Image inpainting aims to fill the undetectable domain, which is applicable to object removal, restoration for corrupted pictures, etc. Factually, we can only use the known information to estimate the undetectable information to obtain an inpainting that integrates into the surroundings. In other words, the inpainting can be viewed as a prediction process.
Diffusion is one of the main ideas for inpainting, which is to fill the undetectable domain using surroundings in a diffusion law. Diffusionbased inpainting such as [1,2,3] is completed via solving a nonlinear diffusion equation. The total variation (TV) inpainting model of Chan and Shen [2] is wellknown for image inpainting and was developed into a CurvatureDriven Diffusions (CDD) inpainting model [3] later by the authors. The TV was used for some regularity and was earlier designed for noise removal with the aim to keep edges of the images by Rudin et al. [4]. The convolutional algorithm proposed by Oliveira et al. [5] is also a diffusion approach.
Diffusionbased algorithms perform well for smooth regions. For textured images, the exemplar matching algorithms perform well; see studies [6, 7], for example. The main idea of the exemplar matching is to search for the most similar patches to fill the undetectable domain. In recent years, deep learning showed prosperities in more challenging inpainting tasks. The Context Encoders (CE) proposed by Pathak et al. [8] suggest generative adversarial networks (GAN) [9] on restoring the complex features and motivated many studies. Yeh et al. [10] adopted the poisson blending [11] to the second training for the generative neural network to enhance the inpainting. Unlike [10] which improves inpainting performance via twice training, Iizuka et al. [12] considered the combination of global and local discriminations simultaneously. For the GANbased inpainting, it is mainly the feature discrimination enhances the performance of the generative model.
In this paper, we propose local smoothnessinformed (regularized) convolutional neural network (CNN) models for image inpainting and then provide a comparative study of them with various versions of the TV model. In contract with the artificially defined convolutional kernels, the learned kernels are determined by image itself. To determine the convolutional kernels, we consider two instructive knowledge. One is the TV. Traditionally, the TVbased inpainting is completed via solving the associated EulerLagrange equation iteratively and is used for noise removal as well. For inpainting without noise removal, a TV term may still perform as a regularization. We propose to study a
By deep learning method, the inpainting is determined by the training data in the detectable domain. Hence, for an image which composes of various gray intensities and structures, it is difficult or even impossible to discover the representation for the global image. Additionally, only relevant data to the surroundings of the undetectable domain contributes to feature estimation. Since the computational time consuming is in proportion to the image size, we consider the local image containing the damaged domain as the input with the aim to mostly use the relevant data and mostly reduce the computational time. In implementation unlike [10], this
Generally, this paper presents the following studies:
A deep learning method for the EulerLagrange equation, which differs from the point based numerical iterative method. The learning method without complex derivatives on the surrounding and fills the domain automatically.
Exploration of the local
Without extra images for learning that saves the training time.
The rest of this paper is organized as follow. In Section 2, we mathematically describe the inpainting problem. In Section 3, we describe the CNN method with TV and our proposed regularization terms and numerical computational details for the image inpainting. Results are presented in Section 4. Discussions and conclusions are stated in Section 5.
With merely a damaged image, inpainting is to reconstruct the information of the damaged parts using the observed data of the image. The inpainting can be mathematically described as follow.
Consider a damaged graylevel image on domain Ω in Fig. 1, divide into an undetectable domain Ω_{1} and a detectable domain Ω_{2}. Denote
For the inpainting problem, we explore a CNN approach with the TV and local smoothness constraints separately. The main idea of the local smoothness preservation is to condition the derivatives of the surroundings around the undetectable domain on the given image at the training stage. The following parts of this section present the designed CNN for inpainting and the computational details.
