Energy-Driven Image Interpolation Using Gaussian Process Regression

Image interpolation, as a method of obtaining a high-resolution image from the corresponding low-resolution image, is a classical problem in image processing. In this paper, we propose a novel energy-driven interpolation algorithm employing Gaussian process regression. In our algorithm, each interpolated pixel is predicted by a combination of two information sources: first is a statistical model adopted to mine underlying information, and second is an energy computation technique used to acquire information on pixel properties. We further demonstrate that our algorithm can not only achieve image interpolation, but also reduce noise in the original image. Our experiments show that the proposed algorithm can achieve encouraging performance in terms of image visualization and quantitative measures.


Introduction
Image interpolation is a very important aspect of image processing and involves the use of a known pixel set to produce an unknown pixel set, resulting in an image of higher resolution 1, 2 .This technique is widely used in remote sensing, aerospace, infrared imaging, low-light level night imagery, and other fields 3-5 .However, maintaining image quality during image interpolation is still a difficult issue 6 .To address this, many image interpolation methods have been proposed.For example, traditional bilinear interpolation computes the unknown pixel value using the location information between the adjacent pixels.This technique does not consider the contents of the image, so edge blurring will occur in the interpolated image 7, 8 .In order to capture image details more clearly, an artifact-free image upscaling method called ICBI 9 has recently been proposed, which uses iterative curvature-based interpolation to obtain a high image quality, but does not take into account underlying local information between image patches.Local image information can be mined according to its structural redundancy characteristic, as proposed by Glasner et al. 10 .This characteristic can lay the foundations for the training and predicting of a statistical model 11, 12 .A statistical model known as Gaussian process regression GPR was first applied in the reconstruction of high-resolution images in 2011 and has been shown to be capable of generating an image with sharp edges by extracting the necessary information from a low-resolution image 13 .However, it should be noted that this method only uses the local structural information for each pixel's neighborhood, so it can still generate unexpected details.To develop the above techniques, we propose here a novel energydriven interpolation algorithm employing Gaussian process regression EGPR Figure 1 .This algorithm not only emphasizes the influence of adjacent pixel properties on interpolated values, but also brings into full play the role of the statistical model.
Our contribution is twofold.Firstly, we propose a framework for both magnification and deblurring in order to fulfill the interpolation task for low-resolution images with low noise.Secondly, we demonstrate an energy-driven approach based on the properties of adjacent pixels within this framework.In addition, we define the processing unit and its properties for better implementation of the EGPR algorithm.
The rest of the paper is structured as follows.Section 2 discusses GPR.Section 3 illustrates the proposed EGPR algorithm.Section 4 presents experimental work carried out to demonstrate the effectiveness of our algorithm.Section 5 concludes the paper.

Gaussian Process Regression
In recent years, GPR has become a hot issue in the field of machine learning and has attracted great academic interest 14-16 .It has many advantages, including its rigorous underlying statistical learning theory, easy regression process implementation, few parameters, and improved model interpretability 17-19 .As a result of these benefits, it has been used in many areas 20-23 ; however, to the best of our knowledge, it has not yet been fully utilized in image interpolation.Rasmussen and Williams 24 defined the Gaussian process and noted in particular that a Gaussian process is completely specified by its mean and covariance functions 2.1 and 2.2 , resp.: where x and x are any random variables.In particular, they could represent n-dimensional input or output vectors.The Gaussian process can be written as follows: There are a variety of covariance functions, of which one of the most commonly used is the squared exponential SE covariance function

2.4
In Gaussian processes, the marginal likelihood p y | X at a point is very useful and is the integral of the likelihood multiplied by the prior probability We can rewrite 2.5 as follows: We can make use of Gaussian identities to obtain 2.7 , in order to compute the log marginal likelihood.The conjugate gradients method has been applied to solve this equation.Using this approach, we can obtain the hyperparameters of the covariance function.Further details of GPR can be found in 24 as follows:

