SUGGESTIVE CONTOURS OVER POINT-SET IMPLICITS

Jo˜ao Proenc¸a, Joaquim Jorge

Department of Computer Systems Engineering, INESC-ID/IST/Technical, University of Lisbon, Portugal

Mario Costa Sousa

Department of Computer Science, University of Calgary, Canada

Keywords:

Implicit surfaces, point-based graphics, non-photorealistic rendering.

Abstract:

This paper presents a system that combines large point- set implicit surfaces with fast line-based rendering.

We devised a new process for extracting suggestive contours quickly by using particles scattered throughout

the surface to identify areas of interest, followed by clustering and line- ﬁtting. Furthermore, we improve on

state-of-the-art methods for extracting silhouettes and feature-lines by harness- ing the descriptive power of

the surface representation. This provides heuristics for fast determination of curvature and allows for the local

regeneration of shape-depicting line elements after editing operations. While visual examples illustrate the

high quality of the drawings obtained with our application, as well as the high detail it can provide for more

complex models, run-times show comparatively higher performance over similar approaches for the same

number of points.

1 INTRODUCTION

Non-Photorealistic Rendering (NPR) methods are

commonly used to depict the shape of three-

dimensional objects in a way similar to the exist-

ing types of artistic drawing techniques. Recently,

most of the work presented on real-time extraction

of lines from 3D objects has focused mainly on

polygonal meshes. However, even though meshes

are visualization-friendly structures, they pose serious

difﬁculties for shape modeling. Namely, memory us-

age increases with both shape size and detail, while it

is difﬁcult to maintain topological consistency with

editing operations. There are other representations

that are better suited for such operations and for which

direct rendering techniques should be devised. One

such representation is the implicit surface, which pro-

vides a simple and ﬂexible mathematical deﬁnition

that derives from a potential ﬁeld function.

In this paper we present techniques to quickly

extract shape-depicting lines from Multi-level Parti-

tion of Unity (MPU) implicits (Ohtake et al., 2003),

which are able to represent complex surfaces from

large sets of points. These techniques were applied

directly to an already existing system (de Araujo and

Figure 1: MPU surface computed with a relative error of

5× 10

−4

. Left: David’s Head model with 59994 particles

scattered over the surface. Right: The same model rendered

with our system using silhouettes, suggestive contours and

feature lines.

Jorge, 2004), which allows users to interactively edit

complex objects deﬁned by MPU implicits. The user

is able to apply modeling operations through a cal-

ligraphic interface that converts 2D strokes into 3D

shape modiﬁcations. The implementation of our tech-

niques in such a system allowed us to use an already

existing application and is justiﬁed by our motivation

to use such rendering methods in an environment that

171

Proença J., Jorge J. and Costa Sousa M. (2008).

SUGGESTIVE CONTOURS OVER POINT-SET IMPLICITS.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 171-180

DOI: 10.5220/0001098301710180

 SciTePress

supports shape editing.

Our approach draws inspiration from methods

presented in (Foster et al., 2005) to construct a parti-

cle system over an implicit surface, extract silhouettes

and feature-lines starting from the positions of these

scattered particles and then using surfels to remove

hidden lines. However, their approach does not sup-

port suggestive contours and uses a different implicit

representation (BlobTree), which supports object def-

inition through constructive solid geometry (CSG).

We were more interested in using the MPU since it

allows representing complex models from very large

sets of points, which can be obtained from 3D scan-

ning of real objects. One of the other advantages

of the MPU implicit is its associated octree structure

(Ohtake et al., 2003) that establishes a 3D cell divi-

sion of the surrounding space of the object, in which

the cell subdivision level is higher in areas of higher

complexity. We highlight the main contributions of

our work below.

• We present a novel technique for extracting sug-

gestive contours from implicit surfaces. This is

done by identifying particles that lie in surface ar-

eas where a suggestive contour exists, clustering

those and applying line-ﬁtting algorithms. This

process works even when the local curvature in-

formation is not well-behaved, making our tech-

nique generalizable to most implicit surface rep-

resentations, not just MPUs.

