## 2 State-of-the-Art
### Tomographic Reconstruction  
The optical measurement of sound pressure can only be done in projection. This means that a two-dimensional tomographic reconstruction must be performed. The solution of this inverse problem can be done using the filtered back projection.  
Figure 1: Measurement principle of the heterodyne Mach–Zehnder interferometer. Laser: \(\lambda=532\) nm, solid state laser; camera: 64 \(\times\) 1024 Pixel, \(f_{B}\): 30 kHz, field of view: 2 \(\times\) 30 mm, frame rate: 120 kHz. Taken from [7].  
The tomographic reconstruction is performed as follows: Let the measurement of a projection be a vector to \(R_{p}(x^{\prime},\alpha,t)\) in a transformed plane \((x^{\prime},y^{\prime})\) from an angular perspective of the rotation angle \(\alpha\) as shown in Figure 2. Using a total number of \(N_{x^{\prime}}\) scan lines and \(N_{\alpha}\) angular scans these vectors give the sinogram with \(N_{x^{\prime}}\cdot N_{\alpha}\) dimensions (see Figure 3).  
Mathematically, a sinogram is the forward operation of the Radon transformation. Thus, a solution of the inverse problem by applying the inverse Radon transformation is needed for the reconstruction of the sound field  
\[p_{\rm recons}(x,y,t)=\int_{0^{\circ}}^{180^{\circ}}R_{p}(x^{\prime},\alpha,t)* h(x^{\prime}){\rm d}\alpha, \tag{7}\]  
with the filter function \(h(x^{\prime})\). Due to the integral operation of the inverse transformation, low-frequency components are weighted higher than high-frequency components. With the filter function this weighting error can be corrected during the reconstruction. This leads to a sharper image and ensures an absolute value reconstruction [9]. Different types of filters can be considered as a filter function. For example, there is the Ram-Lak filter with a linear increase of the amplitude over the frequency, the well-known Hamming filter or the Hanning filter [31]. While the Ram-Lak filter is sensitive to noise, it has provided the best absolute value reconstruction results, which was tested on simulated data. Thus, the Ram-Lak filter will be used for all reconstructions on simulated and measured data in the following investigations for a fair comparison.  
There are prerequisites in the measurement setup to reconstruct a local field. For example, the field to be measured must be scanned from different directions in an angular  
Figure 3: Full angle (FA) sinogram in angular range \([0,180]^{\circ}\).  
Figure 2: Principle of the Radon transformation.  
range of 180\({}^{\circ}\) and high angular resolution [10]. The angle resolution depends on the crucial spatial resolution of the local field. Sparse angular sampling can result in aliasing [10]. Furthermore, a stationary and spatially closed field is required [9]. If these prerequisites are violated, strong artifact formation and additional information loss of the local absolute values will occur. A high angular limitation results in high artifact formation also known as missing cone artifacts [32]. This is caused by an incorrect frequency weighting, as well as the systematic reconstruction deviation of the absolute amplitude, caused by numeric instabilities, compared to the model field [17].