An Automatic Image Inpainting Algorithm Based on FCM

Abstract

There are many existing image inpainting algorithms in which the repaired area should be manually determined by users. Aiming at this drawback of the traditional image inpainting algorithms, this paper proposes an automatic image inpainting algorithm which automatically identifies the repaired area by fuzzy C-mean (FCM) algorithm. FCM algorithm classifies the image pixels into a number of categories according to the similarity principle, making the similar pixels clustering into the same category as possible. According to the provided gray value of the pixels to be inpainted, we calculate the category whose distance is the nearest to the inpainting area and this category is to be inpainting area, and then the inpainting area is restored by the TV model to realize image automatic inpainting.

1. Introduction

Inpainting is a trimming process for the defects and cracks of the art, originated from the art field which was originally the heritage restoration of experts on the basis of being faithful to the original. Image inpainting is a technology for restoring the damaged parts of an image by referring to the information from the undamaged parts to make the restored image look “complete”, “continuous” and “natural”.

The terminology “digital image inpainting” was firstly put forward on the international conference in Singapore in 2000. There are many typical image inpainting algorithms proposed by researchers during the past decade. The BSCB [1] model was presented by Bertalmio, Sapiro, Caselles, and Ballester. The TV (total variation) model [2, 3] was proposed by Chan and his team. The CDD (curvature driven diffusions) model [4, 5] was introduced by Chan T. and Shen J. An isotropic diffusion of fast image inpainting was proposed by Oliverira. A model based on RBF was proposed by Zhou et al. [6]. The p-Laplace inpainting model was calculated from p-Laplace operator by Zhang et al. [7–9]. Wang et al. proposed a new spread function to improve the TV model [10], algorithm based on wavelet decomposition was proposed by Zhang and Dai [11] and so forth. However, these models belong to semi-automatic image restoration algorithms that need to manually determine the repaired area. It is a difficult problem for automatically identifying the repaired area of the image and achieving automatically image inpainting function in the field of image inpainting.

This paper attempts to overcome this drawback of the traditional image inpainting techniques to a certain degree. The proposed algorithm utilizes the fuzzy C-means (FCM) clustering algorithm to automatically identify the damaged area and also combines the TV model to realize image automatic inpainting.

2. Theoretical Basis of FCM Identify Damaged Areas

2.1. Basic Knowledge of the Fuzzy Sets

Fuzzy set theory and fuzzy logic [12] are presented by Zadeh, that is, to use precise methods, formulas, and models to measure and handle phenomena and rules which are vague, incomplete, or incorrect. Membership function is defined as the degree to which an object 𝑥 belongs to a fuzzy set 𝐴, usually recorded as 𝑢_𝐴(𝑥); the scopes of its argument are the all possible objects which belongs to the set 𝐴 (that all the points of the space where the set 𝐴 in) in the range of [0,1]; namely, 0≤𝑢_𝐴(𝑥)≤1. 𝑢_𝐴(𝑥)=1 means that 𝑥 completely belongs to set 𝐴, which is equivalent to the traditional set concept 𝑥∈𝐴. A membership function defined in the space of 𝑋={𝑥} is defined as a fuzzy set 𝐴, or defined as fuzzy subset 𝐴 in the domain of 𝑋={𝑥}. For the fuzzy set 𝐴_∼ which has finite objects 𝑥₁,𝑥₂,…,𝑥_𝑛 can be expressed as 𝐴_∼=𝜇_𝐴𝑥_𝑖,𝑥_𝑖∣𝑥_𝑖∈𝑋. (1)

Fuzzy set theory has played an important role in many applications, such as fuzzy clustering analysis, fuzzy pattern recognition [13], fuzzy synthetic judgments [14], fuzzy decision and forecast [15, 16], fuzzy programming, fuzzy probability [17], and fuzzy statistics [18]. In the problem of clustering, the clusters generated by clustering can be regarded as fuzzy sets; therefore, the value of membership of each sample point belongs to a cluster that is in the interval [0,1].

