Suggestive Contours Eric Jardim ericjardim@gmail.com IMPA - - - PowerPoint PPT Presentation

NPR: Lines

Suggestive Contours

Eric Jardim

ericjardim@gmail.com

IMPA - Instituto Nacional de Matemática Pura e Aplicada

Suggestive Contours – p. 1

Drawing with Lines

• Lines are the base of drawing
• They can convey shape, without being photo-realistic

[Flaxman 1835]

• If you only have lines to draw a scene (no colors, no

shading, etc), which lines should be drawn?

• Which lines an artist would pick to draw this same scene?
• If we have a 3D scene model and a viewpoint, which lines

we should extract from the geometry?

Suggestive Contours – p. 2

Learning with the artists

[Flaxman 1835]

Suggestive Contours – p. 3

Understanding the lines of the artist

• (1) Lines that convey “boundaries” of the geometry
• (2) Lines that convey “curvatures” on the geometry
• (3) Abstract lines (out of our scope)

Suggestive Contours – p. 4

Contours: definition

• Contours (or silhouettes) are the set of points of a surface,

that are “almost invisible” from a given viewpoint

• On a smooth surface, a point p is a countour point if it is

visible and n(p) · v(p) = 0

• On polyhedra, contours are those points where every

neighbourhood have front and back faces near it.

backfacing frontfacing n · v = 0 but is invisible n · v = 0

Suggestive Contours – p. 5

Contours

• Contours are first order curves and were largely studied
• They capture most important features of the geometry, but

not all of them

• This leads us to investigate high order derivative curves

Suggestive Contours – p. 6

Math and aesthetics

[Hilbert and Cohn-Vossen 1952]

• Felix Klein believed that parabolic lines make a connection

between math and aesthetics

Suggestive Contours – p. 7

Motivation

• Look at these curves. They are clearly not contours, at least

not from this viewpoint

• But they might be contours of a nearby view.
• Let us work with the hypothesis that they are “almost

contours”

Suggestive Contours – p. 8

Intuitive definition: almost contours

• In this figure, moving the viewpoint from c to c′, we see two

kinds of new contour points.

• The first (q′) is the contour of q that “slid” over the surface
• The other singular kind (p), that is an inflection of curvature,

will suddenly appear

• We are interested in this second kind, leading us to define

suggestive contours formally

Suggestive Contours – p. 9

Definition 1: local minima of n · v

• We define this point to be part of the suggestive contour if it

is a local minimum of n · v in the direction of w

Suggestive Contours – p. 10

The radial curvature

• Fixed a point p on a smooth surface, we can compute the

local minima of n · v, considering the derivative in the direction of w: Dw(n · v) = Dwn · v + n · Dwv = Dwn · w = −II(w, w) = |w|2κr

• So p is a critical point of n · v ⇔ κr = 0
• Further, p is a local minimum if Dw(Dw(n · v)) > 0

Dw(Dw(n · v)) = Dw(|w|2κr) = |w|2Dwκr

• Thus, if κr = 0, p is a local minimum ⇔ Dwκr > 0
• This leads to a second definition of suggestive contours

Suggestive Contours – p. 11

Definition 2: Zeros of κr where Dwκr > 0

• We have prooved that these two definitions are equivalent

Suggestive Contours – p. 12

Stability

• View vectors almost parallel to the normal vector produces

small w and can produce spurious results

• It is useful to define an angular threshold θc
• Less steep minima are more susceptible to instability and

noise due to errors in curvature estimation

• Small segments of suggestive contours can produce

undesired results

Suggestive Contours – p. 13

Visibility and lining up (1)

Suggestive Contours – p. 14

Visibility and lining up (2)