The Proposed Algorithm
In this paper, we combine the energy-driven approach with GPR to accomplish the task of image interpolation.The proposed algorithm models low-resolution image data as a function of a probability distribution that satisfies a local static Gaussian process.This algorithm framework is shown in Figure 2 and is broadly divided into the training process and prediction process.Firstly, the GPR model can be established using the low-resolution image data.Next, this model is used to predict the unknown pixel values of a high-resolution image by adopting an energy computation approach.Through the above two steps, we produce a high-quality enlarged image.The process is further clarified by the following: where L and H denote the input low-resolution image with a little noise and the output highresolution image, respectively, L denotes the noise-free low-resolution image, H denotes the initial high-resolution image, sdenotes the upsampling factor, and d and f denote the clear transfer function and energy transfer function, respectively.

Training
The following definitions are used in the EGPR algorithm.Definition 3.1.A given image L is divided into many regions of equal size, and each region is defined as a processing unit PU .Each PU is also divided into 3 × 3 overlapping image patches the total number is M .The center of each patch is defined as an output vector Y TR of PU, where Y TR y 1 , y 2 , . . .y M T , while the nearest eight values are defined as an input vector X TR of PU, where Given a total of N pixels in each PU, the pixels are sorted and denoted as I 1 , I 2 . . .I N .I max ave , I min ave , and I ave are defined using the following formulae: where top represents the number of the largest pixels used and below the number of the smallest pixels used.Then I max ave , I min ave , and I ave are called basic properties of PU.
To facilitate the operation of the PU, it is necessary to introduce some properties in advance.
Property 1.Given a number N in each PU, if I max ave 0, then pixel value I i 0, where i ≤ N. Property 2. Given x ij , if x ij a, then its corresponding output vector value is y ij a, where Denoising is the first step in the EGPR algorithm, and we use the following formula 3.4 to obtain noise-free images: where θ and B represent empirical values, and I neighbor represents the adjacent pixel value.
Before applying GPR, we can obtain the particular relationship between the input and output vectors of PU according to Properties 1 and 2. Pixels with this relationship need not be included in the following GPR training, so the predicted values can be directly obtained, saving time and speeding up the EGPR algorithm.
Training plays an important role in the EGPR algorithm, and we adopt a different approach from that used in 13 .Our algorithm contains two processes: training domain establishment and GPR model foundation.In the first stage, we search possible training domains along the four directions of each specific PU.Next, we compute the structural similarity between directions to determine the definite training domain.Inspired by the concept of image SSIM, we define the PU structural similarity as follows.
Definition 3.3 PU structural similarity .Given two processing units P and Q, their structural similarity is defined as where C 1 and C 2 are constants, and the other components are calculated as follows:

3.6
When the search step count reaches the predefined number, or if the PU structure similarity falls below a certain value, the first stage is complete.In the second stage, we apply a Gaussian process prior probability and establish the GPR model with Gaussian noise γ see 3.7 below using the image data from training domains.In 3.7 , "GP " denotes a Gaussian process y g X γ, γ ∼ GP 0, σ 2 n .When aiming to achieve high-quality images, the conjugate gradients method is chosen to obtain the model hyperparameters, including mean, variance, and log marginal likelihood.Notice that different iteration numbers in the conjugate gradients method may lead to different prediction accuracies.Figure 3 shows the interpolation images obtained after 50 iterations and 100 iterations, where it can be seen that the latter is better than the former.

Prediction
Inspired by the ICBI algorithm, we firstly compute the initial pixel value H 2x 1, 2y 1 according to the following formulae:

3.10
Journal of Applied Mathematics 7 However, the pixel value obtained is only a roughly estimated value and needs further refinement.Following 9 , we establish 3.11 to calculate the energy of each interpolated pixel, and the initial estimate can be modified accordingly, where c, e, and i are chosen to adjust the energy contributions from the three parts.E c represents the curvature continuity energy and can be computed with the following formulae:

3.12
where α i i 1 . . . 4 are weight values see 3.13 , and d 1 and d 2 have the same meanings as above.θ is set as the threshold The second energy term E b represents the curvature enhancement energy and can be computed by

