图像修复（Navier-Stokes,FluidDynamics）

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Navier-Stokes, Fluid Dynamics, and Image and Video Inpainting

M. Bertalmio

Computer Eng. Dept.

University Pompeu Fabra

08003 Barcelona, SPAIN

A. L. Bertozzi

Mathematics Dept.

Duke University

Durham, NC 27708

G. Sapiro

Elec. & Comp. Eng. Dept.

Univ. of Minnesota

Minneapolis, MN 55455

Abstract

Image inpainting involves ﬁlling in part of an image or

video using information from the surrounding area. Ap-

plications include the restoration of damaged photographs

and movies and the removal of selected objects. In this pa-

per, we introduce a class of automated methods for digital

inpainting. The approach uses ideas from classical ﬂuid dy-

namics to propagate isophote lines continuously from the

exterior into the region to be inpainted. The main idea is

to think of the image intensity as a ‘stream function’ for a

two-dimensional incompressible ﬂow. The Laplacian of the

image intensity plays the role of the vorticity of the ﬂuid;

it is transported into the region to be inpainted by a vec-

tor ﬁeld deﬁned by the stream function. The resulting al-

gorithm is designed to continue isophotes while matching

gradient vectors at the boundary of the inpainting region.

The method is directly based on the Navier-Stokes equations

for ﬂuid dynamics, which has the immediate advantage of

well-developed theoretical and numerical results. This is

a new approach for introducing ideas from computational

ﬂuid dynamics into problems in computer vision and image

analysis.

1. Introduction

Image inpainting[2, 10, 20, 38] is the process of ﬁlling in

missing data in a designated region of a still or video image.

Applications range from removing objects from a scene to

re-touching damaged paintings and photographs. The goal

is to produce a revised image in which the inpainted re-

gion is seamlessly merged into the image in a way that is

not detectable by a typical viewer. Traditionally, inpainting

has been done by professional artists. For photography and

ﬁlm, inpainting is used to revert deterioration (e.g., cracks

in photographs or scratches and dust spots in ﬁlm), or to add

or remove elements (e.g., removal of stamped date and red-

eye from photographs, the infamous “airbrushing” of polit-

ical enemies [20]). A current active area of research is to

automate digital techniques for inpainting [2, 3, 16, 21, 22].

In this paper, we introduce a novel algorithm for digi-

tal inpainting of still images that attempts to replicate the

basic techniques used by professional restorators. Our al-

gorithm, motivated by a method proposed in [2], involves a

direct solution of the Navier-Stokes equations for an incom-

pressible ﬂuid. The image intensity function plays the role

of the stream function whose isophote lines deﬁne stream-

lines of the ﬂow. After the user selects the regions to be

restored, the algorithm automatically transports informa-

tion into the inpainting region. The ﬁll-in is done in such

a way that isophote lines arriving at the region’s bound-

aries are completed inside. The technique introduced here

does not require the user to specify where the novel infor-

mation comes from. This is done automatically (and in

a fast way), thereby allowing for simultaneously ﬁll-in of

multiple regions containing completely different structures

and surrounding backgrounds. In addition, no limitations

are imposed on the topology of the region to be inpainted.

The only user interaction required by the algorithm is to

mark the regions to be inpainted. Since our inpainting al-

gorithm is designed for both restoration of damaged pho-

tographs and for removal of undesired objects on the im-

age, the regions to be inpainted must be marked by the user.

Our method inherits a mathematical theory already devel-

oped for the ﬂuid equations, including well-posedness and

the design of efﬁcient convergent numerical methods.

1.1. Prior Work

First note that image denoising is different to ﬁlling-in,

since the regions of missing data are usually large. That

is, regions occupied by top to bottom scratches along sev-

eral ﬁlm frames, long cracks in photographs, superimposed

large fonts, and so on, are of signiﬁcantly larger size than

the type of noise treated by common image enhancement

algorithms. In addition, in common image enhancement ap-

plications, the pixels contain both information about the real

data and the noise (e.g., image plus noise for additive noise),

while in our application there is no signiﬁcant information

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planktonlz

2013-11-04

功能一般,图象修复的代码

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