial curvature, were suggested as features of interest. This silhouette, given an arbitrary viewing direction? approach of [3] examined the possibility of having silhou- Answering Query 1 should shed some light on the vi- ettes under a small perturbation of the viewing direction sual importance of this small region or edge. Because sil- and is closely associated with the extraction of parabolic houette curves are visually important, we expect that re- locations. gions (edges) with a high probability of containing (serv- Second order di®erential properties, such as lines of ing as) silhouettes will also be visually important. curvatures and radial curvatures, and third order di®er- A goal of this research is to quantify the visual impor- ential properties that are needed to detect creases, may tance of surface regions based on a probabilistic silhouette be di±cult and numerically unstable to compute, espe- analysis for both polyhedra and continuous freeform ge- cially for piecewise linear polygonal meshes. In this work, ometry. The potential importance of the dihedral angle we show that one can go a long way in detecting features between adjacent polygons has been recognized in the of visual importance with ¯rst order analysis of polyg- past; for a recent example, see [22]. In [14], the edges of onal meshes, taking full advantage of a probabilistic sil- the model are sorted based on their dihedral angle, so as houette analysis. Herein, all features, either soft, rounded to increase the probability of ¯nding the silhouette edges. or sharp, are captured by conducting a global silhouette Interestingly enough, the claim was made in [14], without viewing analysis. proof, that the probability of an edge to be a silhouette The rest of this work is organized as follows. In Sec- is proportional to ¼ ¡ θ where θ is the dihedral angle in tion 2, a probabilistic analysis of silhouette curves is car- radians. Theorem 1 in the next section substantiates this ried out for both polygonal and freeform parametric ge- claim and makes it more precise. ometry. In Section 3, we make a general importance tex- Silhouettes curves are considered important because ture map that can be used in any rendering scheme, con- they convey a shape's visual cues. However, they can fail tinuously prescribing the important surface regions. We to provide signi¯cant shape cues when crucial features then demonstrate its possible use in line-art NPR render- are all front- (or all back-) facing. For example, consider ings. In Section 4, we present some examples and ¯nally, looking at a human face head on. Very few, if any, silhou- we conclude in Section 5. ette curves would be visible from this speci¯c perspective. Researchers who recognized this drawback have searched for ways to augment the silhouette drawings with other 2 Silhouettes and Visual Importance curves or contours that provide more information about the shape. For example, [23] examined a combination of As silhouette curves will play a major role in our forth- silhouette curves, boundary curves, creases that depicts coming discussion, we give an outline of the essence of folds and convex and concave regions. silhouette curves. Consider a C1 continuous surface and Curvature properties, such as lines of curvature [11], let V be the viewing direction. Then, have been examined as \feature-strokes" but the visual cues they provide are not always as intuitive as silhou- De¯nition 1 Point P 2 S(u; v) is a silhouette point if ette curves. While lines of curvature might nicely por- the normal to the surface at P , N, is orthogonal to V , tray the local shape of the geometry, they can also be i.e. hN; V i = 0. confusing, for example, along umbilicals. Further, since We propose an approach for the enhancement of fea- lines of curvature are de¯ned along the entire geometry, tures in NPR-based rendering by analyzing their impor- they do not capture signi¯cant geometry such as that of tance, for both the discrete (polygonal) and continuous creases or ridges. This vital geometry could be sought at domains. Let V be a point on the unit sphere, S2. V will the extremum locations of the principal curvature while denote one random viewing direction out of all possible moving along the line of curvature. This approach to the viewing directions, or points on S2. We de¯ne the impor- computation of creases was used in [24] for registration tance of a surface region as follows. of volumetric data sets, and might be helpful in NPR applications as well. De¯nition 2 The importance of a region (an edge) is Other attempts to exploit curvature properties in- measured by its probability of containing (being) a sil- clude the use of parabolic lines [9], or locations on the houette from a random viewing direction V . surface where the Gaussian curvature vanishes. While parabolic lines are intrinsic to the geometry, they are also This probabilistic view will take into account sharp very sensitive to noise. Small perturbations of the geom- corners as well as emphasize continuous surface regions etry, especially if the geometry is almost developable, are that are highly curved, and hence have a high probability Probabilistic Silhouette Based Importance Toward Line-Art Non-Photorealistic Rendering 3 of being or containing a silhouette. As a result, looking at of this point, the normal to the surface does not change a human face head on, highly curved creases and ridges due to the C1 continuity and the Lipschitz conditions.