# Fast 3D Analytical Affine Transformation for Polygon-Based Computer-Generated Holograms
## 3 Theory
 \tag{11}\]  
where \(u,v,w\) again are spatial frequencies of the global coordinate system, \(f\left(\overset{\rightarrow}{r_{g}}\right)=f(x,y,z)\) is the surface function of the tilted triangle \(\overset{\rightarrow}{r_{g}}{=}(x,y,z)\), \(\overset{\rightarrow}{r_{a}}{=}(s,t,p)\) and the Jacobian determinant \(J=r_{11}r_{22}-r_{12}r_{21}\). Then, the spectrum distribution on the hologram \(F_{HOLO}\) is:  
\[F_{HOLO}(u,v)=J\cdot\mathrm{e}^{-j2\pi(ux_{1}+vy_{1}+wz_{1})}.F_{\Delta}(u_{s},v_{t}), \tag{12}\]  
which has been derived in Appendix C. Note that for an arbitrary tilted polygon, we can set an arbitrary point instead of point \((x_{1},y_{1},z_{1})\) of the polygon. For a different choice, it will change the affine transform matrix, but the Jacobian determinant in Equation (12) also changes at the same time, giving the same result in Equation (12). As shown in Figure 5a, the zero frequency in the global coordinate system is \((u_{0},\,v_{0},\,w_{0})^{t}=\left(0,\,0,\,\frac{1}{\lambda}\right)^{t}\). According to the spatial frequency relationship given by Equation (28) in Appendix C, as shown in Figure 5b, the zero frequency in the affine coordinate system \(\left(u_{s0},v_{0},w_{p0}\right)^{t}\) is \(\left(u_{s0},v_{0},w_{p0}\right)^{t}=R^{-1}(u_{0},v_{0},w_{0})^{t}\):  
\[ \begin{split}\left(\overset{u_{s0}}{v_{t0}}\right)=\left(\begin{array} []{c}r_{11}u_{0}+r_{21}v_{0}+r_{31}w_{0}\\ r_{12}u_{0}+r_{22}v_{0}+r_{32}w_{0}\\ r_{13}u_{0}+r_{23}v_{0}+r_{33}w_{0}\end{array}\right)=\left(\begin{array}[] {c}\frac{r_{31}}{\lambda}\\ \frac{r_{32}}{\lambda}\\ \frac{r_{33}}{\lambda}\end{array}\right).\end{split}  \tag{13}\]  
From the above equation, we can see that the zero frequency in the affine coordinate system has a frequency offset, and this offset will be represented in the reconstruction image with a phase factor. To eliminate this offset, as shown in Figure 5c, we can subtract this frequency offset \(\left(\Delta u_{s},\Delta v_{t},\Delta w_{p}\right)^{t}\) according to  
\[ \begin{split}\left(\overset{u_{s0}}{v_{t0}}\right)=\left(\begin{array} []{c}u_{s0}\\ v_{t0}\\ w_{p0}\end{array}\right)-\left(\begin{array}{c}\Delta u_{s}\\ \Delta v_{t}\\ \Delta w_{p}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ \frac{1}{\lambda}\end{array}\right),\end{split}  \tag{14}\]  
Figure 5: Spectrum shift problem. (**a**) Spectrum in global coordinate system, (**b**) spectrum in affine system and (**c**) spectrum with frequency offsetting. where \(\Delta u_{s}=\frac{r_{11}}{\Delta},\Delta v_{t}=\frac{r_{22}}{\Delta},\Delta w_{p}= \frac{r_{31}}{\Delta}-\frac{1}{\Delta}\). Therefore, the spatial frequencies in the \"offset\" affine coordinate system can be rewritten as  
\[ \begin{array}{c}u\prime_{s}=r_{11}u+r_{21}v+r_{31}w-\frac{r_{31}}{\Delta}\\ v\prime_{t}=r_{12}u+r_{22}v+r_{32}w-\frac{r_{32}}{\Delta}\\ w\prime_{p}=r_{13}u+r_{23}v+r_{33}w-\frac{1}{\Delta}(r_{33}-1)\end{array} . \tag{15}\]  
The spectrum of a tilted triangle on the hologram plane now has been completely analyzed, and we can directly obtain the spectrum distribution through the initial parameters. For a single tilted triangle, we have \(F_{HOLO}(u,v)=F_{HOLO,i}(u,v)\) as in Equation (2), and for a 3D object we use Equation (1). The complex field distribution of a 3D object reconstructed by the hologram can be expressed as the superposition of a tilted triangle complex field on reconstructed image plane \(z=z_{r}\):  
\[f_{re}(x,y,z_{r})=\mathcal{F}^{-1}\Big{\{}e^{-j2\pi\overline{u}z_{r}}\cdot F_ {HOLO}(u,v)\Big{\}}, \tag{16}\]  
where \(z_{r}\) is the distance between the reconstructed image plane \(z=z_{r}\) and the hologram plane \(z=0\). Through the above steps, we can obtain the reconstructed complex field distribution of the 3D object through only one Fourier transform. In the next section, we will verify our proposed method through numerical simulations and optical experiments.