## 1 What is the type of hologram?
### Depth Estimation  
The discussions of Sections 2 and 4 can be combined to provide a rough scene depth estimate for holograms where such information is not accessible, as we indicated in Section 5.1. For microscopic scenes with only a few scattering centers even accurate 3D positioning is possible [16; 25]. As a starting point, we note that the combined wavefields in Figure 7 are in general of unique shape and through analysis it is possible to estimate their focal distances. Figure 9 demonstrates how the phase space footprints shape varies for the hologram of a comparatively shallow object (\(N_{F_{r}}\sim 1650-2250\)). In case a signal shows as impulses (lines parallel to frequency axis), e.g., Figure 9d, the hologram is in focus, and all point-sources located within the depth of focus are already converged to points (Dirac impulses in the 1D cross section). The impulses will be bandwidth-limited by the holograms phase space bounding box. The lateral resolution is limited by the pixel pitch in the object plane and potentially by diffraction--see the Rayleigh criterion in the former section, in the case of the paraxial approximation. These holograms have been recorded as image plane holograms or have been refocused prior to their analysis. Note, that Figure 9d shows the phase space footprint associated with a horizontal center cut through Figure 9a. Curiously, in Figure 9d it is possible to make out individual PSF clusters with the S-method that remain slightly out of focus, due to the sparsity of the point cloud model. Their slope informs about whether they are located in front or behind the current focal plane. This proves helpful in identifying the ordering of multiple sparse objects in depth [25]. A situation which appears frequently in digital holographic microscopy, where usually sparse, multi-component signals are being analyzed. Traditional methods create a depth map relying on exhaustive back-propagations to different scene depths. In case a signal extends over entire spatial domain and appears to be made up of horizontal lines, e.g., Figure 9e, the wavefield does not converge at any finite distance. Such a hologram has been recorded as a Fraunhofer or a (lensless) Fourier hologram. In essence, a Fourier transform is required for refocusing. This representation is especially advantageous in experimental setup as it makes optimal use of the space-frequency bandwidth product \(\text{SPB}_{\text{H}}\) and was proposed for use in [32] under the name \"compact space-bandwidth representation\". Whenever a parallelogram with angles sufficiently different from \(90^{\circ}\) and \(0^{\circ}\) shows, e.g., Figure 9, the focal plane is at some distance \(z>0\) or \(z<0\) depending on the sign of its slope. These holograms have been recorded as Fresnel holograms in the paraxial limit or in the near field with a small frequency bandwidth due to a large pixel pitch. Depending on which, at least the Fresnel method or the rigorous back-propagation need to be used. Finally, if a sigmoidal shape as in Figure 9f is seen, the focal plane is at some distance \(z>0\) and the rigorous method needs to be used for refocusing--usually the angular spectrum method is employed. For finite propagation distances, i.e., in the latter two cases, \(z\) can be estimated from a parametric fit of the central ridge of the footprint or from the footprint of an isolated, bright scatterer. Without aberrations a sigmoidal (Equation (4)) or linear curve (Equation (7)) are to be used. We provide an explicit example in Appendix B. Depth estimation through curve fitting is also the essence of the filter matching technique used in online synthetic aperture radar processing or used to estimate the distances of a scattering target from the radar site in inverse synthetic aperture radar imaging for motion compensation, see [7] p. 805 ff.