## 1 What is the type of hologram?
### Diagnosis of Distortions of Digital Holograms
#### 5.4.3 Compression of Static Holograms  
Another \"distortion\" one might encounter when working with DHs is lossy compression. To minimize coding artifacts and maximize compression efficiency the compression scheme needs to be tailored to the typical phase space footprints of the signals under consideration. In transform-based coding, typically a best tiling strategy of phase space is determined for a given class of signals and a suitable frame or basis is selected based on it. PSRs are a vital instrument in this design process nowadays, and have been in the past. Once such an optimal tiling has been found and a basis representation has been derived, the latter can subsequently be used to reduce the signals to a few essential coefficients and subsequently efficiently compress signals. Whilst the initial motivation for new basis designs does not necessarily have to be sourced in phase space analysis, for some signals, especially for DH, this approach is highly advisable and in line with advances in mathematics made of the past few decades such as: STFT, wavelet, or Gabor schemes. An abundance of work followed the introduction of each of these schemes, with applications to all kinds of one- or two-dimensional signals. See e.g., [52] or more specific to DH [17; 28; 53]. In addition to basis design, PSRs can also be used to simply, localize coding artifacts and identify discarded information for entire coding strategies by comparison of the PSRs of un-/compressed holograms. See for example Figure 10(b), where we present the \"DeepChess\" hologram from the INTERFERE-IV database [46] compressed with H.265/High Efficiency Video Coding (HEVC) in intra-mode at a low bitrate of 0.25 bit per complex sample. The codec was chosen as an example here since it is one of the two anchor codecs proposed for the JPEG Pleno on holography. The codec is applied after a uniform 8 bit quantization of the entire dynamic  
Figure 11: (a,b) Phase space footprint of 1D cross-section of un-/compressed (with H.265/High Efficiency Video Coding (HEVC)) multi-object hologram “DeepChess” from Interfere-IV [46]. (c) Noise in the optical holographic recording “Squirrel” from Interfere-IV [46] of an object with metallic surface.  
range of the real and imaginary channels of the hologram. The highlighted differences can be identified and explained as follows:  
1. reduced signal to noise ratio due to the introduction of discontinuities (impulses) through block based coding, and use of a basis, poorly localized in frequency (discrete cosine transform with short spatial windows \(w\), cf. Section 3.2).
2. added DC term and opposite diffraction orders, due to separate compression of real and imaginary parts, cf. Section 5.5.5.
3.4 missing information due to the bitrate-distortion optimization being tuned for natural images.
6. clipped signal, due to thresholding during rate-distortion optimization.  
Another important characteristic for the compression of especially optically recorded holograms is the amount of measurement noise. The more noise is contained in the signal, the larger its footprint in phase space and the more difficult it is to employ any sparsifying transform. An example of noise visible in phase space is shown on a 100 dB scale in Figure 10(c) for the optically recorded \"Squirrel\" hologram from INTERFERE-IV [46]. The high amounts of measurement noise in this case are a result of the metallic surface of the recorded object. Note, optically recorded incoherent measurement noise is similar to quantizing real and imaginary parts to low bitrates, see Section 5.4.1.