2.2. Theory of FCM Algorithm

Fuzzy C-mean clustering (FCM) [19, 20] is a clustering algorithm that uses membership to determine the degree of each data point which belongs to a cluster. In 1973, Bezdek proposed the algorithm that is an improved algorithm of the early hard C-means clustering (HCM). The fuzzy clustering algorithm uses the initialization method to determine several initial clustering centers and then performs a repeated iteration which continuously adjusts and optimizes the clustering centers to achieve the minimal variance within clusters. It is a method based on the principle of least squares using iterative methods to optimize objective function to get the fuzzy partition of the data sets. Its objective function is defined as min𝐽_FCM(𝑈,𝑉)=^𝑛𝑐_𝑘=1𝑖=1𝑢^𝑚_𝑖𝑘𝑑²_𝑖𝑘, (2) where 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛}∈𝑅^𝑛×𝑝 is a data set that composed of 𝑛 samples of dimension 𝑝, which is divided it into C classes. 𝑥_𝑘 is a 𝑝-dimensional vector; 𝑚∈[1,∞) is the weighted index; when 𝑚=1, the fuzzy C-means clustering is degraded as the classical C-means clustering. Nikhil et al. studies show that the best choice for 𝑚 is in range of [1.5,2.5] and its ideal value is usually set to 𝑚=2; 𝑑_𝑖𝑘 is generally defined as the Euclidean distance, it means the distance between sample 𝑥_𝑘 and center 𝑣_𝑖, that is, 𝑑_𝑖𝑘=‖𝑥_𝑘−𝑣_𝑖‖_𝐴, ‖⋅‖_𝐴 defines as norm of 𝐴, usually 𝐴=2; that is, 𝑑_𝑖𝑘 is 2-norm; 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑐}∈𝑅^𝑐×𝑝 is the set of 𝑐 cluster centers. 𝑣_𝑖 is 𝑝-dimensional vector and denotes the 𝑖th cluster center; 𝑢_𝑖𝑘 is the membership of the 𝑘th pixel that belongs to 𝑖th class. Therefore, the classification results can be expressed as a 𝑐×𝑛-order fuzzy classification matrix 𝑈: 𝑢𝑈=¹¹𝑢₁₂⋯𝑢_1𝑛𝑢₂₁𝑢₂₂⋯𝑢_2𝑛𝑢⋮⋮⋱⋮_𝑐1𝑢_𝑐2⋯𝑢_𝑐𝑛, (3) where 𝑢_𝑖𝑘 need to satisfy the following constraints: (1)𝑢_𝑖𝑘∈[](2)0,1,∀𝑖,𝑘;_𝑐𝑖=1𝑢_𝑖𝑘(3)=1,∀𝑘;0<^𝑐_𝑖=1𝑢_𝑖𝑘<𝑛,∀𝑖. (4)

FCM is to search the optimum number of (𝑈,𝑉) for minimizing the 𝐽_FCM(𝑈,𝑉). Due to the constraint ∑^𝑐_𝑖=1𝑢_𝑖𝑘=1, it introduces a Lagrange multiplier 𝜆 to solve the minimum of the following formula: 𝐽(𝑈,𝑉)=^𝑛𝑐_𝑘=1𝑖=1𝑢^𝑚_𝑖𝑘𝑥_𝑘−𝑥_𝑖²₂+𝜆^𝑐_𝑖=1𝑢_𝑖𝑘−1, (5) where 𝑢_𝑖𝑘=1∑^𝑐_𝑗=1𝑥_𝑘−𝑣_𝑖²₂/𝑥_𝑘−𝑣_𝑗²₂^1/(𝑚−1)𝑣,∀𝑖,𝑘,_𝑖=∑^𝑛_𝑘=1𝑢^𝑚_𝑖𝑘𝑥_𝑘∑^𝑛_𝑘=1𝑢^𝑚_𝑖𝑘,∀𝑘. (6) So the implementation of FCM algorithm is as follows:

Step 1. Select 𝜀>0, set the initial clustering centers 𝑉⁽⁰⁾={𝑣₁,𝑣₂,…,𝑣_𝑐}, and set the number of iteration 𝑙=1.