• We use the MPU cell subdivision information as a

heuristic that in many ways improves the perfor-

mance and precision of both the particle system

simulation and the line extraction process. Indeed,

probing the hierarchical level of each cell allows

us to heuristically determine line lengths for con-

tours, which tend to be shorter in areas of higher

detail (deeper cell levels) and longer in ﬂatter re-

gions (cells closer to octree root).

• We provide support for dynamic line-elements, by

using the MPU octree to allow locally regenerat-

ing particles and view-independent lines at inter-

active speeds after shape editing operations, even

in surfaces with tens of thousands of particles.

We also use the MPU octree to set the initial po-

sitions of particles throughout the surface, which pro-

vides an automatic process for choosing the number

of particles that is adequate for each model without

the need for user intervention, contrary to other meth-

ods (Foster et al., 2005) which use random placement

and require manual adjustments as described in detail

in a previous publication (Proena et al., 2007). All

of these elements come together in one system that

combines the expressiveness of line-based rendering

with the descriptive power of the MPU, to visualize

and edit objects deﬁned by very large sets of points.

2 RELATED WORK

Most of the developed NPR methods that include sil-

houette, feature-line or suggestive contour extraction

focus on polygon meshes and have been gaining mo-

mentum steadily over the past few years (Isenberg

et al., 2003; Gooch et al., 1999; Sousa et al., 2003;

DeCarlo et al., 2003; DeCarlo et al., 2004; Judd et al.,

2007). However, while there was some earlier ef-

forts applied to NPR directly over implicit represen-

trations (Bremer and Hughes, 1998; Elber, 1998) only

in more recent years have we seen a return to the sub-

ject (Plantinga and Vegter, 2003; Foster et al., 2005).

Bremer and Hughes (Bremer and Hughes, 1998)

presented methods for rendering lines over implicit

surfaces, which used ray-intersection to determine

points over the surface and numerical integration pro-

cesses to make them approach the silhouettes and fol-

low them. The same type of ray-driven techniques

were used for positioning short interior strokes and

perform hidden-line removal (HLR). More recently,

Foster et al. (Foster et al., 2005) proposed techniques

that combined these methods with some of the ideas

introduced by Elber (Elber, 1998), such as the us-

age of a Witkin-Heckbert particle system (Witkin and

Heckbert, 1994) for scattering points over the sur-

face instead of the more computationally demand-

ing ray-intersection algorithms. Using a variation of

the Shrink-wrap method (van Overveld and Wyvill,

2004), feature-lines are also extracted from particles

through the identiﬁcation of straddle points over sur-

face faces. An additional HLR method that used sur-

fels was also presented in that paper. In terms of the

Witkin-Heckbert based particle systems, Levet et al.

(Levet et al., 2006) recently presented a technique

that relies on geometry processing for ﬁnding near-

optimal initial positions for particles, which improves

the subsequent simulation time, but their approach

requires more complex calculations in this step than

ours. (Jepp et al., 2006) also extended the develop-

ment of the particle system in (Foster et al., 2005) to

create smarticles that use ﬂocking behaviour for fea-

ture searching and line drawing in implicit surfaces.

In terms of suggestive contours, they are a fairly

recent concept that was introduced by DeCarlo et

al. (DeCarlo et al., 2003; DeCarlo et al., 2004).

These two papers cover the mathematical deﬁnitions

for suggestive contours on a surface and their extrac-

tion from polygonal meshes. (DeCarlo et al., 2003)

presents the formal deﬁnitions and two algorithmic

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

172

approaches for producing rendered images of 3D ob-

jects (one in object-space and the other in image-

space). In (DeCarlo et al., 2004) the authors ex-

tend the previous system to render suggestive con-

tours in real-time, using line-movement analysis tech-

niques and fast algorithms for polygon elimination in

the line-searching process. These papers set the stan-

dard for suggestive contour rendering over meshes,

but there are not many papers that cover their extrac-

tion from other types of representations. (Burns et al.,

2005) provides techniques for silhouette and sugges-

tive contour extraction from volume data using auxil-

iary level-sets, but their methods seem more suited to

volumetric rather than surface-only data such as ours.

(Schmidt et al., 2006) attempts to draw this type of

lines over implicits, but uses a low-resolution mesh

as an intermediate step for the line-extracting algo-

rithms, which can be a limiting factor for some sce-

narios. It was our goal to avoid using this type of ap-

proach because the process of reconstructing the mesh

in CSG operations is not straightforward over an im-

plicit. To the best of our knowledge no published pa-

pers cover suggestive contour extraction from implic-

its without resorting to a mesh.