3.14
The third energy term, E i , represents the isolevel curves smoothing energy and can be computed with where D 2x 1, 2y 1 can be computed as follows:

3.16
Suppose that the low-resolution image L ij is of size m × n and that it is changed to the corresponding interpolated image where "scale" denotes the magnification factor.Then we use the nearest interpolation algorithm for the missing pixels in order to obtain the image Similarly, we partition H ij into overlapping processing units.The eight adjacent pixels of each pixel are treated as GPR model test input data based on the model M which was obtained from the training process.Note that if PU has Property 1 or Property 2, then we can directly obtain the corresponding pixel values.Otherwise, we capture the prediction distribution of unknown pixels in the initial high-resolution image.The joint distribution of the training domain output y and the test output f is given by the following equation: where X denotes the GPR training data matrix, X is the test matrix, and COV X, X is the n × n matrix of covariances.Therefore, we can derive the predictive distribution based on the obtained model M: COV X, X .

3.18
During the prediction of high-resolution image pixels, two rules should be obeyed.Firstly, the PU divided by the initial high-resolution image should correspond to that divided by the low-resolution image.Secondly, the gradient algorithm should satisfy the common positive definite matrix.If not, it will lead to a zero prediction, and the prediction value will need modifying.The modification method can be utilized to maintain the original interpolated pixel value.Finally, we combine all the processing units together in a smooth manner to obtain the high-resolution images without noise.

Experimental Results and Discussion
In this section, we compare the experimental results obtained using the proposed algorithm with those obtained using the bilinear algorithm, GPR algorithm 13 , and ICBI algorithm 9 .Each algorithm was run in MATLAB.In order to evaluate algorithm performance, we first downsampled original high-quality images to acquire low-resolution images.Then we enlarged these low-resolution images by utilizing the different interpolation algorithms and compared the enlarged images with the original high-quality images.In all experiments, we set the PU size to 30 × 30, but this may be increased according to the magnification factor.At the same time, we used zero mean and square exponential functions as the respective mean and covariance functions in the EGPR.The covariance function required two hyperparameters: a characteristic length scale, the default value of which was 0.21, and the standard deviation of the signal, the default value of which was 0.08.In addition, to achieve color image interpolation, we trained and predicted the GPR model separately for each of the R, G, and B channels.
Figure 4 shows the interpolation results from the four algorithms when "scale" was set as 1.Figures 4 a -4 d are comparisons of image 1, and Figures 4 e -4 h are comparisons of image 2. In the enlarged red-bordered region, it can be seen that the bilinear method introduces jaggy effects, the GPR method reduces these jaggy effects, and the ICBI method achieves a clear edge but is still a little blurry.By employing the energy computation based on properties of adjacent pixels, our new method generates a clearer image without noise.
Similarly, Figures 5 and 6 demonstrate the interpolation results with scales of 2 and 3, respectively.From these figures, it can be seen that our method achieved the clearest and smoothest enlarged image of the four methods tested, for example, along edges on the root hand in Figure 6 h .Moreover, the advantages of our proposed algorithm become more enhanced at greater enlargement factors.

Figure 1 :
Figure 1: Overview of our approach for image interpolation.

Figure 2 :
Figure 2: Architecture of the proposed algorithm.

Figure 3 :
Figure 3: Images obtained after adaptation with different numbers of iterations.In a , many black points are observed, each indicating a zero prediction for the pixels.In b , the black points have been eliminated.

Figure 4 :
Figure 4: Comparison of images obtained using four methods, with scale 1. Parts a -d show image 1. Parts e -h show image 2.

Figure 5 :Figure 6 :
Figure 5: Comparison of images obtained using four methods, with scale 2. Parts a -d show image 3. Parts e -h show image 4.

Table 1 :
Comparison of PSNR for the four interpolation methods when applied to test images.

Table 2 :
Comparison of RMS for the four interpolation methods when applied to test images.