IMAGE INPAINTING CONSIDERING BRIGHTNESS CHANGE
AND SPATIAL LOCALITY OF TEXTURES

Norihiko Kawai, Tomokazu Sato, Naokazu Yokoya
Graduate School of Information Science, Nara Institute of Science and Technology

8916-5 Takayama, Ikoma, Nara 630-0192, Japan
norihi-k@is.naist.jp, tomoka-s@is.naist.jp, yokoya@is.naist.jp

Keywords:

image inpainting, image completion, energy minimization

Abstract:

Image inpainting is a tequnique for removing undesired visual objects in images and filling the missing regions with plausible textures. Conventionally, the missing parts of an image are completed by optimizing theobjective function, which is defined based on pattern similarity between the missing region and the rest of
the image (data region). However, unnatural textures are easily generated due to two factors: (1) available
samples in the data region are quite limited, and (2) pattern similarity is one of the required conditions but is
not sufficient for reproducing natural textures. In this paper, in order to improve the image quality of completed texture, the objective function is extended by allowing brightness changes of sample textures (for (1))
and introducing spatial locality as an additional constraint (for (2)). The effectiveness of these extensions is
successfully demonstrated by applying the proposed method to one hundred images and comparing the resultswith those obtained by the conventional methods.

1

INTRODUCTION

Image inpainting is a tequnique for removing undesired visual objects in images and filling the missing
regions with plausible textures. This research can beclassified into two categories. One is a non-exemplarbased method and the other is an exemplar-based
method. The non-exemplar-based methods(A. Levinet al., 2003; C. Ballester et al., 2001a; C. Ballester
et al., 2001b; D. Tschumperlé, 2006; E. Villéger
et al., 2004; M. Bertalmio et al., 2001; M. Bertalmio
et al., 2000; S. Esedoglu and J. Shen, 2003; S. Masnou and J.M. Morel, 1998; T. Chan and J. Shen,
2001; T. Chan et al., 2002) are based on pixel interpolation considering the continuity of pixel intensity.
These methods are effective for small image gaps likescratches in a photograph. However, the resultant image easily becomes unclear when the missing regionis large. Therefore, recently many exemplar-based
inpainting methods have been intensively developed
because they can synthesize complex textures in the
missing region.
Exemplar-based methods basically synthesize textures for the missing region based on pattern simi-

larity that is defined between the missing region andthe rest of the image. Some of the exemplar-based
methods use the distance in the feature space as asimilarity measure. As the feature space, Fourier
space, wavelet domain and eigenspace have beenused (A.N. Hirani and T. Totsuka, 1996; S.D. Raneet al., 1996; T. Amano, 2004). Most of the otherexemplar-based methods simply employ SSD (sumof squared differences)-based pattern similarity measures (A. Criminisi et al., 2004; A.A. Efros and T.K.
Leung, 1999; B. Li et al., 2005; C. Allène and N.
Paragios, 2006; I. Drori et al., 2003; J. Jia and C.
Tang, 2003; J. Sun et al., 2005; N. Komodakis andG. Tziritas, 2006; R. Bornard et al., 2002; Y. Wexler
et al., 2007). Efros et al. (A.A. Efros and T.K. Leung, 1999) have proposed a method that successively
copies the most similar pattern from the data region to
the missing region. Although this method can generate complex textures, the quality of resultant images
is severely affected by the order of texture copy. To
obtain good results with the successive texture copy,
confidence maps such as the number of fixed pixels
in a window, strength of isophotes around the missing
regions and pattern similarity have been used to de-termine the order of texture copy (A. Criminisi et al.,
2004; B. Li et al., 2005; R. Bornard et al., 2002). Although the duplication of similar textures preserves
the local texture continuity in these methods, discontinuous textures are easily synthesized in the completed image. To avoid the ordering problem, recent
inpainting methods employ the iterative global optimization approach (C. Allène and N. Paragios, 2006;
Y. Wexler et al., 2007; N. Komodakis and G. Tziritas,
2006). In these methods, the objective functions that
evaluate the pattern similarity are defined and optimized by using the EM algorithm, Belief Propagation
approach and graph cut approach.
Although the global optimization methods haveobtained good results for many images, unnatural images are still generated due to two factors: (1) available samples in the data region are quite limited, and(2) pattern similarity is one of the required conditions but is not sufficient for reproducing natural textures. Thus, in order to improve the image quality,
these two factors should be considered. There havealready been some attempts at this. For (1), the scaleand orientation of textures have been considered toobtain effective samples (I. Drori et al., 2003). For(2), Sun et al. (J. Sun et al., 2005) and Jia et al. (J.
Jia and C. Tang, 2003) have proposed techniques thatuse explicit constraints for texture boundaries. Thesemethods synthesize textures preserving the edges orboundaries of the texture. However, automatic andeffective determination of these explicit constraints isstill difficult.
In this paper, in order to obtain good results formany images, we employ a new approach different
from conventional ones. For (1), brightness changeof sample textures that has not been considered in theliterature is allowed to obtain effective samples. For(2), the spatial locality of texture pattern is considered
as an implicit constraint that is usually satisfied in a
lot of real scenes. In this study, these ideas are implemented with the framework of energy minimization based on Wexler’s objective function (Y. Wexler
et al., 2007). The effectiveness of our extensions isdemonstrated with comparison of completed images
and subjective evaluation by a questionnaire.