Our approach uses the deﬁnitions for suggestive

contours presented in (DeCarlo et al., 2003) to iden-

tify particles in the MPU surface, create clusters from

those particles and form contour lines by the appli-

cation of line-ﬁtting algorithms to the clusters. This

method was inspired by the work presented by (Barla

et al., 2005), which applies geometric clustering algo-

rithms to line drawing simpliﬁcation. However, their

methods are based on the premise that large sets of

lines are already available for simpliﬁcation, which

is not our case. Furthermore, the technique oper-

ates in image-space, which would make it impossible

to use our surfel-based HLR method for suggestive

contours. Therefore our approach evolved to extract-

ing curves that represent clustered sets of 3D points.

(Gumhold et al., 2001) presented methods for fea-

ture extraction from point clouds, where minimum-

spanning-trees are built over clusters of points and

later trimmed to form the representative lines. Our

main approach to cluster line-ﬁtting adapted this

method to the characteristics of suggestive contours,

but other approaches that ﬁtted smooth curves to sets

of points were also considered detail in Section 5.3.

2.1 The MPU Surface

An implicit surface (Bloomenthal, 1997) is deﬁned

through the potential function f(x) as the set of points

x=(x,y,z) that respect the condition:

f(x) = iso (1)

where iso is a constant value. In our case, where

iso=0, the negative or positive values of f(x) indi-

cate that x is inside or outside of the object volume

respectively. Implicits by their own nature provide

a compact and ﬂexible deﬁnition for highly complex

surfaces and make blending operations easy to apply.

The MPU (Ohtake et al., 2003) is one of the vari-

ous of types of implicit surfaces, which stands out for

providing a method for efﬁcient model construction

from a dense set of control points sampled on the sur-

face of complex objects. Its structure is composed

by three elements: an octree of cubic spatial cells

that cover the object; quadratic functions that approx-

imate the local shape in each cell and weight func-

tions that blend the local functions, thus providing

a precision-controllable approximation to a complex

implicit surface efﬁciently. The construction of an

MPU implicit is guided by the subdivision of the oc-

tree structure, where cells become smaller and more

numerous in areas where the point positions and nor-

mals suggest a higher curvature. We have extended

the model to support interactive shape-editing opera-

tions. The spatial enumeration allows us to mark the

local MPU cells that are involved in an edit operation

and reconstruct the MPU locally from a new set of

control points, which greatly beneﬁts system perfor-

mance (de Araujo and Jorge, 2004).

Figure 2: Shape editing. Top: The user deﬁnes a CSG cut

over a simple object, using the blue 2D stroke. Center: The

MPU surface is locally altered and new particles are gen-

erated in that area. Bottom: Particles become redistributed

after a local simulation, which takes a few seconds.

SUGGESTIVE CONTOURS OVER POINT-SET IMPLICITS

173

3 OVERVIEW

Our methods follow a high-level approach similar to

Foster et al. (Foster et al., 2005), in which particles are

distributed across the surface. Silhouettes, suggestive

contours and feature lines depicting surface disconti-

nuities are then extracted from the relevant particles.

However, while Foster et al. maintained the parti-

cle system in a constant simulation state (the particle

positions are always being updated in a continuous

cycle), we only let the simulation occur for a maxi-

mum number of iterations that are sufﬁcient for an ac-

ceptable distribution of points throughout the surface.

This results in a limited simulation time, after which

a lot of computational resources are freed for the line-

extracting algorithms. We also provide the users with

the possibility of saving the particle system state to

a ﬁle, allowing them to reuse it every time they load

the same object in the application, without having to

endure the simulation process again.

Silhouettes and suggestive contours are view-

dependent elements, so we regenerate them whenever

there is a change in the viewing conditions or when-

ever the particle system is in a simulation state. Fea-

ture lines depicting geometric-dependent discontinu-

ities are view-independent (sometimes they are also

called ridge and valley lines), so we only generate

them once after some iterations of the particle sys-

tem and then apply the appropriate 3D transforma-

tions whenever the viewing conditions change.

