*2.2. Compressed Sensing*

Compressed sensing (CS) is a technique used to reconstruct unknown data from a small number of observed data. In theory, the original data can be estimated when the data is sparse [28]. We considered the problem of reconstructing an image *zh* <sup>∈</sup> <sup>R</sup>*n*<sup>0</sup> with a higher resolution than

the observed low-resolution image *<sup>z</sup><sup>l</sup>* <sup>∈</sup> <sup>R</sup>*m*<sup>0</sup> (*m*<sup>0</sup> <sup>&</sup>lt; *<sup>n</sup>*0). The relationship between the high-resolution image and the low-resolution image can be expressed by Equation (1).

$$\begin{aligned} z^l = \text{SH}z^h = \text{L}z^h, \\ L = \text{SH}\_\prime \end{aligned} \tag{1}$$

where *S* is a downsampling operator and *H* is a filter that lowers the resolution. At this time, since the dimensionality of *zl* is smaller than that of *zh*, the solution cannot be uniquely determined.

Based on the compressed sensing theory, the high-resolution image *zh* is estimated by Equation (2) from the image element *Dh* and sparse representation *a*. The element for reproducing the image is called atom *di* <sup>∈</sup> <sup>R</sup>*n*<sup>0</sup> , and the set of atoms is called the dictionary *Dh* <sup>∈</sup> <sup>R</sup>*n*0×*Nd* .

$$z^h = D\_h a \text{ s.t.}\\\|a\|\_0 \le m\_{0\prime} \tag{2}$$

Using Equation (2), Equation (1) can be expressed as

$$\begin{aligned} z^l = \mathcal{L} z^h &= D \mu\_l \\ D\_l &= \mathcal{L} D\_h. \end{aligned} \tag{3}$$

*z<sup>h</sup>* can be reproduced by *zl* using the sparse representation *a* obtained from Equation (4), in which the sparsity constraint is added to Equation (3).

$$\min\_{a} \|a\|\_{0} \text{ s.t. } \|z^{l} - LD\_{h}a\|\_{2}^{2} \le \varepsilon. \tag{4}$$

Equation (4) can be solved using optimization methods.