2

IMAGE INPAINTING BY
ENERGY MINIMIZATION

Figure 1 shows the flow of the proposed method.
First, a user manually selects regions to be repaired
such as physically damaged regions and undesired object regions in an image (a). Next, initial values are
given to the missing regions (b). Finally, selected re-Start(a) Input target region(b) Initial values are given(c) Update of pixels
by minimizing energy
No
Is energy converged?
Yes
End

Figure 1: Procedure of the proposed method.

Data region Φ

window Wx̂Ω′Ω

Missing
regionwindow W

x

Figure 2: Missing and data regions in an image.

gions are completed by minimizing the energy function (c). In the following sections, first, we describe
the conventional objective function which is defined
based on pattern similarity SSD in Section 2.1. The
new energy function considering brightness change
and spatial locality is then defined in Section 2.2, andfinally Section 2.3 describes the inpainting procedurethat minimizes the energy function.

2.1 Energy function based on patternsimilarity
This section briefly describes the SSD-based objective function for image inpainting originally proposedby Wexler et al. (Y. Wexler et al., 2007). Here, although the original objective function is defined as aprobability density function, we redefine this objective function as an equivalent energy function.
As illustrated in Figure 2, first, an image is divided
into region Ω including the missing region Ω selectedby a user and the data region Φ, which is the rest of
the image. The plausibility in the missing region Ω isdefined by using image patterns in the data region Φ.
Here, Ω is the expanded area of the missing regionΩ in which there is a central pixel of a square window W of size NW (where NW is a constant) overlapping the region Ω. The energy function that represents
the plausibility in the missing region is defined as theweighed sum of SSD between the pixels around the
pixel x in region Ω and those around the pixel x̂org inregion Φ as follows:∑ wx SSDx x̂org 

Eorg

xΩ

(1)

¼

where x̂org in the data region Φ denotes the pixel
around which the pattern is the most similar to thataround x in the region Ω , and SSDx x̂org  is defined
as follows.
SSDx x̂org ∑ I x  p  Ix̂org  p2



(2)

pW

Here, I x represents the intensity of pixel x. The
pixel x̂org for minimizing E org is decided as follows.
forg x

x̂org

argmin SSDx x 

(3)

x¼ Φ

Note that the weight wx is set as 1 if x is inside of
the region Ω  Ω because pixel values in this region
are fixed: otherwise wx is set as cd (d is the distancefrom the boundary of Ω and c is a constant) because
pixel values around the boundary have higher confidence than those in the center of the missing region.
In Wexler’s work (Y. Wexler et al., 2007) , themissing region is completed by calculating the pixel
value I x in the missing region and the position of the
pixel x̂org that minimizes the energy function E org .

2.2.1 Pattern similarity considering brightnesschangeThe similarity measure SSD is defined as:
SSD x x̂∑ I x  p  αxx̂ I x̂  p2



(6)

pW

To allow the brightness change, we introduce an intensity modification coefficient α xx̂ . In this paper,
we employ the ratio of average pixel values aroundthe pixels x and x̂ as the modification coefficientαxx̂ . However, an unnatural image is easily generated
if large brightness change is approximated by linear
transformation. Therefore, we limit the range of thevalue αxx̂ where brightness change can be approximated linearly as given in Eq. (7).

αxx̂



1  D (βxx̂  1  D)
(1  D  ββ 1  D)xx̂xx̂1  D (β  1  D)
xx̂

(7)

where D is a constant (0  D  1) and β xx̂ is defined
as follows:



βxx̂



∑qW I x  q2∑qW I x̂  q2(8)

2.2 Energy function extended byconsidering brightness change andspatial locality

The modification coefficient α xx̂ makes brightness ofgenerated textures smooth while preserving texture
patterns and enables utilization of the texture that has
different brightness but the same pattern.

