Illustrating Smooth Surfaces Aaron Hertzmann Denis Zorin New York University Presented by Mark Blackburn, Fall 2005

Illustrating Smooth Surfaces

Aaron Hertzmann

Denis ZorinNew York University

Presented by Mark Blackburn, Fall 2005

Organization

1. Motivation

2. Contributions

3. Related Work

4. Process

5. Results

6. Conclusion & Future Work

Motivation

• Silhouette drawing often insufficient for drawing objects that are complex or free-form.

• A smooth object may have no visible silhouette lines except the outer silhouette and all the information inside the silhouette may be lost (cf. Figure 1)

• Adding lines can convey the complexity of the shape

• Line drawing often conveys info better than a photograph. Figure 1

Contributions

Three Algorithms1. Silhouette detection

2. Cusp detection and segmentation

3. Computing smooth direction fields

Related Work

• Image-space vs. Object-space NPR• Silhouette detection

– Randomized algorithm (fast, but does not guarantee to find all surfaces)

– Gauss map (only works in orthographic projection)

• This paper presents an algorithm that is fast, deterministic, and is applicable to orthographic as well as perspective projections

Related Work

• Computing smooth direction fields:– Parametric lines on NURBS

• Parameterization does not exist for many types of surfaces

– Principal Curvature Hatching• Can’t be reliably or uniquely computed at many surface points

– Intersections of the surfaces with planes• Requires segmentation of the surface into parts where different

groups of planes are used

• Plane orientations computed using skeletons relate only indirectly to the local surface properties

Process OverviewInput: Polygonal Mesh

(Stage 1) Hatch Direction Field: Defines a view-independent cross-field that can be used later to generate hatches

(Stage 2) Silhouette Curve computation: Computes all curves, creases, cusps, and determines the visibility and segmentation of all

silhouette curves

(Stage 3) Hatch Generation: Divides surface into four levels of brightness (highlights, midtones, shadows, and undercuts) and generates

hatches for each area of the surface

Output: Illustrated Image

Stage 1:Hatch Fields

Determine hatch direction fields for entire surface

• Choice of direction field algorithm guided by these three observations:

1. Surface geometry is rendered best by principal curvature directions on cylindrical surfaces

2. Isoparametric lines work well as curvature directions when a parameterization exists and is close to isometric

3. Artists tend to use relatively straight hatch lines, even when the surface has wrinkles. Smaller details are conveyed by varying the density and the number of hatch directions.


Simple requirements for hatching fields• In areas where the surface is close to parabolic, the

field should be close to principal curvature directions

• On the whole surface, the integral curves of the fields should be close to geodesic

• If the surface has small details, the field should be generated using a smoothed version of the surface


Four steps to determine hatch direction field1. (Optional) Create a smoothed copy of the

original mesh (User specified)2. Identify areas of surface which are

sufficiently close to parabolic3. Initialize field over the surface by computing

principal curvature directions4. Fix the quasi-parabolic areas and *optimize*

the rest of the vertices.


(a) Silhouettes alone do not convey the interior shape of the surface.

(b) Raw principle curvature directions produce an overly-complex hatching pattern.

(c) Smooth cross field produced by optimization.


(d) Hatching with the smooth cross field.

(e) Very smooth cross field produced by optimizing all directions.

(f) Hatching from the very smooth field.

Stage 2: Silhouette Curves

Definition

• A silhouette drawing includes only the images of the most visually important curves on the surface: boundaries, creases, silhouette lines, and self-intersection lines


• Problem: There are significant differences between the silhouettes of smooth surfaces and their approximating polygonal meshes

• For polygonal meshes, complex cusps, where several silhouette chains meet, are stable (do not disappear when the viewpoint is perturbed)

• For smooth surfaces, the only type of stable singularity is a simple cusp.


Examples


• Silhouette detection:– First, we find the silhouette set– Silhouette set is the set of points p of the surface such

that: g(p) = (n(p) • (p-c)) = 0– We compute an approximation to g(p) by computing

the true surface normal and g(p) for each vertex and then linearly interpolating the values of the function.

– The zero set of this function will consist of line segments inside each triangle of the polygonal approximation.

Stage 2: Silhouette Curves• Cusp Detection: cusps are the points where the

tangent plane is parallel to the view direction• Cusp function:

C(p) = K1(v • w1)2 +K2(v • w2)2

K1, K2Principle curvatures

v View dir (Viewpoint – surface point)

w1, w2Principle curvature directions


• Cusps are contained in the intersection set of the two families of curves: the zero set of g(p) and zero set of C(p). Cusps = { g(p) = 0 C(p) = 0}

Silhouette curve

Cusp zero set


• Optimization: Eliminate need to traverse entire mesh.

