## 2 Methodology of Multi-Scale Subset and Subvoxel Shifting in DVSP  
### Theory of Multi-Scale Subset and Subvoxel Shifting in DVSP  
The theory of DVSP has been described previously [32] and is only briefly presented here for easy reference. Assume that a reference volumetric image and a deformed volumetric image of an object are acquired by a CT system (i.e., either in nano- or microscale), these two volumetric images are subsequently divided into volumetric subsets with arrays of \(16\times 16\times 16\) voxels or \(32\times 32\times 32\) voxels and then 'compared'. The cross correlation with the POF is operated in the frequency domain (\(\xi\), \(\eta\), \(\zeta\)) as follows:\[\overline{G}(\xi-u,\eta-v,\zeta-w) =\Im\left\{\frac{H_{1}\left(f_{x},f_{y},f_{z}\right)H_{2}^{*}\left( f_{x},f_{y},f_{z}\right)}{\sqrt{\left|H_{1}\left(f_{x},f_{y},f_{z}\right)H_{2} \left(f_{x},f_{y},f_{z}\right)\right|}}\right\}\] \[=\Im\left\{\left|H_{1}\left(f_{x},f_{y},f_{z}\right)\right|\exp \left\{j\left[\phi_{1}\left(f_{x},f_{y},f_{z}\right)-\phi_{2}\left(f_{x},f_{y}, f_{z}\right)\right]\right\}\right\}\]  
where \(\overline{G}(\xi-u,\eta-v,\zeta-w)\), an expanded impulse function located at \((u,v,w)\), is the discrete 3D cross correlation between the two volumetric subsets. \(\Im\) denotes the Fourier transform, with \(H_{1}\left(f_{x},f_{y},f_{z}\right)\) being the Fourier transform of \(h_{1}(x,y,z)\), representing the gray distribution function of a subset from the reference volumetric image, and \(H_{2}\left(f_{x},f_{y},f_{z}\right)\) being the Fourier transform of \(h_{2}(x,y,z)\), representing the gray distribution function of the corresponding subset from the deformed volumetric image. In addition, \({}^{*}\) denotes the complex conjugate. \(\left|H\left(f_{x},f_{y},f_{z}\right)\right|\) and \(\phi\left(f_{x},f_{y},f_{z}\right)\) are the spectral amplitude and phase fields, respectively. In Equation (1), the term \(\exp\left\{j\left[\phi_{1}\left(f_{x},f_{y},f_{z}\right)-\phi_{2}\left(f_{x}, f_{y},f_{z}\right)\right]\right\}\) may be viewed as a POF, which can give a good balance between the peak sharpness and the noise tolerance in the correlation theory. As noted in reference [34], the random error of DVSP depends on subset size and micro structure pattern of objects. Furthermore, if the displacement between the corresponding subsets is large, the increase in the nonoverlapping area would cause an increase in decorrelation and thus result in an enhanced random error. On the other hand, in the case of images, \(\overline{G}\) in Equation (1) is only displaying a delta-like function if \(u,v\) and \(w\) are integers. Non-integer translations between two subsets cause the peak in \(\overline{G}\) to spread across neighboring voxels, subsequently degrading the quality of the displacement estimate. To identify sub-voxel investigation, the common approach is to apply cubic spline interpolation. The accuracy of these interpolation methods is highly dependent on the shape of the \(\overline{G}\) function near the peak. In the 2D method, a multi-scale and coarse-fine subset calculation process and a subpixel shifting technique have been applied to decrease the error caused by decorrelation and non-integer translations [41, 42]. These techniques are hereby introduced into DVSP, and we call this new algorithm Multi-scale and Subvoxel shifting Digital Volumetric Speckle Photography (MS-DVSP). The essence of the multi-scale and coarse-fine calculation process is to minimize the nonoverlapping area under one voxel. In the coarse calculation process, the largest size of the subset (\(2^{n}\times 2^{n}\times 2^{n}\)) should not be larger than the size of the region-of-interest (ROI) in the reference volumetric image, and the integer voxel of displacement value (\(u_{0},v_{0},w_{0}\)) is obtained by the DVSP method.