CNN has been designed since the 1980s and was later diversely developed, for example, the wellknown CNN developed by Fukushima [13] and Zhang et al. [14]. We utilize CNN to the graylevel image inpainting and design the CNN with the same sized output to the input for every layer to avoid the data loss. The CNN accepts one damaged image and outputs one inpainted image; the image is convoluted by the kernels and activated from input to output. The kernels stride over the image by one step to complete the convolution for the whole image and the Rectified Linear Units [15] is used as the activation operator. The main computational details and architecture of the CNN are as follow.
Consider a CNN (denoted as
Training the CNN aims to learn the kernels and biases using the loss function presented in Section 3.2.
Let
The variational problem to find a solution
The TV on the whole domain Ω was initially for noise removal. For image inpainting without noise removal to the partial unknown domain, it mainly to enhance the connectivity between the corrupted domain and the detectable domain. In this paper, we explore a local
Denote
In this section, we present the computational details of deep learning models mentioned in previous section for image inpainting. The integral is discretely represented in a summation and derivatives are approximated by finite differences. Avoiding data loss, we pad zero sides to the boundaries of the image matrix.
Denote
Computing the gradient for the TV, we tend to use the Sobel operator [18] {
Computing the second order partial derivatives, we consider the convolutions twice by the operators
A
Near the boundary of Ω_{1}, it is not convenient to compute derivatives with the operators
Let
We adopt the Adam [19] algorithm to update the parameters. Computational process is stated with loss function
SmoothnessInformed Deep Learning for Image Inpainting
1:  Given required parameters 
2:  Initialize parameters 
3:  
4:  Produce estimation: 
5:  Compute the loss:

6:  Update the parameters using Adam algorithm on TensorFlow [20]. 
7: 
We present the implementation details and results of the smoothnessinformed deep learning for image inpainting in this section. The algorithm is implemented in Python. We test these models using photos captured by ourself in Fig. 3, the images from [16] for comparison in Fig. 4. The shape of the damaged domain and size are referenced to [10, 16]. For the damaged blocks in Fig. 3, we size the input image as 40 × 40 and size the damaged domain as 24 × 24. For the damaged blocks in Fig. 4, we size the input image as 60 × 60 and size the damaged domain as 32 × 32. The inpainting results in Figs. 5 and 6. We evaluate the inpainting by the Peak Signal to Noise Ratio (PSNR) [21], Structural Similarity (SSIM) [22], which are defined by
In this study, we designed a smoothnessinformed CNN approach for image inpainting, the TV and the local
Quantitative evaluation for the inpainting in Fig. 2/Fig. 3
8/5  13.0833/12.9284  0.9979/0.7977  
8/5  13.1586/13.6270  0.9983/0.9960  
8/5  13.1694/13.9076  0.9987/0.9985  
8/5  13.1419/13.6265  0.9966/0.9871  
8/5  13.1984/14.5362  0.9974/0.9990 
The inpainted images in Fig. 5 show that the local
The deep learning algorithm for the TV inpainting model conducts better PSNR and SSIM values for the same image in comparison to (may influenced by inpainting regions and numbers) the Split Bregman algorithm in [16].
Only detectable data of an image to be used for learning by our method, which is not limited by the training dataset. The smoothnessinformed CNN is applicable to both smooth and some texture images with appropriate detectable data in the damaged image, and can deal with large size as studied in [10, 16] without numerous images for training. The local
This study mainly to show that, the deep CNNbased image inpainting, without TV, local
SmoothnessInformed Deep Learning for Image Inpainting
1:  Given required parameters 
2:  Initialize parameters 
3:  
4:  Produce estimation: 
5:  Compute the loss:

6:  Update the parameters using Adam algorithm on TensorFlow [ 
7: 
Quantitative evaluation for the inpainting in Fig. 2/Fig. 3
8/5  13.0833/12.9284  0.9979/0.7977  
8/5  13.1586/13.6270  0.9983/0.9960  
8/5  13.1694/13.9076  0.9987/0.9985  
8/5  13.1419/13.6265  0.9966/0.9871  
8/5  13.1984/14.5362  0.9974/0.9990 
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