Foster et al. treat the implicit representation as a

black-box, i.e. their methods only query the potential

function to extract its value and gradient, making their

approach usable with almost any type of implicit.

While our approach requires the same mathematical

information, we exploit the spatial information pro-

vided by the octree, thus leveraging on the data struc-

tures associated with the MPU. This includes using its

octree for particle and line proximity, instead of using

a regular grid as in their method.

Our system supports modeling operations via a

calligraphic interface (de Araujo and Jorge, 2004).

After an editing operation, we mark the affected oc-

tree cells and start a new particle system simulation

conﬁned to those cells. This makes the regenera-

tion of particles much more efﬁcient, because only

the local particles are affected. This is illustrated in

Figure 2, in which we can clearly see the redistribu-

tion of particles after editing the surface shape. Since

feature-lines are view-independent elements, they are

also locally regenerated. While our work was mainly

concerned with rendering implicit surfaces, it is im-

portant to ensure that shape editing is still possible

and efﬁcient with our techniques, because it is one

of the main motivations for using implicits instead of

other representations, especially in the MPU where

the local reconstruction of surface areas is directly

supported in CSG and blending operations.

Finally, to perform hidden line removal we use a

surfel approach, as in (Foster et al., 2005). Surfels are

oriented ellipses or circles which are used for point-

based rendering of surfaces (Pﬁster et al., 2000). Our

approach uses circular textures inside quadrilateral

polygons positioned slightly behind each particle us-

ing the surface normal for orientation. These are ren-

dered as white disks and have a radius set by the dis-

tance to the nearest particle. Although this method

does not guarantee a total occlusion for all cases, it is

very effective as long as we have a good distribution

of particles throughout the surface.

4 THE PARTICLE SYSTEM

Our particle system improves on the approach of

Foster et al., which is based on Witkin and Heck-

bert’s (Witkin and Heckbert, 1994) model. While

their methods place particles at random initial posi-

tions, we use the MPU features to obtain a better ini-

tial distribution as described in (Proena et al., 2007),

which we recall here to help in understanding what

folows. Our objective is to achieve denser concentra-

tions of particles in areas of higher surface curvature.

Since octree cells are smaller and more numerous in

those areas, we create k random particles in each cell,

where k is ﬁxed (usually k = 1), and obtain an initial

distribution that is already close to our ﬁnal objective,

thus saving a considerable number of simulation it-

erations. We obtain a set of n particles, P

,i ∈ [1, n],

chosen from the set of MPU control-points that al-

ready lie on the surface. Each of these particles is

registered in the corresponding octree cell, enabling

us to use fast algorithms for point-proximity determi-

nations, using nearby-cell inspection.

The particle system simulation process is com-

posed by a set of iterations where, at each step ev-

ery particle suffers attraction and repulsion forces and

moves accordingly. The calculation of these forces, as

well as the movement of each particle, is performed

in the same way as in (Foster et al., 2005), with the

difference that we compute the repulsion distribution

factor δ

in a different way. This value directly inﬂu-

ences the repulsion force and controls the local den-

sity of particles. Foster et al. calculate this using

Hessian values extracted from the potential function

to determine the local mean curvature. This poses a

signiﬁcant computational burden; we avoid this alto-

gether by using the octree cell depth information on

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174

Figure 3: 10000 particles scattered over the surface of the

Bunny model, after running 20 iterations of the particle sys-

tem simulation. Red points correspond to octree cells of

higher depth, green points to lower depth.

the MPU. The depth indicates the cell subdivision de-

gree and therefore is a good heuristic for determining

local curvature, which yields a simpler formula for δ

= weight(

− d

min

max

− d

min

) (2)

where d

is the cell depth of particle P

and d

max

and d

min

are the maximum and minimum cell depths

where particles exist in the model (these are deter-

mined during the initial particle generation step). The

weight function is a quadratic B-Spline that sets a

higher value for δ

if the particle is in one of the lower

depth cells (less curved regions of the surface) and

vice versa. The practical effect of using this technique

is that it is easy and fast to obtain good results where

the particles become well distributed over the surface

to reﬂect local curvaturewith comparativelyless steps

and a lower cost per particle (see Figures 1 and 3).

