1. SIGGRAPH 2007 Tilke Judd Frédo Durand Edward Adelson

2.  Introduce a new definition of feature lines to express three-dimensional shape based on perceptual observations. ◦ Human perception is sensitive to the variation of shading.  Little affected by lighting and reflectance modification  Focus on normal variation ◦ View-dependent lines better convey smooth surfaces  Define view-dependent curvature  Apparent ridges as the loci of points that maximize a view- dependent curvature

3.  An artist will often use a line drawing to convey an object’s shape in a manner that is independent of BRDF and lighting. ◦ Not photorealistic ◦ Line drawing captures the essential visual properties in a compact and abstract manner.

4.  Line drawing is a popular topic in NPR ◦ Where do you put the lines?  Bounding contour, silhouette  Discontinuities of surface  Bumps, dips, and undulations of varying geometry  Not very clear how to depict with lines

5.  The object is rendered to form an image or geometric buffer to be processed by image processing methods such as edge detection  The result is often visually pleasing ◦ Suffers from low precision due to the loss of 3D scene information ◦ Unsuitable for additional processing  Saito and Takahashi 1990; Decaudin 1996; Hertzmann 1999; Pearson and Robinson 1985; Lee et al. 2007

6.  Finds curves that have special properties in terms of the differential geometry of the surface  Captures important object properties ◦ Do not make natural looking line drawings ◦ Locked to the object surface ◦ Tend to look overly sharp  Koenderink 1990; Ohtake et al. 2004; Interrante et al. 1995

7. OriginalRidge & Valleys

8.  It is also possible to define curves that have view dependence ◦ Silhouette - contours ◦ Suggestive contours [DeCarlo et al. 2003, 2004]  Locations which are almost contours, and correspond to true contours in nearby viewpoints  Curvature in view direction = 0  Fails to capture convex regions

9. OriginalSuggestive Contours

10.  Motivated by perceptual considerations  Human observers are highly sensitive to line-like and edge-like features ◦ Ex) Points of high luminance variation. ◦ These locations tend to be stable across different choices of BRDF and illumination  Our features are view dependent ◦ The lines are drawn at the same places that “line detectors” and “edge detectors” are likely to fire on the 2D rendered image of the same object  Where do you put the lines? ◦ Draw a line when the surface normal is changing at a locally maximal rate with respect to image position ◦ “Apparent ridges”

11. OriginalApparent Ridges

12.  Line is to be drawn independent from BRDF and illumination ◦ Surface normal - Parabolic lines  Blocky  Not appropriate to be used in animations ◦ View dependent stable conditions  Observations of Flemming et al. [2004]  Local orientation structure of rendered objects was similar across multiple choices of BRDF and environment map ◦ Rapid luminance changes occur at points where the angle of the surface normal is changing rapidly  Points in the image with maximal view-dependent curvature will usually contain maximal luminance gradients

13.  n(m) : the outward facing unit normal to the surface at a point m  Curvature operator S at point m ◦ S(r) = D r n  Directional derivative of the normal along vector r in the tangent plane

14.  n(m) : the outward facing unit normal to the surface at a point m  Curvature operator S at point m ◦ S(r) = D r n  Directional derivative of the normal along vector r in the tangent plane  Symmetric 2 x 2 matrix  Principal curvatures are eigenvalues of S  Eigenvectors of S are principal directions  Positive curvature – ridges / Negetive – valleys  Higher-order derivative must be negetive to ensure the ridge is a maximum of curvature

16.  View-dependent curvature is how much the surface is seen to bend from the viewpoint ◦ Q(s) = D s n’  Where n’(m’) = n(P-1(m’))  M : Object in 3d space  V : Screen Plane  m’∈ V  P : Parallel projection which maps points m ∈ M to m’∈ V  We do not project the normal on to the screen space  Since we are motivated by the shading which is based on the object space normal

18.  Given a choice of basis (r1, r2) for the tangent plane and (s1, s2) for the screen plane takes tangent vectors at m to vectors in V

19.  Q = SP -1 ◦ Where the basis of the tangent plane chosen for expressing S and P are the same ◦ Define maximum view-dependent curvature as ◦ Or

21.  Adds view dependency to the traditional definition of curvature ◦ Where the object normal points towards to the screen, curvature and the view-dependent curvature are the same ◦ Where the object turns away from the screen plane, the view-dependent curvature becomes much larger  View-dependent principal direction is shifted towards the view vector

