# A Feasibility Study for a Hand-Held Acoustic Imaging Camera
## 5 Array Optimization Methodology
### Problem Parameterization  
We consider a planar array of \(N\) microphones located at positions \(\mathbf{r}_{n}=(x_{n},y_{n})\) in the \(xy\) plane (Figure 10). Acoustic sources at frequency \(f\) impinge on the array from angles \(\theta\) and \(\phi\). The goal is to generate high-resolution, low-artefact acoustic images over the  
Figure 8: The _Dual Cam_ board is based on an FPGA that reads continuously from the I2S MEMS microphones (TDK/InvenSense); using a programmed DMA the read microphones values are stored in a RAM buffer. In our new proof of concept (POC) the processor can easily read data from RAM and redirect them to the USB port. Figure 9: Conceptual framework in the microphone array simulation. signal band \([f_{\min},f_{\max}]\). The array geometry and frequency-dependent beamformer filter coefficients \(\mathbf{w}(f)=[w_{1}(f),\ldots,w_{N}(f)]^{T}\) are jointly optimized to maximize directional sensitivity. The complex beam pattern \(B(\theta,\phi,f)\) encodes the array response at each look angle and is parameterized as:  
\[B(\theta,\phi,f)=\sum_{n=1}^{N}w_{n}(f)e^{-j2\pi f\mathbf{(\theta,\phi)}\cdot \mathbf{r}_{n}} \tag{3}\]  
where \(\mathbf{\hat{r}}(\theta,\phi)\) is the source position unit vector. The expression depends on both the layout \(\mathbf{r}_{n}\) and filter coefficients \(w_{n}(f)\) which are microphone-specific filters to be optimized. The expression has directionality dependence on both the layout \(\mathbf{r}_{n}\) and filter responses \(w_{n}(f)\). To allow joint optimization, a cost function \(J(\mathbf{w},\mathbf{r})\) is formulated that balances simultaneously directional focus, artefact suppression, frequency coverage, and robustness. It incorporates an idealized unity gain beampattern \(B_{0}(\theta,\phi,f)\) at the look direction and minimizes the deviation from this response over angle-frequency space. Regularization terms manage overall beamformer gain and robustness. The optimization determines array configurations and filters customized for the imaging application. With the beam pattern expressed as \(B(\theta,\phi,f)=\mathbf{w}^{T}(f)\mathbf{V}(\theta,\phi)\), the filter coefficients \(\mathbf{w}(f)\) can be analytically extracted from the cost function into a closed-form solution \(\mathbf{w}_{\text{opt}}(f)=\mathbf{R}^{-1}(f)\mathbf{q}(f)\). For the array layout optimization with \(\mathbf{w}_{\text{opt}}(f)\) fixed, simulated annealing avoids poor local minima. Iterative stochastic perturbations to microphone locations \(\mathbf{r}_{n}\) are accepted probabilistically based on the cost function to enable escaping local minima. After sufficient iterations, the array geometry converges to enhance directionality. To improve robustness, the cost function is averaged over possible microphone gain and phase errors modelled as random variables. This penalizes configurations with low white noise gain, minimizing sensitivity to imperfections. The expected beam pattern \(E[B(\theta,\phi,f)]\) is incorporated to account for errors; this optimization framework (Figure 9) allows the designing of array geometries and filters customized for compact, robust acoustic imaging over desired frequency bands. The resulting unconventional configurations maximize power focused on look directions while minimizing off-axis contributions and artefacts using small apertures. The cost function \(J(\mathbf{w},\mathbf{r})\) balances several competing objectives:  
1. Directional focus. Minimizing deviation of the achieved beam pattern \(B(\theta,\phi,f)\) from the ideal unity gain pattern \(B_{0}(\theta,\phi,f)\) at the look direction over angle-frequency space. This is quantified by the integral term: \[\iiint_{\Theta,\Phi,F}|B(\theta,\phi,f)-B_{0}(\theta,\phi,f)|^{2}d\theta d \phi df\]  
Figure 10: Cartesian coordinates system and steering angles (\(\theta_{0}\), \(\phi_{0}\)). Artefact suppression. Minimizing the beam pattern gain away from the look direction, incorporated through: \[\iiint_{\Theta,\Phi,F}|B(\theta,\phi,f)|^{2}d\theta d\phi df\]
3. Frequency coverage.