1. Introduction

Photoacoustic imaging requires light transmission and optical absorption of a target, which undergoes local heating and thermal expansion and generates a sound wave [1,2]. This sound wave is subsequently detected with conventional ultrasound receivers. Spatial coherence is a measurement of the similarity of the recorded pressure distribution across the aperture of the acoustic receivers, and this measurement can be leveraged to overcome one of the primary limitations of photoacoustic imaging — i.e., limited optical penetration. This limitation exists because photoacoustic signals are not generated in the absence of optical absorption. Similarly, in the presence of low optical absorption at depth, shallower regions that experience greater absorption, tend to dominate photoacoustic images created with image reconstruction techniques that display differences in signal amplitudes. These limitations are particularly evident with recent advances in smaller and portable low-energy light sources, such as pulsed laser diodes [3–7] or light emitting diodes [8–10].

Photoacoustic image reconstruction techniques that utilize the spatial coherence of recorded pressure distributions include coherence factor (CF) weighting to mitigate sidelobe-associated clutter [11]. The combination of CF weighting and the minimum variance beamforming has also been shown to enhance both spatial resolution and contrast [12]. Similarly, CF and synthetic aperture focusing have been used together to improve lateral resolution and signal-to-noise ratios (SNRs) [13]. A SNR-dependent CF was additionally introduced for adaptive side lobe suppression in photoacoustic imaging [14].

Short-lag spatial coherence (SLSC) imaging shares many of the same benefits of the coherence-based imaging techniques described above, but it differs by directly displaying spatial coherence information rather than using coherence as a weighting metric for amplitude-based images. SLSC imaging was initially developed for ultrasound imaging [15] and later applied to photoacoustic imaging of targets ranging from brachytherapy seeds in phantoms, ex vivo tissue, and in vivo canine prostates [16–18] to blood vessel phantoms consisting of either India ink solution [19,20] or flexible rubber rods [5,21,22]. This technique was also applied to image graphite rods and larger spherical inclusions created from a graphite-titanium-gelatin mixture, both surrounded by gelatin [23]. These applications of SLSC imaging consistently demonstrated improvements in contrast in high-noise imaging environments, particularly in the presence of low laser fluence and when targets were located at large distances from the light source. SLSC is primarily useful in imaging tasks aimed at identifying the presence or absence of a photoacoustic signal (e.g., interventional imaging to identify tool tips or brachytherapy seeds). Similar to the CF weighing methods described above, SLSC images have also been weighted with conventional delay-and-sum (DAS) photoacoustic images to reduce acoustic clutter and improve DAS photoacoustic image quality [24], thus providing additional promise for amplitude-dependent tasks, such as spectral unmixing.

While initial experimental results with SLSC imaging have shown outstanding promise when compared to both DAS and Fourier-based reconstruction methods [25], the theoretical basis for coherence-based photoacoustic imaging is still being explored [19,20], and current models do not simulate various noise conditions that are observed in experimental data. The absence of noise models in coherence-based photoacoustic theory limits accurate modeling of the observed benefits of averaging coherence data measured from multiple acquisitions (particularly when using low-energy laser sources [5]). In addition, the lack of noise models is expected to be responsible for many of the deviations observed between noiseless theoretical spatial coherence functions and experimental data [19,20].

The purpose of this paper is to explore noise models that mimic observations from previously acquired experimental data. In particular, spatial coherence functions from brachytherapy seeds of length 4.5 mm and 0.8 mm outer diameter exhibit a familiar linear decrease in the short-lag region, but the surrounding noise regions have a degree of randomness in the coherence measurements that is not modeled with existing theory [16]. Similarly, photoacoustic data acquired with pulsed laser diodes have low coherence at lag one and this coherence rises with averaging [5]. These two trends are not described with existing photoacoustic spatial coherence theory [19,20]. Therefore, this paper introduces, develops, and tests the limits of two noise models that mimic these two experimental observations that are not included in noiseless models of theoretical spatial coherence. These noise models are then evaluated in the context of SLSC imaging.

