*2.4. Notation*

In the proposed method, four features are extracted from low-resolution images, and the set of features is called the feature map. We used the gradient map proposed by Yang et al. [13] as the feature map. Let *x high <sup>i</sup>* <sup>∈</sup> <sup>R</sup>*p*2×<sup>1</sup> be a patch of size *<sup>p</sup>* <sup>×</sup> *<sup>p</sup>* extracted from a high-resolution image, and *xlow <sup>i</sup>* <sup>∈</sup> <sup>R</sup>4*p*2×<sup>1</sup> be a set of four patches of size *p* × *p* extracted from the feature map. The *i*th training data patch *xi* <sup>∈</sup> <sup>R</sup>5*p*2×<sup>1</sup> is defined as *xi* <sup>=</sup> ⎡ ⎢⎢⎢⎢⎣ *x high i xlow i* ⎤ ⎥⎥⎥⎥⎦. *<sup>X</sup>* <sup>=</sup> <sup>+</sup> *x*1, ··· , *xNt* , <sup>∈</sup> <sup>R</sup>5*p*2×*Nt* , *<sup>X</sup>high* <sup>=</sup> <sup>+</sup> *x high* <sup>1</sup> , ··· , *x high Nt* , <sup>∈</sup> <sup>R</sup>*p*2×*Nt* , and *<sup>X</sup>low* = <sup>+</sup> *xlow* <sup>1</sup> , ··· , *xlow Nt* , <sup>∈</sup> <sup>R</sup>4*p*2×*Nt* represent the set of patches of the training data, high-resolution training data, and low-resolution training data, respectively, where *Nt* is the number of training data. *x high <sup>i</sup>* indicates the mean value of the intensity values of the *i*th training data patch *x high <sup>i</sup>* . *D* = *Dhigh <sup>D</sup>low* = <sup>+</sup> *d*1, ··· , *dNd* , , *<sup>D</sup>* <sup>∈</sup> <sup>R</sup>5*p*2×*Nd* is called a dictionary and *di* <sup>=</sup> ⎡ ⎢⎢⎢⎢⎣ *d high i dlow i* ⎤ ⎥⎥⎥⎥⎦ is the *i*th atom of the dictionary where *Nd* is the number of atoms. *d high <sup>i</sup>* <sup>∈</sup> <sup>R</sup>*p*2×<sup>1</sup> is a high-resolution dictionary atom of size *p* × *p*, and *dlow <sup>i</sup>* <sup>∈</sup> <sup>R</sup>4*p*2×<sup>1</sup> is a low-resolution dictionary atom of size *<sup>p</sup>* <sup>×</sup> *<sup>p</sup>*. All the atoms are arranged in raster scan order. The high-resolution dictionary *Dhigh* <sup>∈</sup> <sup>R</sup>*p*2×*Nd* and the low-resolution dictionary *<sup>D</sup>low* <sup>∈</sup> <sup>R</sup>4*p*2×*Nd* are defined as *Dhigh* <sup>=</sup> <sup>+</sup> *d high* <sup>1</sup> , ··· , *d high Nt* , and *<sup>D</sup>low* = <sup>+</sup> *dlow* <sup>1</sup> , ··· , *<sup>d</sup>low Nt* , , respectively. *<sup>Y</sup>low* <sup>∈</sup> <sup>R</sup>4*p*2×*Npatch* indicates the feature map of the low-resolution input image to be reconstructed, where *Npatch* is the number of patches of the input image. *Isr* <sup>∈</sup> <sup>R</sup>*p*2×*Npatch* represents the reconstructed high-resolution image. *Isr* <sup>∈</sup> <sup>R</sup>*p*2×*Npatch* represents the mean value of the intensity values of the reconstructed high-resolution image patch. The sparse representations are denoted as <sup>α</sup> <sup>∈</sup> <sup>R</sup>*Nt*×*Npatch* and <sup>β</sup> <sup>∈</sup> <sup>R</sup>*Nd*×*Nt* , and <sup>λ</sup> represents the sparsity regularization parameter.
