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An image inpainting technique based on the fast marching method
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Abstract. Digital inpainting provides a means for reconstruction of small damaged portions of an image. Although the inpainting basics are straightforward, most inpainting techniques published in the literature are complex to understand and implement. We present here a new algorithm for digital inpainting based on the fast marching method for level set applications. Our algorithm is very simple to implement, fast, and produces nearly identical results to more complex, and usually slower, known methods. Source code is available online.
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Vol.9,No.1:25—36
An Image Inpainting Technique Based on
the Fast Marching Method
Alexandru Telea
Eindhoven University of Technology
Abstract. Digital inpainting provides a means for reconstruction of small dam-
aged portions of an image. Although the inpainting basics are straightforward, most
inpainting techniques published in the literature are complex to understand and im-
plement. We present here a new algorithm for digital inpainting based on the fast
marching method for level set applications. Our algorithm is v ery simple to im-
plement, fast, and produces nearly identical results to more complex, and usually
slower, known methods. Source code is available online.
1. Introduction
Digital inpainting, the technique of reconstructing small damaged portions of
an image, has received considerable attention in recent years. Digital inpaint-
ing serves a wide range of applications, such as removing text and logos from
still images or videos, reconstructing scans of deteriorated images by remov-
ing scratches or stains, or creating artistic effects. Most inpainting methods
work as follows. First, the image regions to be inpainted are selected, usu-
ally manually. Next, color information is propagated inward from the region
boundaries, i.e., the known image information is used to fill in the missing
areas. In order to produce a perceptually plausible reconstruction, an in-
painting technique should attempt to continue the isophotes (lines of equal
gray value) as smoothly as possible inside the r econstructed region. In other
words, the missing region should be inpainted so that the inpainted gray value
and gradient extrapolate the gray value and gradient outside this region.
© A K Peters, Ltd.
25 1086-7651/04 $0.50 per page
26 journal of graphics tools
Several inpainting methods are based on the above ideas. In [Bertalmio 00,
Bertalmio 01], the image smoothness information, estimated by the image
Laplacian, is propagated along the isophotes directions, estimated by the im-
age gradient rotated 90 degrees. The Total Variational (TV) model [Chan
and Shen 00a] uses an Euler-Lagrange equation coupled with anisotropic dif-
fusion to maintain the isophotes’ directions. The Curvatur e-Driven Diffusion
(CCD) model [Chan and Shen 00b] enhances the TV method to drive dif-
fusion along the isophotes’ directions and thus allows inpainting of thicker
regions. All above methods essentially solve a Partial Differential Equation
(PDE) that describes the color propagation inside the missing region, subject
to various heuristics that attempt to preserve the isophotes’ directions. Pre-
serving isophotes is, however desirable, never perfectly attained in practice.
The main problem is that both isophote estimation and information propaga-
tion are s ubject to numerical diffusion. Diffusion is desirable as it stabilizes
the PDEs to be solved, but leads inevitably to a cetain amount of blurring of
the inpainted area.
A second type of methods [Oliveira 01] repeatedly convolves a simple 3 ×
3 filter over the missing regions to diffuse known image information to the
missing pixels.
However impressive, the above methods have several drawbacks that pre-
clude their use in practice. The PDE-based methods require implementing
nontrivial iterative numerical methods and techniques, such as anisotropic
diffusion and multiresolution schemes [Bertalmio 00]. Little or no informa-
tion is given on practical implementation details such as various thresholds
or discretization methods, although some steps are mentioned as numerically
unstable. Moreover, such methods are quite slow, e.g., a few minutes for
the relatively small inpainting region shown in Figure 1. In contrast, the
convolution-based method described in [Oliveira 01] is fast and simple to im-
plement. However, this method has no provisions for preserving the isophotes’
directions. Hig h-gr adient image areas must be selected manually before in-
painting and treated separately so as not to be blurred.
We propose a new inpainting algorithm based on propagating an image
smoothness estimator along the image gradient, similar to [Bertalmio 00]. We
estimate the image smoothness as a weighted average ov er a known image
neighborhood of the pixel to inpaint. We treat the missing regions as level
setsandusethefastmarchingmethod(FMM)describedin[Sethian96]to
propagate the image information. Our approach has several advantages:
• it is very simple to implement (t he complete pseudocode is given here);
• it is considerably faster than other inpainting methods–processing an
800 ×600 image (Figure 1) takes under three seconds on a 800 MHz PC;
• it produces very similar results as compared to the other methods;
Telea: An Image Inpainting Technique 27
a)
b)
Figure 1. An 800 × 600 image inpainted in less than three seconds.
• it can easily be customized to use different local inpainting strategies.
In Section 2, we describe our method. Section 3 presents several results,
details our method’s advantages and limitations in comparison to other meth-
ods, and discusses possible enhancements. Source code of a sample method
implementation is available online at the address listed at the end of the paper.
2. Our Method
This section describes our inpai nting method. First, we introduce the math-
ematical model on which we base our inpainting (Section 2.1). Next, we
describe how the missing regions are inpain ted using the FMM (Section 2.2).
Finally, we detail the implementation of inpainting one point on the missing
region’s boundary (Section 2.3).
2.1. Mathematical Model
To explain our method, consider Figure 2, in which one must inpain t the
point p situated on the boundary ∂Ω of the region to inpaint Ω.Takeasmall
neighborhood B
ε
(p)ofsizeε of the kno wn image around p (Figure 2(a)). As
described in [Bertalmio 00, Oliveira 01, Chan and Shen 00a], the inpainting
of p should be determined by the values of the known image points close to
p, i.e., in B
ε
(p). We first consider gray value images, color images being a
natural extension (see Section 2.4). For ε small enough, we consider a first
order approximation I
q
(p)oftheimageinpointp,giventheimageI(q)and
gradient ∇I(q) values of point q (Figure 2(b)):
I
q
(p)=I(q)+∇I(q)(p − q) . (1)
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