2. Theory

2.1. Background

Our current theory states that the normalized spatial coherence, 𝒞(u), of a lateral cross section of a photoacoustic target at the imaging depth, z, is equal to a scaled Fourier transform of the square of the product of the initial acoustic pressure distribution, A, and the distribution of optical absorbers, χ, in the lateral dimension of the imaging plane, x, as defined by the equation:

One pixel in a photoacoustic SLSC image is then created by integrating the spatial coherence function up to a specific short-lag value, M, over the multiple frequencies within the −6 dB bandwidth of the ultrasound probe:

2.2. Noise models derived from empirical observations

The following descriptions of our noise models are based on two observations of spatial coherence functions obtained from experimental data (i.e., they are phenomenological models), whereas Section 2.3 provides a mathematical foundation for our observations. The first observation is that there is randomness in the coherence estimates that appears to be independent of the lag value. The second observation is that the coherence at lag 1 for pulsed laser diodes is significantly lower than the lag-zero value, and this lag-one coherence increases with temporal averaging. These two observations are illustrated in Fig. 1, and we hypothesize that they originate from two sources of noise that can be modeled independently.

 

Fig. 1 Examples of experimental data showing (a) random noise in coherence functions [16] and (b) increased lag-one coherence when averaging signals received from a pulsed laser diode, where T represents the number of frames that were averaged before calculating the coherence functions [5]. These figures were reproduced with permission from Refs. [5,16].

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We model the random noise in experimental coherence functions, 𝒩, by adding random values drawn from a standard, zero-mean, normal distribution to the theoretical coherence functions described by Eq. (1). The standard deviation of the distribution is σ_N, which indicates that 99.7% (i.e., approximately 100%) of the noise values are within the range:

The additional noise observed with low-energy lasers is modeled as a delta function that is scaled by a noise-to-signal ratio (NSR) defined as:

2.3. Mathematical foundations to support noise models

Although Eq. (7) was derived based on empirical observations, it can also be derived by first noting that random variations in the fluence distribution incident on a target F (e.g., caused by scattering or random photon travel) contribute to noise in the coherence estimates. These random variations have a direct impact on the initial pressure distribution, A, of Eq. (1). Therefore, we can describe the pressure distribution containing source-related noise, A′, as the sum of a noiseless term, A, and an uncorrelated noise term, N_A, which we model as drawn from a normal distribution, such that:

Eq. (9) reduces to the sum of two Fourier transforms, each multiplied by a phase term:

The product of the phase term and the first Fourier transform is proportional to Eq. (1). We define the product of the phase term and the second Fourier transform as proportional to 𝒩, which simplifies to:

Next, we note that normalized spatial coherence in the presence of source-related noise can also be defined in terms of the normalized correlations between two signals S_i at location i on an array and S_i+m at a location separated by m elements from location i:

According to Eq. (13), 𝒞_noise is equal to 1 when m = 0. After assuming that the system-related noise and received signals are uncorrelated when m ≠ 0, Eq. (13) can be simplified to:

Based on the combination of Eq. (13) and Eq. (15) and the definition of u provided in Eq. (3), as P_N increases, 𝒞_noise remains as 1 when u = 0, and 𝒞_noise is a lower-amplitude version of 𝒞′(u) when u ≠ 0, such that [26]:

After substituting Eq. (11) into Eq. (17), the resulting expression can be simplified to:

We use the mathematical description provided by Eq. (18) to relate physical terms to the noise models described in Section 2.2. Specifically, 𝒩 describes the fluence variations (which occur due to scattering and random photon travel, regardless of the laser energy), and $\frac{{𝒞(u)|}_{m=1}}{𝒞(0)}$ is equal to the ratio between the additive noise power on each channel and the signal power (i.e., the noise-to-signal power ratio, $\frac{P_N}{P_S}$). This ratio is related to the receiver electronics sensitivity, which dictates the NSR that can be achieved with a particular photoacoustic system configuration (including a photoacoustic system that uses low-energy energy light sources). Fig. 1(b) demonstrates that temporal averaging of signals from a low-energy laser decreases this NSR.

3. Methods

3.1. Simulation methods

Theoretical SLSC images were created from a simulated 2D phantom with a circular target of high optical absorption surrounded by a low optical absorption background, using the theory described in Section 2. The magnitude of the initial pressure distribution inside and outside of the target, across the lateral dimension of the imaging plane, (i.e., A), was equal to 5.93 μJ/cm³ and 0.13 nJ/cm³, respectively. Specifically, the distribution was obtained based on the following parameters: an average fluence F = 130 mJ/cm² incident on the lateral dimension of the imaging plane, the Grüneisen parameter, Γ, set to 0.144 and 0.81 inside and outside of the target, respectively, and the optical absorption, μ_a, set to 111 cm⁻¹ and 0.1 cm⁻¹ inside and outside of the target, respectively. The expected value of the optical absorber distribution, χ, was modeled as a constant value of 1. The target diameter was varied from 2 mm to 20 mm in increments of 2 mm, unless otherwise noted.