Step 2. The elements of membership matrix 𝑈^(𝑙) are calculated as 𝑢^𝑙_𝑖𝑘=^𝑐_𝑗=1𝑥_𝑘−𝑣_𝑖2^(𝑙)𝑥_𝑘−𝑣_𝑗2^{(𝑙)2/(𝑚−1)−1},∀𝑖,𝑘. (7) If 𝑑^(𝑙)_𝑘𝑗=‖𝑥_𝑘−𝑣_𝑗‖₂^(𝑙)=0, then 𝑢^(𝑙)_𝑘𝑗=1, 𝑢^(𝑙)_𝑘𝑗=0, 𝑘≠𝑗.

Step 3. Calculate the new cluster center set 𝑉^(𝑙+1), whose elements are 𝑉^(𝑙+1)=∑^𝑛_𝑘=1𝑢^(𝑙)_𝑖𝑘𝑥_𝑘𝑥_𝑘∑^𝑛_𝑘=1𝑢^(𝑙)_𝑖𝑘𝑥_𝑘^𝑚,∀𝑘. (8)

Step 4. If ‖𝑣^(𝑙+1)−𝑣^(𝑙)‖<𝜀, then stop; otherwise 𝑙=𝑙+1 and go to Step 2.

3. The TV Model

3.1. Model Establishment

To establish a normal formula of variation image inpainting model 𝐽_𝑟(𝑢)=_Ω𝑟||𝜆∇𝑢𝑑𝑥𝑑𝑦+^𝐷2_Ω𝑢−𝑢₀²𝑑𝑥𝑑𝑦, (9) where 𝜆_𝐷=𝜆⋅𝑙_𝐷(𝑥)=0,𝑥∈𝐷𝜆,𝑥∈Ω⧵𝐷, (10)𝑢(𝑥,𝑦) is the image gray function, shorthand for 𝑢, 𝑢₀ is the defect image grayscale function, 𝐷 is the repaired area, Ω is the entire image area, Ω⧵𝐷 is the known image field, and 𝜆 is the Lagrange operator, the equation 𝑑𝛿=𝑑𝑥𝑑𝑦 is a double integral.

In (9), we have set the energy functional of TV model 𝑟[|∇𝑢|]=|∇𝑢|; that is, 𝐽_𝑟(𝑢)=_Ω||𝜆∇𝑢𝑑𝑥𝑑𝑦+^𝐷2_Ω𝑢−𝑢₀²𝑑𝑥𝑑𝑦. (11)

After calculating out the extreme value of formula (11), we can obtain Euler-Lagrange equation of TV model as follows: −∇⋅∇𝑢|∇𝑢|−𝜆𝑢−𝑢₀=0. (12)

3.2. Discretization of the Model

Application of the half-point difference method to calculate the diffusion format of the partial differential equation (12).

As shown in Figure 1, 𝑂 is the inpainting pixel point, and Λ₀={𝑁,𝑆,𝑊,𝐸} means four neighborhood points set of 𝑂, and Λ={𝑛,𝑠,𝑤,𝑒} means the four half-pixel points. Denote the objective pixel 𝑂 by 𝑢(𝑖,𝑗), Λ₀={(𝑖,𝑗+1),(𝑖,𝑗−1),(𝑖−1,𝑗),(𝑖+1,𝑗)}. In order to avoid |∇𝑢| turning to 0 and take the value of |∇𝑢|_𝜀=𝜀²+|∇𝑢|² or |∇𝑢|_𝜀=𝜀+|∇𝑢|, if 𝜀 is small enough, it will not affect the performance of TV inpainting model.

Figure 1 

Inpainting pixel points and neighbor fields.

Let 𝑣=(𝑣¹,𝑣²)=∇𝑢/|∇𝑢| and take performance about central difference numerical method for divergence, ∇⋅𝑣=𝜕𝑣¹+𝜕𝑥𝜕𝑣²≈𝑣𝜕𝑦^1𝑒−𝑣¹_𝑤ℎ+𝑣²_𝑛−𝑣²_𝑠ℎ. (13)

The length ℎ=1 and then solve the approach values of 𝑣¹_𝑒, 𝑣¹_𝑤, 𝑣²_𝑛, and 𝑣²_𝑠: 𝑣¹_𝑒=1∇𝑢_𝑒𝜀𝜕𝑢𝜕𝑥_𝑒≈1∇𝑢_𝑒𝜀𝑢_𝐸−𝑢_𝑂ℎ,∇𝑢_𝑒𝜀≈1ℎ𝜀²+𝑢_𝐸−𝑢_𝑂²+𝑢_𝑁𝐸−𝑢_𝑆𝐸+𝑢_𝑁−𝑢_𝑆4². (14)

