Research on Image Super-resolution Reconstruction Algorithm Based on Least-Squares Interpolation (Matlab Code Attached)

✳️ 1. Introduction

Image super-resolution reconstruction technology originated from the single-frame image reconstruction method constructed by Harris and Goodman in the 1960s; in the early 1980s, Tsai and Huang first proposed the super-resolution reconstruction of sequence or multi-frame images. And an approximation method based on the frequency domain is given; since the 1990s, the research on super-resolution image reconstruction methods has made a breakthrough and has become more and more practical. According to the number of low-resolution images that can be used, super-resolution reconstruction algorithms can be divided into: 1) High-resolution estimation of sequence images, that is, combining multiple low-resolution images of the same scene to obtain high-resolution images; 2) Single image High-resolution estimation of images, from a low-resolution image to a high-resolution image. In addition, single image super-resolution algorithms can be divided into spatial domain reconstruction algorithms and transform domain reconstruction algorithms according to different scopes; sequence image super-resolution reconstruction algorithms can also be divided into constraint-based algorithms and learning-based algorithms.

✳️ 2. Least squares image interpolation theory and Matlab processing

In this algorithm, the image to be interpolated is divided into overlapping blocks with a size of M×M, and the goal of the algorithm is to obtain an interpolated image of each block with a size of N×N. Assume that the relationship between the interpolated low-resolution block and the interpolated high-resolution block is given by

where: $g_{i,j}$和 ${\hat{f}}_{i,j}$M 2 × 1 { {M}^{2}}\times 1 at (i,j) respectively$M^2\times 1$ and$N^2\times 1$ Sequentially arranged low resolution image (LR) and high resolution image block (HR); W is$N^2\times M^2$ weight matrix, this matrix must be adaptive for each block.

Therefore, it is necessary to obtain the weight matrix W. In order to ensure the least squares solution with the minimum mean square error (MSE), the relationship model between the current LR block and the original HR block can be considered, which is given by

The matrix D is $N^2\times M^{2-}$ dimensional, minimize the following cost function:

The above equation expresses to minimize the MSE between the downsampling of the LR block to be interpolated and the HR block, namely

The above formula is processed in Matlab as follows:

norm(g_ij-D*f_ij0)>bound

Differentiating the weight matrix on both sides at the same time has:

The weight matrix can be obtained by the following equation

The above formula is processed in matlab as follows:

W=W+u*D'*(g_ij-D*f_ij0)*g_ij';

✳️ 3. Experimental verification of super-resolution reconstruction based on least squares interpolation

In the simulation experiment, the original image is down-sampled first, then white Gaussian noise (AWGN) is added to simulate the degradation model of the low-resolution image, and finally, the original image is reconstructed by bilinear interpolation on the low-resolution image. The results are as follows: As shown in Figure 1, at this time, the peak signal-to-noise ratio between the reconstructed image and the original image is 29.1124.

Figure 1 Reconstruction results based on least squares interpolation

✳️ 4. References

[1] S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky et al. 2005. Adaptive Least Squares Acquisition of High Resolution Images, Int. J. Information Acquisition, 2, 45–53.

[2] S. E. El-Khamy, M. M. Hadhoud, M. I. Dessouky et al. 2006. Efficient Solutions for Image Interpolation Treated as an Inverse Problem, J. Information Sci. Eng., 22, 1569–1583.

[3] H. I. Ashiba, K. H. Awadalla, S. M. El-Halfawy et al. 2011. Adaptive Least Squares Interpolation of Infrared Images, J. Circuits Syst. Signal Proc., 30, 543–551.

✳️ 5. Matlab program acquisition and verification

The above demonstration example is implemented by Matlab code. Before obtaining the Matlab code, carry out targeted verification experiments . Please private message the blogger.

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Introduction to the blogger: The research direction involves intelligent image processing, deep learning, convolutional neural network and other fields. He has published many SCI papers and has rich experience in scientific research. Any questions related to algorithms , programs , and scientific research can be exchanged and discussed in private messages .

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