The simulated ultrasound array had 128 elements, 0.3 mm pitch, and frequencies of 3 MHz to 7.25 MHz in 0.25 MHz increments. The short-lag values, M, ranged from M = 5 to 20 (which corresponded to 3.9% to 15.6% of the 128-element aperture). The values of σ_N were 0, 0.1, 0.2, 0.5, and 1.0, and the NSR values were 0, 0.25, 0.50, 0.75, 1.0, and 1.5, unless otherwise stated.

3.2. In vivo liver data

We acquired data from a hepatic blood vessel in an in vivo porcine liver and created spatial coherence functions and SLSC images of the blood vessel using previously described methods based on Eq. (12) [16]. Our photoacoustic imaging system included an Alpinion E-Cube 12R ultrasound scanner connected to an Alpinion L3-8 linear array transducer (Alpinion, Seoul, Korea), which was synchronized with a Phocus Mobile Nd:YAG-based laser (Optoek Inc., Carlsbad, CA) operating at 750 nm. A 5 mm diameter fiber bundle coupled to the laser output transmitted an incident energy of 40 mJ per pulse to the tissue surface. A laparotomy was performed to gain access to the liver, and the ultrasound probe and fiber bundle were both in direct contact with the liver tissue. This study was approved by the Johns Hopkins Animal Care and Use Committee.

3.3. Performance metrics

The relationships among signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and contrast measurements in theoretical SLSC images were quantified as functions of short-lag value, M, and target diameter, d. To calculate these image quality metrics for theoretical and in vivo images, two regions of interest (ROIs) were identified inside and outside the target at the same axial depth. Rectangular ROIs were selected to include the maximum signal amplitude, and a second ROI of the same size at the same depth was located laterally to the right of the target. The ROIs were 1.6 mm (axial) × 2.7 mm (lateral), and the distance between the right edge of the target ROI and left edge of its matching lateral ROI was 1.5 cm. The SNR, CNR, and contrast were defined as:

Spatial coherence functions, SLSC image cross-sectional profiles, contrast, CNR, and SNR measurements obtained with our theory were compared to experimental results published in six previous papers [5,16–18,22,23] and to the in vivo data described in Section 3.2 in order to assist with theory validation and to offer explanations for previous trends across a wide range of data types (i.e., phantoms, ex vivo tissue, and in vivo). It would therefore be helpful to view the results presented in this paper alongside the experimental results presented in Refs. [5,16–18,22,23].

4. Results

4.1. Simulation results

Fig. 2(a) shows the SLSC image of a circular target with a 5 mm diameter. The coherence functions corresponding to selected points of interest inside, on the edge, and outside off the target are shown in Fig. 2(b)–(d), with the locations of the points of interest marked in the SLSC image. For each of these locations, 𝒩 was varied to assess the modeled effects of fluence-related noise on the spatial coherence functions. The fluence-related noise model that closely resembled the noise observed in previous experiments had σ_N values ranging from 0.5 to 1.

 

Fig. 2 (a) Theoretical SLSC image with markers at the outside (yellow), on the edge (red), and inside (blue) of a 5 mm diameter target. Corresponding coherence functions at the marker locations are shown for the (a) inside, (b) edge, and (c) outside locations when σ_N is 0, 0.5, and 1.0 for the a 5 mm diameter target.

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For visual comparison of the effect of σ_N on simulated SLSC images, Fig. 3 shows SLSC images created with various σ_N values. Qualitatively, the images generally demonstrate increased randomness of the coherence inside and outside of each target as σ_N increases for each short-lag value, M. The increased randomness is expected to reduce image SNR and CNR and introduce more randomness in corresponding contrast measurements. In addition, contrast appears to be reduced as M increases from 5 to 20. The axial line plots shown in Fig. 3 were taken from a lateral line corresponding to the target center in the associated SLSC images. Results from three M values were combined into one plot for each noise level. These axial line plots are later used to evaluate both qualitative and quantitative relationships among contrast and M.

 

Fig. 3 SLSC images of a 5 mm diameter target when σ_N is (a) 0, (b) 0.5, and (c) 1.0, displayed with short-lag values of M = 5 and 20 and the corresponding axial line plots taken at the lateral position corresponding to the target center.

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Fig. 4 shows the effect of including the noise model that simulates the lower coherence observed with low-energy laser sources. The first column contains no variations in fluence (i.e., σ_N = 0). In this first column, as NSR increases from 0 to 1, the coherence functions inside the target start to exhibit the low coherence that is observed in experimental data acquired with pulsed laser diodes rather than Nd:YAG lasers. The coherence functions obtained with the Nd:YAG laser is more representative of the results obtained with NSR = 0 (see Fig. 1(a)). Previous work has also shown that averaging signals from multiple laser firings enables us to recover similar spatial coherence to that measured with Nd:YAG lasers (see Fig. 1(b)), and the results in Fig. 4 show that this effect of averaging can be modeled by decreasing the NSR value.

