Fast Image Super-resolution with Sparse Coding

In this paper, we introduce a novel fast image reconstruction method for super-resolution (SR) base on sparse coding. This method combine online dictionary learning and a fast sparse coding way, both of which can improve the efficiency of the reconstruction process and ensure the image visual quality. The new online optimization algorithm for dictionary learning based on stochastic approximations, which can drastically advance the learning speed, especially on millions of training samples. Meanwhile, we trained a neural network to speed up the reconstruction process, which based on iterative shrinkage-thresholding algorithm (ISTA), we called learned iterative shrinkage-thresholding algorithm (LISTA). It would produce best approximation sparse code with some fixed depth. We demonstrate that our approach can simultaneously improve the image fidelity and cost less computation.


Introduction
High-resolution image is valuable in many social areas. Such as reconstruct the high-resolution image from low-resolution image which take in Criminal Investigation, get high-accurate CT, MIR, Ultrasonic wave to help doctor diagnose disease. Additionally, this is also play an important role in HDTV, Video Rip and Military occasion.
The main contribution of this paper is combining the online dictionary learning and approximating sparse coefficient together to reconstruct the High-resolution image, so that the reconstruction rate is splendidly speedup. Learning the dictionary by online method process one element of the millions of training samples at a time, which based on stochastic approximations (Aharon & Elad, 2008). Instead of optimize the l1-norm optimization for accurate sparse coefficients, we employ a feed-forward neural network to construct approximated sparse coefficients(Gregor & LeCun, 2010) .

Fast image super-resolution
This section describe the two key component of SRIR learn the over-complete dictionary and find sparse coefficient. Online dictionary learning proposed an iterative algorithm that solves eqn. by efficiently minimizing at each step a quadratic surrogate function of the empirical cost over the set of constraints (Mairal, Bach, Ponce, & Sapiro, 2009). We find the approximation sparse coefficient by train a non-linear, feed-forward neural network instead of find the accurate solution(Gregor & LeCun, 2010). 1.1 Online dictionary learning According to we talked above, the dictionary learning can be get by optimize the energy function: 9 Return DT (learned dictionary).
We supposed the training set is composed of i.i.d samples of a distribution p(x), every loop it draws one element from the training set. It alternatively solve D by classical sparse coding and D by minimizing the following function: The quadratic function This function aggregates the previous information by the process of computing D .
1 Repeat;2 For j=1 to k do 3 Update the j-1 column to optimize for eqn. : Here, L is a constant, which is larger than the largest eigenvalue of Here, zi is the accurate sparse coefficient of yi which from traditional sparse method. And 3: In order to make the fair comparison, we adopt the same training set and initial D for both methods. a) Visual result: fig. 1(a)-(d) compared the results of Bicubic interpolation method with our fast image super-resolution four test images. From the left to right are the original HR images, the reconstructed images by Bicubic interpolation method and the reconstructed images by fast super-resolution method. From the results, we can easily see that our recovery are closed to the original images, the edge is more sharper than the Bicubic recovery, such as the moustaches of the cat, the texture of the freckles of the girl the texture of the leaves and the hair of lena. Here, I and K are respectively represent the original image and our recovery.
The PSNR is defined as: From table II we can see that our algorithm speedup the reconstruction process significantly. In addition, we can ensure both of the PSNR and the cost of compute time, since our algorithm adopt the approximate sparse coefficients and train the online dictionary. From fig. 2 for statuary image, Yang s algorithm has the higher PSNR than our algorithm, but both of two images almost have the same visual quality.

Conclusions
In this paper, we propose a novel approach to reconstruct the high-resolution image based on online dictionary learning and approximate sparse coefficient.The preliminary experiment prove that our algorithm not only ensure the visual quality but faster than other state-of-art approaches. However, our algorithm is not have the optimal recovery than other SR approaches, so, our future research is to train accurate dictionary to represent the sparse coefficients.