• Contour and suggestive contours generators are the

respectively solutions to n · v = 0 and κr = 0

• They meet at the ending contours, that are points of

potential visibility change of contours

• Also, they meet with G1 continuity, producing seamlessly

aligned curves

• Visibility can occur locally or by occlusion. The first can be

computed with κr = 0, the other with traditional techniques

Suggestive Contours – p. 15

Contour algorithms

• Generally, there are two algorithms for computing contours:

an image-space and an object-space algorithm

• Image-space algorithm
• Rely on image processing of the n · v or depth map
• Easy, image precision results, no control over the lines
• Object-space algorithm
• Find zero crossings of n · v on the mesh, segment

joining, optional parametrization, visibility

• Harder to implement, control over the lines

Suggestive Contours – p. 16

Suggestive contour image-space algorithm

• Definition 1 leads to an image-space implementation
• Find valleys of n · v maps
• Alternative solution:
• Approximate n · v by shaded diffuse light source
• Find steep valleys (stable suggestive contours)
• Take a pixel i and collect intensities of pixels in r radius
• Label a pixel i a valley if:
• No more than s percentage of pixels are darker than i
• pmax − pi exceeds a fixed threshold d

where pi is the intensity of i and pmax is the max intensity of the r-radius neighbourdhood

• Apply median filter of r radius, to remove irregularities
• To avoid discretization effects scale s by 1 − 1

r and d by r

Suggestive Contours – p. 17

Suggestive contour object-space algorithm

• Definition 2 leads to an object-space implementation
• Find zero crossings of κr and trim the results with the

devative test and angular threshold (Dwκr > 0 and θc)

• Estimate principal directions and curvatures on the vertices

and apply κr = κ1cos2φ + κ2sin2φ, φ = angle (κ1, w)

• To find Dwκr we can compute

Dwκr = III(w, w, w) + 2Kcot(θ) where III is ∂II

∂u ∂II ∂v

• a 2 × 2 × 2 tensor and |w| = 1
• To avoid errors in curvature, apply Dwκr

|w|

> td

• Segments shorter than ts threshold are discarded

Suggestive Contours – p. 18

Results: image-space

r = 4, s = 0.2 and d = 0.25

Suggestive Contours – p. 19

Results: object-space

θc ∈ [20, 30] degrees, td ∈ [0.02, 0.08] and ts = 2

Suggestive Contours – p. 20

Results

contours, image-space, object-space

Suggestive Contours – p. 21

Results: object-space and image-space

80K polygons, 500K polygons

Suggestive Contours – p. 22

Results

contours, valleys, suggestive contours

Suggestive Contours – p. 23

Parabolic lines and negative gaussian curvature

• Parabolic lines are boundaries of elliptic and hyperbolic

regions of the surface

• Suggestive contours only appear on regions of hyperbolic

regions (K < 0)

Suggestive Contours – p. 24

Experiments with derivative tests

(a) κr = 0, Dwκr > 0 (b) H = 0, DwH > 0 (c) K = 0, DwK > 0

Suggestive Contours – p. 25

Elliptic surfaces

contours, ridges, modified suggestive contours κr = 1/2

• Suggestive contours not appear on elliptic regions (K > 0)
• But finding positive crossings can yield interesting results

Suggestive Contours – p. 26

Demo

Suggestive Contours Demo

Suggestive Contours – p. 27

Final considerations

• Suggestive contours indeed extend ordinary contours
• Perceptual and aesthetic research
• Deal with noise and instability
• Promising and much work to do

Suggestive Contours – p. 28

References

[1] D. DeCarlo, A. Finkelstein, S. Rusinkiewicz, A. Santella, "Suggestive Contours for Conveying Shape", SIGGRAPH 2003 [2] S. Rusinkiewicz, D. DeCarlo, A. Finkelstein, Line Drawings from 3D Models, SIGGRAPH 2005 course notes [3] A. Hertzmann, D. Zorin, "Illustrating smooth surfaces", SIGGRAPH 2000 [4] A. Hertzmann, "Introduction to 3D Non-Photorealistic Rendering: Silhouettes and Outlines", SIGGRAPH 99 [5] do Carmo, M. P ., "Elementos de Geometria Diferencial", 1971 [6] John Flaxman Odissey, www.bc.edu/bc_org/avp/cas/ashp/flaxman_odyssey.html

Suggestive Contours – p. 29