*2.5. Normalized Noise Power Spectrum*

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The normalized noise power spectrum (NNPS) was measured to quantify the noise level in the homogeneous volume of interest (VOI) of the poly methyl methacrylate (PMMA) background. The three-dimensional (3D) NPS was measured as described in Figure 4. The eight different VOIs without interference of any structure with the size of 150 × 150 × 45 (300 × 300 × 90 for high-resolution reconstruction) were selected for measuring the 3D NPS. Each sub-volume overlapped with others to evaluate the radially and symmetrically distributed noise property (location independent noise pattern) [21].

Each mean subtracted sub-volume patch was Fourier transformed, absolute squared, and ensemble averaged to yield the power spectrum as follows [22]:

$$\text{NPS}\{f\_{\lambda'}, f\_{y'}, f\_z\} = \frac{1}{2} \frac{d\_{\lambda} d\_{y} d\_{z}}{N\_{\lambda} N\_{y} N\_{z}} \left\langle \left| \mathcal{F} \left[ S(i, j, k) - \overline{S} \right] \right|^{2} \right\rangle,\tag{4}$$

 <sup>−</sup> ℱ[∙] where *fx*, *fy*, and *f<sup>z</sup>* are spatial frequencies (mm−<sup>1</sup> ), *<sup>d</sup>x*, *<sup>d</sup>y*, and *<sup>d</sup><sup>z</sup>* are pixel sizes (mm), *<sup>N</sup>x*, *<sup>N</sup>y*, and *<sup>N</sup><sup>z</sup>* arethe numbers of voxels in the sub-volume patch, <sup>F</sup> [·] is the fast Fourier transform operator, and *<sup>S</sup>*(*i*, *<sup>j</sup>*, *<sup>k</sup>*) and *S* indicate each voxel value and the mean intensity of the sub-volume patch, respectively. The 1D NNPS can be derived by radially averaging the 3D NPS [23].

**Figure 4.** A schematic illustration for deriving 3D NPS.
