## Appendix B Estimating Depth through Linear Curve Fitting--An Explicit Example  
We will demonstrate the depth estimation of the scene center on the example of Figure 9c using 1D phase space analysis and linear curve fitting. Since we evaluated the PSR on a horizontal 1D cross-section of the hologram, we will utilize the \(\xi,f_{\xi}\) domain for our estimation. For convenience, we restate all parameters in use. The square hologram of size \(N=8192\), had a pixel pitch of \(p=1\)\(\upmu\)m, and was recorded with \(\lambda=532\) nm light.  
The S-method was evaluated with a spatial Hann window \(w\) of length \(M=256\), of maximal overlap of 255 px, and a frequency filter window \(v\) of length \(L=5\), which was defined in Fourier space as a Hann window. The resulting phase space image was of size \(F\times S\), thus allowing to resolve \(F=256\) distinct frequency and \(S=7937\) distinct spatial channels. The origin of pixel coordinates within the PSR image is chosen to be in the top-left corner.  
The scene center is defined by the central ridge of the phase space footprint of the signal. Using standard signal processing we extract two points on the linear fit that central ridge.  
The obtained points have the following interpolated coordinates \((\xi_{1}^{px},f_{\xi,1}^{px})=(1~{}px,20.30~{}px)\) and \((\xi_{2}^{px},f_{\xi,2}^{px})=(7937~{}px,243.77~{}px)\). These coordinates are converted to \(\xi\) in m and \(f_{\xi}\) in m\({}^{-1}\) via  
\[\xi=g(\xi^{px}):=\xi^{px}\frac{Np}{S\lambda}-\frac{Np}{2\lambda}\quad\text{ and}\quad f_{\xi}=h(f_{\xi}^{px}):=\frac{f_{\xi}^{px}-\xi}{pM}\,. \tag{1}\]  
Although the Fresnel number of the hologram at the scene center is \(\sim\)2232, the Fresnel approximation is sufficient to estimate depth, as the pixel pitch is sufficiently large as to induce a strong band-limitation revealing only the approximately linear part around \(f_{\xi}=0\) of the sigmoid of the spherical PSF, compare with Figure 3c. As \(z\) influences only the slope of the linear instantaneous frequency function we choose the following ansatz:  
\[\frac{-1}{\lambda z}\overset{!}{=}\frac{f_{\xi,2}-f_{\xi,1}}{\xi_{2}-\xi_{1}}\,, \tag{2}\]  
where the left-hand side is obtained by differentiation of Equation (7) after \(\xi\). The right hand side is the slope of the linear fit in phase space. Combining Equations (1) and (2) we can estimate \(z\) as  
\[z=-\frac{\xi_{2}-\xi_{1}}{\lambda\left(f_{\xi,2}-f_{\xi,1}\right)}\approx\frac {g\left(\xi_{2}^{px}-\xi_{1}^{px}\right)}{\lambda h\left(f_{\xi,2}^{px}-f_{ \xi,1}^{px}\right)}\approx 1.48\,\text{cm}\,. \tag{3}\]  
The error of the obtained depth can be estimated using the total differential for our estimator of \(z\) and the standard-deviation of the polynomial fit. Accounting for the width of the spatial window \(v\), we actually obtain \(z=1.48\pm 0.19\) cm which is in good agreement with the correct \(z=1.41\) cm. For larger resolutions the error decreases fast. For multi-object holograms the ridge estimates get more complicated, but holographic scene segmentation [30] prior to the analysis can help. On a final note, also the lateral position along \(x\) of the scene center wrt. the optical axis can be determined with higher precision via Equation (1), the linear phase space fit for \(f_{\xi}\), and the ansatz \(f_{\xi}\overset{!}{=}0\). If the same is repeated for a vertical cross-section through the middle of the hologram, an estimate of the full 3D position of the center of the object can be obtained.