5 SUGGESTIVE CONTOURS

Suggestive contours (DeCarlo et al., 2003) are sets

of points that, for a speciﬁc camera position, are

seen in a nearby viewpoint (at a radial distance less

than 90 degrees) that are not in correspondence with

silhouette-points of any (radially) closer viewpoint.

What this deﬁnition informally means is that if we ob-

serve a suggestive contour in a speciﬁc viewing sce-

nario, a silhouette will appear in that same area with

a small change in the viewing position.

The processes described by DeCarlo et al. for sug-

gestive contour extraction in object-space use a deﬁ-

nition that relies on the directional derivativeof the ra-

dial curvature of the surface. This has to be calculated

from principal curvature values extracted from the ob-

ject’s surface, which are usually computed from the

Hessian of the potential function f(x) in an implicit.

It is hard to ﬁnd implicit representations that yield fast

and numerically stable Hessian values and the MPU

is no exception. It becomes particularly difﬁcult to

obtain these values near discontinuities, where it is

highly probable to ﬁnd a suggestive contour, because

the potential function of the MPU is not well behaved.

Therefore, our approach relies on another deﬁnition

for suggestive contours mentioned by DeCarlo et al.

in their image-space algorithm, namely the set of min-

ima of the dot product between the view vector and

the surface normal in the direction of the projection of

the view vector onto the tangent plane at x. This deﬁ-

nition only requires evaluating gradient values ∇f(x)

from the implicit to obtain the surface normal.

Our suggestive contour extraction process begins

with identifying the particles that lie in suggestive

contour areas. We then form clusters of particles close

to one another and apply line-ﬁtting algorithms to

each cluster to obtain the suggestive contours. In the

next three sections we describe each of these methods

in detail.

5.1 Identifying Suggestive Contour

Particles

Figure 4: The positions x

and x

−

are calculated over the

planar surface deﬁned by n and x

, in the direction of v.

The blue normal vectors are computed by consulting the

gradient of the potential function in x

and x

−

The normalized view vector v for a speciﬁc sur-

face position x and a view position c is deﬁned as:

v =

(x− c)

k (x− c) k

(3)

Consider w as the projection of v onto the surface tan-

gent plane at x with normal n, obtained from the nor-

malized gradient ∇ f(x)/ k ∇f(x) k. Suggestive con-

tours can be deﬁned as the set of minima of (n· v) in

the direction of w (DeCarlo et al., 2003).

Using this deﬁnition, we identify if a particle lies

close to a suggestive contour by estimating if (n · v)

experiences a local minimum at its position. Consider

the two points x

and x

−

deﬁned by:

= x

+ k

w (4)

SUGGESTIVE CONTOURS OVER POINT-SET IMPLICITS

175

−

= x

− k

w (5)

where x

is the position of the particle P

and k

is scale-dependent and controls the displacement be-

tween x

and x

−

(for the tested models we usually

use k

= 0.5, Figure 4). Also consider that dp

, dp

and dp

−

are the (n· v) values (obtained by consulting

the gradient ∇f(x)) at x

, x

and x

−

respectively). If

the following inequalities are veriﬁed:

< dp

(6)

< dp

−

(7)

we can conclude that there is a local minimum be-

tween x

and x

−

and, if k

is small enough, x

is a fair

estimation of that minimum. Additionally we enforce

the stability threshold mentioned in (DeCarlo et al.,

2003), which excludes areas where the view vector is

almost normal to the surface:

< cos

−1

(dp

) (8)

where θ

is a user-deﬁned scalar. If the inequalities

(6), (7) and (8) are veriﬁed, we consider that P

be-

longs to a suggestive contour.

We use the minima of (n · v) to identify sugges-

tive contour points in object-space, instead of the ze-

ros of radial curvature that are used by DeCarlo et

al., since the latter does not seem to work reliably on

MPUs because it is discontinuous over blending func-

tions. This is a new application of this deﬁnition for

suggestive contours and it also allows for their extrac-

tion from implicit surfaces without using an auxiliary

structure such as a polygonal mesh.