In this study, we extend the original energy function
Eorg defined in Eq. (1) considering brightness change
and spatial locality of texture patterns. Concretely,
we introduce a modification coefficient to allow linear
brightness change of the texture pattern. For considering spatial pattern locality, the cost function basedon distance between the pixel in the missing region
and the corresponding pixel in the data region is also
added to the original energy function. The extended
energy function is defined as follows:

2.2.2 Spatial locality of textureE∑ wx SSD x x̂ wdisSDx x̂



(4)

xΩ¼

where SSD x x̂ represents the pattern similarity
considering brightness change, and SDx x̂ means
the cost term for the spatial locality. w dis is the weight
representing the strength of spatial locality. x̂ is determined as the pixel position that minimizes the extended energy function E:
x̂

f x

argminSSD x x  wdis SDx x  (5)x ¼ Φ

In the following sections, definitions of SSD x x̂
and SDx x̂ are detailed.

Spatial locality of a texture pattern is defined by using
a sigmoid function:

W 
(9)
1  eK xx̂X0 
where K and X0 are constant and  W  is the numberof pixels in a window. This cost function is defined
based on the assumption that the probability of similar texture existence for a certain pixel is uniformly
high for the object region where the pixel exists. On
the other hand, outside the object region, the probability can be assumed to be uniformly low. Note thata constant-sized object region is currently assumed in
Eq. (9) because we could not know the range of theobject in the missing region. By adding the constraintof spatial locality, even when the deformation of texture pattern exists around the target region, appropriate textures that exist near the target region are preferentially selected. Thus, textures are less likely to blurby not selecting blurry textures of low frequency far
from the missing region.
SDx x̂2.3 Update of pixel values and window
pairs for energy minimization
The energy function E defined in Eq. (4) is minimized
by using a framework of greedy algorithm similar to
Wexler’s EM approach (Y. Wexler et al., 2007). In
our method, we pay attention to the fact that the energy function E for each pixel can be treated independently if similar pattern pairs (x, x̂) calculated by
Eq. (5) can be fixed and the change of coefficient α xx̂
in the iteration is very small. Thus, we repeat the following two processes until the energy converges: (I)update of pairs of windows for fixed pixel values, and(II) parallel update of all the pixel values in the missing region for fixed similar pairs of windows in the
missing region and the data region.
In process (I), we update all the similar patternpairs of windows in the missing region and in the dataregion fixing the pixel values calculated in the process(II). Concretely, the update of the pair of windows canbe performed by calculating SSD and SD that satisfy
Eq. (5) and determining the position x̂.
In process (II), we update all the pixel values I x
in the missing regions in parallel by minimizing theenergy defined by Eq. (4). In the following, we describe the method for calculating the pixel values I x
for fixed pairs of windows. First, the energy E is resolved into the element energy E x for each pixel inthe missing region. As shown in Figure 3, the target pixel to be updated is x, and the pixel position
inside a window can be expressed as x  p (p  W )and is corresponded to f x  p by Eq. (5). Thus,
the position of the pixel corresponding to the pixel
x is f x  p  p. Now, the element energy E x
can be defined in terms of the pixel values of x andf x  p  p, the coefficient αxx̂ and the Euclid distance between x and f x as follows:E x∑ wxp I x  αxp f xp I  f x  p  p2

pW

 wdis 1  e KWx x̂ X   0 



(10)

The relationship between the energy E for all of themissing region and the element energy E x for eachpixel can be written as follows:E∑ E x C



(11)

xΩ

C is the energy of pixels in the region Ω  Ω , and istreated as a constant because pixel intensities in theregion and all pairs of windows are fixed here. By
differentiating E with respect to I x in the missing
region, the requirement for minimizing the energy EΦf (x +p) −p

Ω′
x

f

x +p

f (x + p)
most similar windows

Figure 3: Relationship between pixels in energy calculation.

can be obtained as follows:∂E x
∂E∂I xk  x∑
∂I
xk 
Ω

0

(12)

Here, if it is assumed that the change of intensity
modification coefficient α xx̂ is smaller than that ofthe pixel intensity I x k , we can obtain the following
equation.
∂αxi x0 xi x j  Ω x  Φ(13)∂I x j 
From this equation, the equation ∂E x∂I x k  0(x  xk ) is formed. Thus, we can minimize the energy
E by calculating I x k , which satisfies the following
equation.
∂E xk ∂E0
(14)∂I xk ∂I xk 
By generalizing Eq. (14) to all the pixels in the missing region, each pixel value I x in the missing region
can be calculated as follows:∑pW wxp αxp f xp I  f x  p  pI x (15)∑pW wxp
Eq. (15) gives an approximate solution when Eq. (13)is satisfied. We can obtain a good solution as the energy converges because the value of intensity modification coefficient αxx̂ converges as I x converges.
Additionally, in order to avoid local minima efficiently, a coarse-to-fine approach is also employed.
Specifically, an image pyramid is generated and theenergy minimization processes (i) and (ii) are repeated from higher-level to lower-level layers successively using a certain size of window. Good initialvalues are given to the lower layer by projecting results from the higher layer. This makes it possible todecrease computational cost and avoid local minima.
In the lowest layer (original size), the energy minimization process is repeated while reducing the sizeof the window, and it enables reproduction of moredetailed textures.
¼