22.  The loci of points at which the maximum view- dependent curvature q1 assumes a local maximum in the principal view-dependent curvature direction t1 ◦ Which is the loci of points where D t1 q1 = 0 ◦ Keep only the maximum points by selecting those whose higher order derivative is negative  View-dependent curvature is always positive ◦ Still captures ridge-like and valley-like features ◦ Can distinguish between features by the sign of the object space curvature

24.  We leverage standard techniques to estimate the curvature S at each point on the mesh ◦ Estimating curvatures and their derivatives on triangle meshes  Rusinkiewicz 2004; Symposium on 3D Data Processing, Visualization, and Transmission ◦ Q = SP -1 ◦ Approximate a perspective camera using a local parallel projection for P  Projection line for each vertex to be the line between the viewpoint and the vertex

25.  We estimate D t1 q1 using finite differences ◦ Average the finite difference between p and the two w points  w, w’ – points on the edges of triangles adjacent to p in the direction t 1  View-dependent curvature of w points  Linear interpolation

26.  To make t 1 field consistent across the mesh, we flip t 1 to point in the direction of the positive derivative, where view-dependent curvature is increasing

27.  t 1 of both vertices along an edge point in the same direction ◦ No zero crossing  If not, ◦ Interpolate the location of the zero crossing using the values of the derivatives at each vertex

28.  We only want to draw lines at maxima ◦ Drop a perpendicular from each vertex to the zero crossing line ◦ If the positive t1 at each vertex makes an acute angle with the perpendicular, then the zero crossing is a maximum

29.  The result yields a lot of lines ◦ we want points where the magnitude of the view- dependent curvature is locally a maximum ◦ AND which this maximum has a high value  Eliminate lines based on a threshold of the view-dependent curvature ◦ Scaled by the feature size of the mesh (average edge length)

30.  Ridges and valleys and suggestive contours are quick to compute ◦ Curvature is stable across viewpoints ◦ Can be precomputed for an object  Unoptimized code on a 2.33 Ghz Intel Core 2 Duo Mac ◦ Real time for small meshes ◦ ~1.5 seconds for 50,000 polygon meshes ◦ ~9 second for 250,000 polygon meshes  A limitation of apparent ridges is that they involve higher-order derivatives, which makes them prone to numerical noise in digital meshes

31.  For fair comparison ◦ used constant stroke width and sharp cutoffs for line ends ◦ Thresholded each image to match the number of gray pixels per image  Contours are located where the normal is perpendicular to the view direction ◦ view-dependent curvature approaches a maximum of infinity because of projection  so contours are extracted as apparent ridges  Some of other methods does not  must be combined with contours

32.  Apparent ridges share the same definition with ridges and valleys modified by a projection ◦ Ridges and apparent ridges are similar when the effect of projection is small ◦ Differs on the part where the object turns away from the viewer  Ridges and valleys are fixed on a object, they can appear as artificial surface markings and produce a boxy look  Apparent ridges are also defined in cases where ridges are ill-defined

33. Occluding ContourApparent Ridges

34.  Suggestive contours look at an extremum of the normal ◦ Apparent ridges look at the extremum of the normal variation  Look at curvature in the direction of the view vector ◦ While our approach look at curvature in the direction of t1 ◦ These directions are defined differently, but sometimes they align  Find where the curvature in the direction is 0 ◦ While we find where the curvature is maximum

35.  It is hard to tell which method is clearly better ◦ The methods are significantly different  Suggestive contours and apparent ridges trade off situations where they draw single or double lines

36.  Both suggestive contours and apparent ridges have an attribute of extending contour lines  Some important features in convex regions of an object are not conveyed by suggestive contours

37.  Given head-on illumination with a single light source at the viewpoint and lambertian shading ◦ Suggestive contours are drawn in the shaded areas of an object  Only make sense given a certain shading setup ◦ Apparent ridge lines are drawn where are important independent of the light direction.

40.  Did a Monte-Carlo experiment where a diffuse surface is rendered from a given viewpoint with thousands of random lighting configurations ◦ The average output of a Canny edge detector [Canny 1987] on those thousands of images matches remarkably well the lines extracted with our technique.

41.  Similar experiment with real photographs with flash illumination ◦ Canny edge detection ◦ Apparent ridges of 3D scan of the object ◦ we can see that the two extraction approaches agree

42.  Introduced apparent ridges for non- photorealistic line drawings  Produce visually pleasing line drawings ◦ Capture important information about an object’s shape ◦ independent of a specific lighting situation ◦ Where ridges and valleys do well, apparent ridges appear in similar locations  If ridges and valleys look boxy, apparent ridges modify them to be more perceptually pertinent ◦ Related to, but distinct from, suggestive contours  View dependent  Both produce pleasing images, but in many cases apparent ridge images are more appealing