• Duality Map: Map surface M M’

• M’ can be obtained by mapping each point of M to a homogeneous point N = [n1, n2, n3, - (p • n)] where n = [n1, n2, n3,0] is the unit normal at p.


• If we let C = [c1, c2, c3, c4] be our viewpoint in the homogeneous form, then the silhouette of the surface consists of all points p for which C is in the tangent plane at that point.

• Finding the surface is reduced to the problem of intersecting a plane with a surface for which many space-partition-based acceleration techniques are available.


Yellow arrow vector c from origin to viewpoint

Yellow circle intersection of plane ┴ to c , through origin and unit

sphere

Cyan Points a silhouette point and its dual

Dual Map

Stage 2: Silhouette CurvesPlanar Projections


Fast Silhouette Detection1. For each vertex p, with normal n, we compute

the dual position N = [n1, n2, n3, - (p • n)]

2. Normalize each dual position N using l∞-norm

3. Each triangle of dual mesh is assigned to a list for every 3D space in which it has a vertex

4. An octree is constructed for each 3D face and the triangles assigned to this face are placed in the octree

Stage 3: Hatch Generation

• Surface is separated into four levels of hatching:

Level Hatching

Highlights & Mach Bands None

Midtones Single hatching

Shadowed Regions Cross-Hatching

Undercuts Dense Cross-Hatching


(a) 3 Densities: highlights, midtones, and shadows, corresponding to 0, 1, and 2 directions of hatches

(b) Undercuts

(c) “Mach bands.” Undercuts and Mach bands increase contrast where surfaces overlap.


• Basic rules:– If there is an undercut, on the other side of the

silhouette from a fold, a thin area along the silhouette on the fold side is not hatched

– Undercuts are densely hatched– Hatches are approximately straight– Optionally, hatch thickness within each density

level can be made inversely proportional to lighting


• Hatching has several user tunable parameters:1. Basic hatch density

2. Hatch density for undercuts

3. Threshold for highlights

4. Threshold for transition from single to cross hatch

5. Max hatch length

6. Max hatch deviation from initial direction

• Varying these parameters has considerable effect both on the appearance of the images and the time required by the algorithm


• Hatch placement process:1. Identify Mach bands and undercuts

2. Cover single and cross-hatch regions with cross hatches, and add extra hatches to undercut regions

3. Remove cross-hatches in the single hatch regions, leaving only one direction of hatches


• Identifying Mach Bands– Step along each silhouette and boundary curve– Use ray test near each curve point to determine

if fold overlaps another surface– Undercuts and Mach Bands are indicated in a

2D grid by marking every grid cell within a small distance of the fold on the near side of the surface as a Mach Band


• Cross-Hatching:– Create evenly spaced cross hatches on a surface– Creates a queue of surface curves– Dequeue each curve and seed it with cross-hatches by

tracing direction of field– Hatches are also seeded parallel to other hatches

(distance determined by tunable param)– Hatches are seeded perpendicular to all curves– A hatch continues along a curve until it terminates in a

critical curve, deviates from direction more than ‘Max’ param, or until it comes near a parallel hatch


• Hatch Reduction– Once we have cross-hatched all hatch regions,

we remove hatches from the single hatch regions until they contain no cross-hatches

– Algorithm implicitly segments the visible single-hatch regions into locally consistent single hatching fields

– Use a breadth-first traversal of hatches.

Results

ResultsMathematical surfaces

Results

Results

Conclusion

• Advantages:– Beautiful images that mirror pen-and-ink illustrations– Technique can be used in orthographic or perspective

projections– User tunable

• Disadvantages:– Performance varies significantly based on parameters

and model (seconds to minutes)– Parameters have to be chosen carefully which may take

several iterations to get desired result

Future Work

• Improvements should be made to the silhouette visibility algorithm

• Performance speedups are possible

• Reduce the number of user parameters

• Hatch reduction algorithm could be more robust

References

• Aaron Hertzmann and Denis Zorin, "Illustrating Smooth Surfaces," Siggraph 00, pp. 517 - 526.

Documents

Illustrating Smooth Surfaces Aaron Hertzmann Denis Zorin New York University Presented by Mark Blackburn, Fall 2005