## 2 Materials and Methods
### SVD Denoising  
Received radio-frequency signals had a strong trigger-induced noise that spans from 0 to 0.50 \(\upmu\)s corresponding to a distance from 0 to 0.75 mm to the sensor surface (Figure 2). This signal was relatively consistent across different scanning positions and manifested as vertical lines in the raw signals matrix (Figure 2). One way to avoid the interference of the noise is to move the focal plane away from this region (greater than 0.75 mm depth), however, this approach sacrifices the achievable spatial resolution with a multimode fibre. Here, SVD was used to address this issue. SVD has been demonstrated to remove laser-induced noise and background thermal noise in photoacoustic imaging [41; 42]. Briefly, the 2D raw signals matrix X (time-varying signals acquired at different spatial locations) was decomposed with SVD as:  
\[\mathbf{X=USV^{T}} \tag{1}\]  
where S is the diagonal matrix of singular values that are sorted in declining order, and U and V are matrices of the left- and right-singular vectors, respectively. The raw signals matrix is assumed to comprise three main components as:  
\[\mathbf{X=X_{t}+X_{pa}+\varepsilon} \tag{2}\]  
where X\({}_{t}\) represents the trigger-induced noise, X\({}_{pa}\) is the photoacoustic signal generated from tissue, and \(\varepsilon\) is the residual thermal noise. Since electrical triggers caused strong and almost constant noises across all A-line signals, these compositions have relatively high coherence and thus tend to be acquired in the singular vectors with the largest singular values. Similarly, the thermal noise can be considered random, and corresponds to the singular vectors with small singular values. So, \(X_{\text{pa}}\) can be preserved by assigning the first n and the last k singular values as zero in the new diagonal matrix of singular values \(S_{\text{a}}\) as:  
\[S_{a}(i)=\left\{\begin{array}{c}0,\;1\leq i\leq n\\ S(i),\;n<i\;\leq k\\ 0,\;i>k\end{array}\right. \tag{3}\]  
and the filtered ultrasound signals can be reconstructed via:  
\[\mathbf{X^{\prime}=US_{a}V^{T}} \tag{4}\]  
where \(X^{\prime}\) is the denoised signals matrix, each column of which represents an A-line photoacoustic signal.  
Similarity matrices of the spatial vectors were used to determine which singular values and vectors to be removed as described in Ref. [43]. Here, correlation coefficients of left-singular vectors were calculated and formed a matrix with each coefficient representing the similarity between each two vectors. Due to different coherence associated with \(X_{\text{t}}\), \(X_{\text{pa}}\) and \(\varepsilon\), the similarity matrix is expected to be separated into 3 distinct areas. In the case shown in Figure 2, the vectors that contained \(X_{\text{pa}}\) (in the orange box) had observable correlation coefficients, whilst the 1 st vector showed coherence only to itself represented \(X_{\text{t}}\), and other vectors referred to random noise \(\varepsilon\) (in the green box). Thus, we kept only the singular vectors corresponding to the singular vectors in the orange box in the new diagonal matrix \(S_{\text{a}}\) to reconstruct ultrasound signals.  
Figure 2: Workflow of singular value decomposition denoising to retrieve photoacoustic signals that were significantly distorted by the trigger-induced noise. A set of A-line signals received by the sensor form a raw signals matrix. After singular value decomposition of this matrix, the correlation coefficients between left-singular vectors are calculated and grouped into three sub-areas. The first sub – area is the first vector which only has coherence to itself and corresponds to the trigger-induced noise. The second sub-area (orange box) contains vectors with coherence to neighborhood vectors and corresponds to photoacoustic signals. The third sub-area (green box) that corresponds to small singular values and shows no coherence to other vectors contains random noises.