5.2 Clustering Points

After identifying suggestive contour particles, we

cluster those particles that are close together on the

surface. Our approach then builds a k-nearest neigh-

bor graph of particles for each cluster. We start by tak-

ing one arbitrary particle P and searching for neigh-

bors in 3D space within a ﬁxed radius. This radius

is multiplied by the δ

value of the respective octree

cell to conﬁne this search to a more limited space in

areas of high curvature. From those nearby particles

we select the k nearest (we normally use k = 3) and

those become the neighbors of P in the graph. For

each neighbor we repeat the nearby particle search,

skipping particles that have already been inspected.

The cluster graph becomes complete when the search

yields no new neighbors. We create additional clus-

ters by applying this process to other particles that

have not been inspected yet. Because of the dynamic

nature of the search radius, it is possible to have par-

ticles from one cluster ﬁnding nearby particles from

another cluster. In this situation we merge the two

clusters.

5.3 Line Fitting

After clustering particles, we ﬁnally apply a line ﬁt-

ting algorithm to extract lines that represent each clus-

ter topology. Our approach is inspired by the methods

presented in (Gumhold et al., 2001) and starts by cre-

ating a minimum-spanning-tree (MST) of the cluster

graph, using the optimized Kruskal algorithm. This

process minimizes the geometric distance between

points.

Since the obtained MST can have a large num-

ber of short branches, we must apply a simpliﬁca-

tion process to extract suggestive contour lines. To

achieve this, we have two options. The ﬁrst is to com-

pute the longest path in the MST and use the cor-

responding edges as the suggestive contour. To ﬁnd

this path we take an arbitrary particle P and ﬁnd the

farthest particle P

through breadth-ﬁrst search. We

then ﬁnd the particle P

′

farthest from P

. The longest

path becomes the one connecting those two particles.

The second approach is to trim the tree by removing

short branches. To perform this trimming step, we

go through all of the particles in the MST that have

more than two connections to other particles (inter-

section nodes) and analyze each incident branch. If

the branch leads to another intersection node, it re-

mains intact. We removethe branch if it leads to a tree

leaf node and is composed by less than k

trim

(usually

three) particles. In either approach, the extracted lines

undergo a ﬁnal smoothing step.

6 SILHOUETTES AND FEATURE

LINES

As with the particle system, our silhouette and

feature-line extraction methods are similar to (Fos-

ter et al., 2005) with four notable improvements. In

what follows, we provide enough background to ex-

plain the base techniques and highlight our changes.

To determine silhouettes, we start by identifying par-

ticles P

with position x

that verify the following in-

equality:

· n

| < k

(9)

where k

is controlled by the user (we use k

= 0.05

for the majority of the presented models), v

is the

view vector and n

is the surface normal at x

. This

inequality sets off a surface area where particles are

considered to be close to the real silhouette (let’s call

it silhouette area).

From each identiﬁed particle we build the silhou-

ette polyline iteratively by a numerical integration

process. We use the predictor/corrector method pre-

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176

Figure 5: Extracting suggestive contours. 1: suggestive contour particles. 2: two clusters are identiﬁed and MSTs are built.

3: MST trimming approach. 4: MST longest-path approach. 5: using the longest-path approach and smoothing the lines, we

obtain the suggestive contours.

sented by Foster et al., performing two point move-

ments at each iteration. The ﬁrst follows an estimated

silhouette direction with a ﬁxed step size, while the

second results from two correction vectors computed

to ensure that the line stays close to the silhouette and

on the surface. Our approach uses a dynamic step

size, by multiplying with the δ

value of the MPU

octree cell (described in Section 4) where the point

movement is taking place. This provides better pre-

cision to the silhouette polyline in areas of higher

curvature, because the respective line segments be-

come smaller, while improving performance in areas

of lesser curvature, where the segments become larger

(Improvement 1). The process stops if either (1) we

detect that the silhouette is looping or (2) whenever

it reaches the vicinity of another silhouette line or (3)

when the following inequality is not veriﬁed:

· n

| < k

· k

(10)

This inequality is similar to (9), with the k

parame-

ter added, which expands the silhouette area where

the curve may develop. By doing this, we can set

a smaller k

value to limit the number of identiﬁed

silhouette particles and allow the silhouette building

process to have a wider area of expansion (by using

= 2 or k

= 3). Since it is very common to have

particles that yield the same silhouette line, this tech-

nique improves the overall performance while main-

taining quality and continuity of the extracted curves

(Improvement 2). After all the silhouettes have been

computed, we perform a 2D analysis to chain pairs

of polylines that are within a certain distance of each

other and follow similar orientation.