Similarly we can get 𝑣¹_𝑤=1∇𝑢_𝑤𝜀𝜕𝑢𝜕𝑥_𝑤≈1∇𝑢_𝑤𝜀𝑢_𝑊−𝑢_𝑂ℎ,𝑣²_𝑛=1∇𝑢_𝑛𝜀𝜕𝑢𝜕𝑦_𝑛≈1∇𝑢_𝑛𝜀𝑢_𝑁−𝑢_𝑂ℎ,𝑣²_𝑠=1∇𝑢_𝑠𝜀𝜕𝑢𝜕𝑦_𝑠≈1∇𝑢_𝑠𝜀𝑢_𝑆−𝑢_𝑂ℎ. (15)

So −∇⋅∇𝑢||𝑣∇𝑢=−∇⋅𝑣=¹_𝑤−𝑣¹_𝑒+𝑣²_𝑠−𝑣²_𝑛=_{𝑄∈Λ𝑂,𝑞∈Λ}1∇𝑢_𝑄𝜀𝑢_𝑄−𝑢𝑖,𝑗, (16) where 𝑄∈Λ_𝑂, 𝑞∈Λ means that when 𝑄 is a pixel of Λ_𝑂, then 𝑞 is the corresponding half pixel of Λ; as 𝑄 take 𝐸, correspondingly 𝑞 take 𝑒. So formula (12) can be remarked as _{𝑄∈Λ𝑂,𝑞∈Λ}1∇𝑢_𝑄𝜀𝑢_𝑄−𝑢𝑖,𝑗+𝜆_𝐷𝑖,𝑗𝑢₀𝑖,𝑗−𝑢𝑖,𝑗=0. (17) So, 𝑢=∑𝑖,𝑗^{𝑄∈Λ𝑂,𝑞∈Λ}1/∇𝑢_𝑄𝜀∑_{𝑄∈Λ𝑂,𝑞∈Λ}1/∇𝑢_𝑄𝜀+𝜆_𝐷𝑢𝑖,𝑗^𝑄+𝜆_𝐷𝑖,𝑗∑_{𝑄∈Λ𝑂,𝑞∈Λ}1/∇𝑢_𝑄𝜀+𝜆_𝐷𝑢𝑖,𝑗⁰𝑖,𝑗, (18) where 𝜔_𝑞=1∇𝑢_𝑄𝜀,ℎ_𝑄=𝜔_𝑞𝜆_𝐷(𝑂)+∑_{𝑄∈Λ𝑂,𝑞∈Λ}𝜔_𝑞,ℎ_𝑂=𝜆_𝐷(𝑂)𝜆_𝐷(𝑂)+∑_{𝑄∈Λ𝑂,𝑞∈Λ}𝜔_𝑞. (19) So formula (18) can be remarked as: 𝑢𝑖,𝑗=_{𝑄∈Λ𝑂,𝑞∈Λ}ℎ_𝑄𝑢_𝑄+ℎ_𝑂𝑢_𝑂𝑖,𝑗, (20) where ∑_{𝑄∈Λ𝑂,𝑞∈Λ}ℎ_𝑄+ℎ_𝑂=1; apparently, the inpainting objective pixel value 𝑢(𝑖,𝑗) adds the weight ℎ_𝑄 to be repaired through neighbor field point 𝑢_𝑄.

Use Gauss-Jacobi iteration method to search the optimum: 𝑢^(𝑛)𝑖,𝑗=_{𝑄∈Λ𝑂,𝑞∈Λ}ℎ_𝑄^(𝑛−1)𝑢_𝑄^(𝑛−1)+ℎ_𝑂^(𝑛−1)𝑢_𝑂𝑖,𝑗, (21) where 𝑛 is the number of iterations. According to the discretization process of TV model, the process of TV inpainting model mainly is shown in Algorithm 1.