3

EXPERIMENTS

In this section, four images that have different characteristics are selected from one hundred images asshown in Figure 4(a). The missing regions for theseimages are given in Figure 4(b). Figure 4(c) indicatesthe resultant images by our implemented Wexler’s
method described in Section 2.1. Figure 4(d) shows
the images completed by the proposed method.
Image (I) includes little brightness change underconstant illumination and little pattern change in thesame object region around the missing region. It canbe confirmed that images generated by the conventional and proposed methods are natural. The subjective difference is very small.
Image (II) includes little pattern change withoutcomplex textures but large brightness changes under nonconstant illumination around the missing region. By the conventional method, the resultant image
looks unnatural because sudden intensity changes appear at the seat and the seat back. By allowing bright-

ness changes of sample textures, the sudden intensitychange is suppressed by the proposed method.
Image (III) includes little brightness change under constant illumination but large pattern change dueto the various sizes and shapes of objects around themissing regions. In this image, although the samekinds of textures apparently exist around the missing regions, texture pattern greatly changes due to thedifferent sizes of stones. A part of the missing regions is blurred in white by the conventional method
because the SSD-based similarity is sensitive to thepattern changes especially for high-frequency components, and thus inappropriate textures are selected forthe missing regions. It should be noted that there exists spatial locality of texture pattern such as watercolor and stones around the missing region in image(III). By considering spatial locality of texture pattern, neighboring textures are preferentially selected
and thus the missing region is completed successfullyby the proposed method.
Image (IV) includes large brightness change undernonconstant illumination conditions and texture pattern continuously changes due to the perspective projection effect. In the resultant image of the conventional method, an unnatural image is generated dueto the blurs on the textured area with black squares
and the discontinuous brightness changes at the walland floor. On the other hand, in the proposed method,by using the constraint of the spatial locality of textures, neighboring textures are selected for completion of the missing region and windows of the post
are reproduced in Figure 4(d). In addition, by allowing brightness changes of sample textures, brightnesschange inside the missing region becomes more natural than that by the conventional method.
Next, we have compared the conventional andproposed methods with respect to computational cost.
Table 2 shows the processing time of the conventional and proposed methods. The proposed method
requires about three to five times as much time as theconventional method. This is because the computational cost for calculating intensity modification coefficients and cost function considering spatial locality
is increased.

Table 1: Parameters in experiment.

Table 2: Processing time.

To demonstrate the effectiveness of our extensionsof the objective function, we have applied five kindsof image inpainting methods including the conventional methods and the proposed method to one hun200 pixels). First, the effecdred images (200tiveness of the proposed method is demonstrated bycomparing the characteristic results of the proposed
method and our implemented Wexler’s method (Y.
Wexler et al., 2007). Next, by the subjective evaluation based on a questionnaire using our implemented
Wexler’s method, Criminisi’s method (A. Criminisi
et al., 2004) and the proposed method, the effectiveness of our method is objectively assessed.
In these experiments, we used a standard PC(CPU: Xeon 3.2 GHz, Memory: 8 GB) and each parameter in the energy function was set as shown in
Table 1. Here, the missing region was manually specified, and the average pixel value of the boundary of
the missing region is given as an initial value in the
missing region.

3.1 Comparison of inpainted images

Window size

Nw

Weight for distanceParameter in sigmoid function

wdis
K
X0
D

Range of coefficient α

max 9 9min 3 3
120
0.4
20
0.1

Image (I)Image (II)Image (III)Image (IV)Conventional
method
2’17”
3’17”
3’28”
5’25”

Proposedmethod
8’45”
12’15”
18’59”
18’32”

(a)

(b)

(c)

(d)

Image (I): little change in brightness and pattern

(a)

(b)

(c)

(d)

Image (II): large change in brightness

(a)

(b)

(c)

(d)

Image (III): change in texture pattern

(a)

(b)

(c)

(d)

Image (IV): large change in brightness and continuous deformation of texture patternFigure 4: Image inpainting for four representative images: (a) original image, (b) missing region specified manually, (c)resultant image by conventional method, and (d) resultant image by our method.