For feature-line extraction, we identify particles

that lie in the relevant areas of the surface and build

lines from their positions through numerical integra-

tion methods. We begin by discovering pairs of par-

ticles (straddle points) close to each other, which

present a large difference between their surface nor-

mals. Foster et al. performed this step in the particle

system simulation phase, by comparing the normal of

each particle before and after its movement. Formally,

for a certain particle P

with initial surface normal n

Figure 6: Silhouettes, Feature Lines and Suggestive Con-

tours drawn over the Cow model.

and normal n

′

after the movement, we consider the

two respective positions x

and x

′

to be straddle points

if:

angle(n

′

) > k

(11)

where angle(v

) returns the angle between two

vectors and k

is the threshold angle (we usually use

= 0.15).

While this method is usually effective and ex-

tracts enough straddle points for a good visual re-

sult, there are situations where it may fail to ﬁnd

feature-lines due to our optimizations in the parti-

cle system. Indeed, particles which are initially set

with near-optimal densities throughout the surface

(see Section 4) can sometimes lead to a simulation

stage where there will be very small changes in their

positions. While this provides good performance, it

may limit opportunities to identify straddle points.

Therefore, we devised an alternative method to solve

this problem which we apply when the Foster et al.

method fails. It consists in searching for nearby points

within a ﬁxed radius of each particle and comparing

normals to ﬁnd pairs that verify Inequality (11) (Im-

provement 3).

After identifying straddle points, we move them

along the respective faces separated by the feature,

while trying to maintain a parallel direction to it.

While this is performed, we estimate intermediate

points that lie near the feature, to yield the feature

SUGGESTIVE CONTOURS OVER POINT-SET IMPLICITS

177

Figure 7: This caption has one line so it is centered.

Figure 8: The Phlegmatic Dragon model rendered using silhouettes, feature-lines and suggestive contours in our system.

polyline. As in the silhouette case, we use the δ

value

of the current MPU octree cell to dynamically adjust

the step of the movement to the local curvature (Im-

provement 4). The process stops when Inequality (11)

is not veriﬁed or when another feature-line is detected

in the vicinity of the straddle points.

7 RESULTS AND DISCUSSION

Our techniques for extracting silhouettes and feature-

lines proved to be very effective as long as the MPU

surface provides enough shape precision. This usu-

ally depends on the number and density of points in

the object dataset. We are also able to produce sug-

gestive contours for the objects, using our particle

identiﬁcation and line-ﬁtting techniques.

For the two approaches that we used for ﬁtting

lines onto the suggestive contour MST (Section 5.3),

the longest-path approach is guaranteed to extract sin-

gle continuous line from each cluster, which makes

it usually the best option to avoid artifacts caused

by MST branching. However, when there are sug-

gestive contours that almost intersect, the MST trim-

ming approach is usually able to correctly distinguish

the intersecting lines when the respective clusters are

merged together. This is in fact one of the major dif-

ﬁculties to overcome in the whole process: there is

always a certain probability that two or more distinct

suggestive contours will be covered by the same par-

ticle cluster because of their proximity. Therefore,

we allow the user to select the appropriate method for

each model.

Table 1: Performance results for our system. Each row in-

dicates the model, number of points in the dataset, number

of points in the particle system, frame-rate with silhouettes

and feature-lines (in frames per second) and frame-rate with

suggestive contours additionally.

Model Dataset Particle w/o Sugg. w/ Sugg.

Points System Contours Contours

Bunny 69451 10000 3.4 1.9

Cow 92864 8334 3.4 1.8

Igea 268686 13665 3.7 1.4

Armadillo 345944 28360 1.6 0.5

Dragon 480076 26984 1.8 0.6

David 827181 59994 0.9 0.2

Table 1 presents the performance results obtained

for the objects depicted in Figures 1, 6, 7, 8 and 9.

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178

Figure 9: Renderings using silhouettes, feature-lines and suggestive contours. Left: The Igea model. Center: The Armadillo

model. Right: The Armadillo model showing a particle distribution.

These results were gathered using a 3.6GHz Pentium

IV with 2 gigabytes of RAM and a NVIDIA Quadro

FX 3400 graphics card, running Windows XP SP2.