Initialized 𝑢0(𝑖,𝑗), generally we set 𝑢0(𝑖,𝑗)=𝑢0(𝑖,𝑗);  For  𝑛=1,2,…   For  𝑖=1,2,…,𝑀    For  𝑗=1,2,…,𝑁    By the use of formula (19), calculate 𝜔𝑃,ℎ𝑃,ℎ𝑂 respectively,    the (𝑖,𝑗) as the center of them, then use formula (21) to calculate.    𝑢(𝑛)𝑖,𝑗=∑𝑄∈Λ𝑄,𝑞∈Λℎ𝑄(𝑛−1)𝑢𝑄(𝑛−1)+ℎ𝑂(𝑛−1)𝑢0(𝑖,𝑗)    End   End  End

Algorithm 1 

4. Computer Experiment

4.1. FCM Experimental Analysis

In the FCM experiment in this paper given gray values of the pixel which in the repaired areas, we can calculate the Euclidean distance of it with each clustering center concluded by formula (8), the cluster which has the minimum distance is the repaired area 𝐷.

In this experiment, it is crucial to select the appropriate values of these parameters, namely, the number of image clustering (C value), the initial value of the clustering center (V), C, and the clustering center initial value selected which directly influence whether the defect area can be accurately identified. Generally apply the gray value distribution of the gray histogram of the image to be repaired to determine the value of C and V.

We discussed the experimental results of FCM to identify scratches, text, and big block inpainting areas

4.1.1. Scratch Inpainting

In Figure 2(a) there is identification effect of areas to be repaired; Figure 2(b) is gray histogram of Figure 2(a).

(a)

(b)

(c)

Figure 2 

(a) Identification effect of areas to be repaired with C = 6. (b) The gray histogram of image to be repaired. (c) Identification effect of areas to be repaired with C = 8.

As Table 1 has shown in the experiment, we set C = 6; to initialize clustering centers 𝑉=[0,50,100,125,175,255] and update the clustering centers, obtained clustering center 1 is 21.6442, and the clustering center 2 is 69.3399, the clustering center 3 is 108.8676, the clustering center 4 is 154.7014, the clustering center 5 is 192.2138, and the clustering center 6 is 217.2253. Apparently the area to be repaired in cluster 1 identifying the scratches of the repaired area was shown in Figure 2(a), the pixel in the image to be repaired where it close to black gray value will impact identification of the repaired areas from the figure.

Table 1 

Clustering centers table.

When set C = 8, identification effect of areas to be repaired was shown in Figure 2(c).

Apparently, the effect of Figure 2(c) is better than that of Figure 2(a); at this time, we initialize the clustering center 𝑉=[0,20,50,75,100,125,175,255] and the clustering center 1 is 9.6953, the clustering center 2 is 45.918, the clustering center 3 is 78.5321, the clustering center 4 is 107.6517, the clustering center 5 is 139.0719, the clustering center 6 is 169.5539, the clustering center 7 is 196.3286, the clustering center 8 is 218.8029. Area to be repaired which has been in Cluster 1, as shown in Tables 1 and 2, the cluster center value of 1 is closer to the color of the area to be repaired (gray value is 0) than the cluster center value of 1 when C = 6. Therefore, for the greater value of C, the identification is more accurate.

Table 2 

Clustering centers table.

4.1.2. Text Inpainting

In Figure 3(a), there is the identification result of the areas to be repaired; Figure 3(b) is gray histogram of Figure 3(a).

(a)

(b)

Figure 3 

(a) Identification effect of areas to be repaired with C = 5. (b) The gray histogram of image to be repaired.

As Table 3 shows, in the experiment, we set C = 5 initialize clustering center 𝑉=[0,75,150,200,255], obtained the clustering center 1 is 52.0827, the clustering center 2 is 85.3504, the clustering center 3 is 119.696, the clustering center 4 is 171.9693, and the clustering center 5 is 254.6589. Apparently, the area to be repaired in cluster 5 identifying the scratches of the repaired area was shown in Figure 3(a).

Table 3 

Clustering centers table.

4.1.3. The Big Block Area Inpainting

In Figure 4(a), there is the identification result of areas to be repaired; Figure 4(b) presents gray histogram of Figure 4(a).

(a)

(b)

(c)

Figure 4 

(a) Identification effect of areas to be repaired with C = 7. (b) The gray histogram of image to be repaired. (c) Identification effect of areas to be repaired with C = 10.