3.2 Evaluation by a questionnaire
In this section, completed images for one hundred images by five kinds of inpainting methods are subjectively evaluated by 37 subjects. The subjects are men
and women in their twenties and all of them often
use computers. This experiment aims to illustrate theeffectiveness of the proposed method objectively by
evaluating the resultant images by subjects.
3.2.1 Evaluation method
The subjects were requested to access the web pagefor questionnaire evaluation and evaluate the 500 resultant images for 100 input images by giving a scoreof 1 to 5. In this experiment, images were completed by five methods: our implemented Criminisi’s
method (method A) that is the representative of successive synthesis methods, our implemented Wexler’s
method (method B), proposed method allowing only
brightness change (method C), proposed method considering only spatial locality (method D), and proposed method considering both brightness change andspatial locality (method E). On the evaluation web
page, the resultant images generated by five methods
were arranged in random order so that subjects could
not know the relationship between image and method.
The evaluation criteria were that the lowest score 1
was for an image that could not be used and the highest score 5 was for an image that was natural enoughto use for personal homepages or magazines.
3.2.2 Results and discussion
The average score for the 100 resultant images andthe number of images that obtained the highest scoreare shown in Table 3 for each method. 1 Table 3 showsthat the average score of the resultant images by the
proposed method (method E) is higher than that by theconventional methods (methods A and B) and the inpainted images by the proposed method are scored as
the best most frequently. In this experiment, scores ofthe proposed method (method E) and the conventional
methods (methods A and B) were also compared byusing the t-test with a 5% significant level. In the
result, significant difference was observed between
these scores. Therefore, the proposed method can beverified to be better than the conventional methods
A and B. In addition, both method considering only
brightness change (method C) and method considering only spatial locality (method D) obtained higherscores than methods A and B and the significant difference was also observed by using the t-test with a
1 100 input images and resultant images are shown on

the web page [http://yokoya.naist.jp/research/visapp/](a) image with missing
region.

(b) result by ourimplemented Wexler’s
method.

(c) result by our
method (wdis 120).

(d) result by our
method (wdis 2700).

Figure 5: Example images for which both the proposed andconventional methods have a problem.

5% significant level. This means that the introductionof each factor to the energy function is clearly effective.
It should be noted here that there exist some images for which both the proposed (method E) andconventional (method B) methods have a problem asshown in Figure 5. The average scores for both methods are less than 2 points. In this image, due to thefairly large change in texture patterns, the texture in
the missing region blurs despite the consideration of
spatial locality. Here, when the weight for spatial locality was increased (wdis 2700) a more natural image was generated as shown in Figure 5 (d). The com-

Table 3: Average score and the number of best-scoring images from 100 images.

Method A
Method B
Method C
Method D
Method E

Average of
score
2.21
3.24
3.39
3.42
3.60

Number ofbest-scoring images
7
7
21
20
45

pletion with a single weight coefficient does not always work well, and thus it is necessary to determine
the parameter adaptively considering the characteristics of the image in order to obtain good results formany images containing complex textures.

4

CONCLUSION

C. Ballester, V. Caselles, J. Verdera, M. Bertalmio, and G.
Sapiro (2001b). A Variational Model for Filling-In
Gray Level and Color Images. In Proc. ICCV, pages
10–16.
D. Tschumperlé (2006). Curvature-Preserving Regularization of Multi-valued Images Using PDE’s. In Proc.
ECCV, volume 2, pages 295–307.
E. Villéger, G. Aubert, and L. Blanc-Féraud (2004). Image
Disocclusion Using a Probabilistic Gradient Orientation. In Proc. ICPR, volume 2, pages 52–55.
I. Drori, D. Cohen-Or, and H. Yeshurun (2003). FragmentBased Image Completion. In Proc. SIGGRAPH2003,
pages 303–312.

In this paper, the objective function for image inpainting is extended to acquire natural images. To obtain
good results, two factors were considered: (1) brightness change of sample textures was allowed, (2) spatial locality was introduced as a new constraint. By
considering these two factors, the missing region wascompleted successfully for many images. In experiments, we have demonstrated the effectiveness of our
method by comparing the resultant images of the conventional and proposed methods. In addition, by aquestionnaire evaluation using 37 subjects, we haveverified that the proposed method could obtain goodresults for more images than the conventional method.
In experiments, parameters such as the size of window and the weight in the energy function were decided empirically. In future work, we should establish
a method to decide optimum parameters.

N. Komodakis and G. Tziritas (2006). Image Completion
Using Global Optimization. In Proc. CVPR, pages
442–452.

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