All of the frame-rates were obtained after the particle

system simulation process had been completed (usu-

ally the simulation comprises 20 iterations).

The number of particles in the particle-system is

always determined by the MPU implicit structure,

since we place one particle per octree cell. This

strategy proved to be very effective for all of the

tested models and also becomes automatic, without

any need for the user to adjust the amount of particles

needed for a correct surface coverage. As the results

suggest, the number of particles is not proportional to

the number of dataset points, because it is also inﬂu-

enced by the surface topology.

The performance results show that it is possible

to achieve interactive frame-rates with medium com-

plexity objects, even while extracting suggestive con-

tours. However, the dimension of the particle system

and the complexity of the model usually affect per-

formance, since they inﬂuence the number of implicit

function evaluations that are made and the associated

computational overhead. This also explains why the

suggestivecontour extraction has such an inﬂuence on

frame-rate, since it implies a great increase in function

evaluations. Nevertheless, our approach compares fa-

vorably to (Foster et al., 2005) in that, for a similar

frame-rate, we are usually able to render models with

six times the number of particles, while includingsug-

gestive contours which are not handled by their ap-

proach. We should also mention that, while implicit-

based approaches such as ours and the one from Fos-

ter et al. are currently less efﬁcient than mesh-based

ones, it is important to constantly improve these ren-

dering methods because the descriptive ﬂexibility that

an implicit provides is vital in surface modeling and

cannot be obtained when we use a polygonal mesh.

In terms of visual results, we verify that, for higher

point densities in the dataset, the MPU can more ac-

curately represent the surface and the overall line-

extracting process yields better results. This explains

why the David’s Head and Dragon models seem to

present the most impressive depictions (Fig. 7 and 8).

There are however some important remarks to be

made in terms of our system’s limitations. Since

many of our methods are based on Foster et al., we

inherited some of the dependency on parameter set-

ting by the user, namely in terms of adjusting the re-

pulsion between particles and some silhouette extrac-

tion parameters to the scale and topology of the 3D

model. Another limitation is the need for almost com-

plete point clouds to achieve good results. These are

usually obtained from 3D scanning of real objects and

many times only certain portions of the overall sur-

face are scanned. Although our particle system can

deal with occasional holes in the MPU representation,

it cannot effectively distribute points over an incom-

plete surface. Our technique also does not guarantee

that all silhouettes, suggestive contours and feature

lines are extracted from each surface, since the search

for the relevant surface areas, for each viewing posi-

tion, depends on the existence of strategically placed

particles in those areas. This dependencyis evenmore

important for suggestive contours, since they are ob-

tained from particle clusters. This results in unstable

curves that become noticeable as the camera moves.

8 CONCLUSIONS AND FUTURE

WORK

We have presented techniques to extract silhouettes,

suggestive contours and feature-lines directly from

MPU implicits, which are able to represent surfaces

from large sets of points. Our methods beneﬁt from

the spatial information provided by the MPU struc-

ture, enabling us to automatically place particles

throughout the surface and adjust the particle system

and line extracting algorithms to the local shape. This

implicit representation also provides good support for

SUGGESTIVE CONTOURS OVER POINT-SET IMPLICITS

179

shape modiﬁcations by allowing us to locally regener-

ate visual elements affected by edits. We improve on

state of the art techniques for line-based rendering of

implicits and introduce a method for suggestive con-

tour extraction from this representation.

Our techniques provide precise and expressive de-

pictions of very complex objects, especially the more

detailed and complete datasets. Our results show bet-

ter performance levels in comparison to similar sys-

tems for the same number of particles. Some areas for

further improvementremain. Among them, new ways

for deﬁning the MPU implicit near the edge areas of

the surface might allow us to draw more precise and

continuous feature-lines. Some work also remains to

be done to improve frame-coherent suggestive con-

tours.

ACKNOWLEDGEMENTS

The models are courtesy of the Digital Michelangelo

Project 3D Model Repository (David’s Head), the

Stanford 3D Scanning Repository (Bunny, Cow and

Armadillo), Cyberware (Igea) and UTIA, Academy

of Sciences of the Czech Republic, and CGG, Czech

Technical University in Prague (Phlegmatic Dragon).

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