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首页图像修复(Navier-Stokes, Fluid Dynamics)
图像修复(Navier-Stokes, Fluid Dynamics)
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更新于2023-03-03
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一种图像修复的方法--Navier-Stokes, Fluid Dynamics(基于流体动力学的方法),很好,希望大家下载!
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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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