As Table 4 has shown, in the experiment, we set C = 7, to initialize clustering center 𝑉=[10,20,50,100,125,175,255], update the clustering center, finally obtained the clustering center 1 is 22.732, the clustering center 2 is 60.23, the clustering center 3 is 93.2463, the clustering center 4 is 124.5873, the clustering center 5 is 162.8123, the clustering center 6 is 194.4558, and the clustering center 7 is 217.9387. Apparently, area to be repaired in cluster 1 identifying the scratches of the repaired area was shown in Figure 4(a); the pixels in the image to be repaired nearly black will be misclassified as noise, so if the grayscale values of the area to be repaired are similar to the gray values ​​of the known area, they will be misclassified.

Table 4 

Clustering centers table.

When we set C = 10, the extracted area to be repaired was shown in Figure 4(c).

Apparently, the result of Figure 4(c) is better than Figure 4(a); at this time, to initialize clustering center 𝑉=[0,10,20,30,40,50,100,125,175,255], update the clustering center, finally obtained the clustering center 1 is 12.2282, the clustering center 2 is 39.6536, the clustering center 3 is 63.6337, the clustering center 4 is 85.1273, the clustering center 5 is 106.6592, the clustering center 6 is 129.8529, the clustering center 7 is 156.3771, the clustering center 8 is 179.4131, the clustering center 9 is 198.9847, and the clustering center 10 is 219.6205. At this time, Area to be repaired which has been in Cluster 1, as shown in Tables 4 and 5, the cluster center value of 1 is closer to the color of the area to be repaired (gray value is 0) than the cluster center value of 1 when C = 7. Therefore, the greater value of C and the initial value of clustering center are more appropriate, and the more accurate identification results can be obtained.

Table 5 

Cluster center table.

4.2. Realization of the Image Automatic Inpainting

Let 𝜆=0, 𝜀=0.0001 with improved signal-to-noise (ISNR) as the objective metric standard of the quantity of the inpainting image, the definition of ISNR as ISNR=10⋅log₁₀∑^𝑀_𝑥=1∑^𝑁_𝑦=1𝐼̂𝑥,𝑦−𝐼⁰𝑥,𝑦²∑_𝑥=1∑_𝑦=1𝐼̂𝑥,𝑦−𝐼⁰𝑥,𝑦², (22) where 𝐼(̂𝑥,𝑦) is the original gray image, 𝐼⁰(𝑥,𝑦) is the damaged image, and 𝐼(𝑥,𝑦) is the recovered image.

Combining TV model inpainting algorithm with FCM identifying the damaged areas, we have discussed the automatic inpainting effect of scratches and big block damaged areas as follows.

4.2.1. Scratch Inpainting

In the following experiments, 𝑛 is the number of iteration, 𝑇 is the inpainting time, and ISNR is improved signal-to-noise.

Figures 5(a) and 5(b) show that the automatic inpainting model has good inpainting effect on fine scratches and coarse scratches and can accurately identify the different kinds of image noise. Figure 5(a) is black noise, and Figure 5(b) is white noise.

(a)

(b)

Figure 5 

(a) 𝑛=50, 𝑇=0.0832, ISNR = 60.648. (b) 𝑛=50, 𝑇=0.0685, ISNR = 53.0085.

4.2.2. The Big Block Area Inpainting

Comparing Figures 5 and 6, it shows that the automatic image inpainting model may generate blurred edges for big block damaged area, not as good as small inpainting areas.

(a)

(b)

Figure 6 

(a) 𝑛=50, 𝑇=0.0598, ISNR = 35.5889. (b) 𝑛=50, 𝑇=0.1054, ISNR = 10.0905.

5. Conclusions

FCM clustering is an unsupervised clustering technique applied to classify images into clusters with similar properties. It utilizes the distance between pixels and cluster centers in the repaired area to compute the membership function. Experiments showed that the inpainting effect of automatic image inpainting model was decided by the inpainting model, identifying the repaired area that depends on FCM algorithm. When the selected clustering number C and the selected clustering center initial value are appropriate, FCM can accurately identify the repaired area. As combined with the inpainting model, we can realize automatic digital image inpainting function.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.