[其它] Freeform Vector Graphics with Controlled Thin-Plate Splines

晃晃

1023 主题	3 听众	359 积分

设计实习生

Rank: 2

纳金币: 335582
精华: 0

电梯直达

楼主

发表于 2011-12-28 11:01:49 |只看该作者 |倒序浏览

Freeform Vector Graphics with Controlled Thin-Plate Splines

Mark Finch John Snyder Hugues Hoppe

Microsoft Research

Abstract

Recent work deﬁnes vector graphics using diffusion between col-

ored curves. We explore higher-order fairing to enable more nat-

ural interpolation and greater expressive control. Speciﬁcally, we

build on thin-plate splines which provide smoothness everywhere

except at user-speciﬁed tears and creases (discontinuities in value

and derivative respectively). Our system lets a user sketch discon-

tinuity curves without ﬁxing their colors, and sprinkle color con-

straints at sparse interior points to obtain smooth interpolation sub-

ject to the outlines. We reﬁne the representation with novel con-

tour and slope curves, which anisotropically constrain interpolation

derivatives. Compound curves further increase editing power by ex-

panding a single curve into multiple offsets of various basic types

(value, tear, crease, slope, and contour). The vector constraints are

discretized over an image grid, and satisﬁed using a hierarchical

solver. We demonstrate interactive authoring on a desktop CPU.

Keywords: bilaplacian/biharmonic PDE, slope/contour curves

1 Introduction

Traditional vector graphics ﬁlls each closed shape independently

with a simple color function. Recent work applies more global

and powerful Laplacian interpolation between diffusion curves with

colors on each side [Orzan et al. 2008; Jeschke et al. 2009].

A Laplacian solution yields a membrane function which is “as-

constant-as-possible”. Its low-order smoothness objective has

drawbacks as illustrated in Figure 2. The solution is smooth only

away from constrained points. Value constraints yield tent-like

responses at isolated points and form creases along curves. The

Laplacian objective is also incompatible with derivative constraints,

because it already seeks zero ﬁrst-derivatives everywhere in all

directions. Only a higher-order notion of smoothness supports

sparse constraints on directional derivatives.

Our approach builds on thin-plate splines (Tps) [Courant and

Hilbert 1953], which deﬁne a higher-order interpolating function

that is “as-harmonic-as-possible”. This smoothness objective

overcomes previous limitations (Figure 2). Thin-plate splines

have been applied in several areas including geometric modeling

[e.g. Welch and Witkin 1992; Botsch and Kobbelt 2004; Sorkine

and Cohen-Or 2004; Botsch and Sorkine 2008], computer vision

[Terzopoulos 1983], and machine learning [Bookstein 1989]. They

have also been adapted to allow discontinuity control with explicit

tears and creases [Terzopoulos 1988]. We extend these controls and

demonstrate their usefulness in vector graphics authoring.

In the simplest case, an artist sketches some outlines (tears) without

ﬁxing their colors, and speciﬁes color constraints at a few interior

points or curves to obtain a smooth color wash within the outlines.

This ink-and-paint ordering of tasks is similar to hand drawing.

The result is then reﬁned by adding creases, contour curves, slope

curves, and critical points. These features increase editing power

by anisotropically constraining interpolation derivatives (e.g. along

or across the curves, or in both directions).

In addition to the basic curves, we introduce compound curves, with

user-assigned widths, for more complex effects. These internally

yield several offset curves, of possibly different types. For instance,

a value-slope curve juxtaposes the two basic types to create a

smooth ridge-like feature. A wide contour uses two offset contours

to form a constant-colored strip without ﬁxing its color. Other

combinations produce a variety of interesting and useful results.

We demonstrate a prototype systembased on these ideas. Like other

variational approaches such as diffusion curves, our system is easy

to use and supports “freeform” input based on a general network of

curves, which we augment with points. Smoother interpolation and

more ﬂexible constraints enhance naturalness and editing power

and produce rich results from a compact input (Figure 1).

Our contributions include:

Extension of the diffusion curves framework to beneﬁt from

higher-order interpolation and general discontinuity control.

A discretized least-squares kernel for accurate modeling of

crease curves.

Contour and slope curves that constrain derivatives anisotropi-

cally for intuitive control.

A variety of compound curve types for added expressiveness.

Discontinuity-aware upsampling for improved accuracy in a

multiresolution setting.

全文请下载附件：