 

Fig. 4 Spatial coherence functions inside and outside a 5 mm diameter target shown with NSR values of 0, 0.25, 0.5, 0.75, and 1 and σ_N values of (a) 0, (b) 0.1, and (c) 0.2.

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When outside of the target, decreasing the NSR value increases the absolute value of the coherence that is measured when lag ≥ 1, relative to that measured with NSR = 0, which is not observed experimentally [5]. However, the addition of fluence-related noise, σ_N, in the second and third columns of Fig. 4 better represents our experimental observations, particularly when NSR > 0. In these cases, the increasing NSR has minimal impact on the variations observed in the coherence functions obtained outside of the target. This behavior is consistent with previously observed experimental measurements (e.g., Fig. 1(b)), indicating that both NSR and σ_N in Eq. (7) must be greater than zero to adequately model the noise observed in experimental data. Thus, the fluence-related noise (i.e., σ_N) adds randomness to the coherence measured at all lags and this randomness increases as the value of σ_N increases and affects both the coherence measured within and outside of the photoacoustic target.

For visual comparison of the effect of both σ_N and NSR on simulated SLSC images, Fig. 5 shows SLSC images created with Eq. (18) replacing the 𝒞(u) term in Eq. (4) for various σ_N and NSR values and a short-lag value of M = 5. Qualitatively, as NSR increases, the images in Fig. 5 do not appear to be affected, which is likely due to NSR only adjusting lag 1 relative to lag 0 and the coherence functions are always normalized to 1 at lag 0. However, similar to Fig. 3, as σ_N increases, the randomness of coherence inside and outside of the target increases, for both values of NSR. These results indicate that the fluence-related noise (i.e. σ_N) decreases SLSC image SNR and CNR, while noise due to a combination of system receiver electronics, low-energy laser sources, and minimal or no averaging (i.e., noise represented by NSR) have minimal impact on SLSC image contrast, which is generally consistent with experimental observations (e.g., Fig. 3 in Ref. [5]). However, this trend was not observed in Table 1 of Ref. [23], which shows increasing contrast with averaging. The difference between these trends likely occurs because of differences in the implementation of averaging. Signals were averaged prior to computing normalized cross correlations to form SLSC images in Ref. [5], while SLSC images were averaged in Ref. [23].

 

Fig. 5 SLSC images for NSR values of 0.5 and 1.5 and σ_N values of (a) 0, (b) 0.5, and (c) 1.0 for a 5 mm diameter target and M = 5.

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Table 1. Previously reported contrast, CNR, and SNR measurements from experimental SLSC images and corresponding σ_N and NSR values required to achieve these measurements. Lag is reported as a percentage of the receive aperture

View Table

Fig. 6 shows changes in the theoretical SLSC images as target size increases in the absence of a noise model. Qualitatively, contrast decreases as target size increases, which can also be appreciated by axial line plots through the central lateral position of each target in the SLSC images. Fig. 7(a) shows these axial line plots for the same target sizes shown in Fig. 6 but with added noise (i.e., σ_N = 0.1 and NSR = 0.5). Fig. 7(b) shows similar line plots with a higher magnitude of added noise (i.e., σ_N = 1.3 and NSR = 0.5). In both noise cases, as target size increases, a decrease in contrast is observed. In addition, the coherence signal difference between the target boundaries and target center increases as target size increases. Qualitatively, the line plots for the higher noise levels seem to match line plots obtained in both experimental and in vivo data [18,22], particularly in the background of the image (i.e., outside of the target). This similarity can be confirmed by comparing Fig. 7(b) with Fig. 9(b) in [22] and Fig. 5 in [18].

 

Fig. 6 SLSC Images with target diameters of (a) 1, (b) 4, (c) 6, (d) 8, and (e) 10 mm, created with M = 5 and no noise in the model.

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Fig. 7 Axial line plots taken at the lateral location corresponding to the target center for 1 mm, 4mm, 6mm, 8 mm, and 10 mm diameter circular targets with noise values of (a) σ_N = 0.1 and NSR = 0.5 and (b) σ_N = 1.3 and NSR = 0.5 and M = 5. Target sizes of 4 mm and 8 mm were omitted from the line plots in Fig. 7(b) to assist with plot readability. The vertical lines show axial boundaries of the ROIs used to measure contrast, CNR, and SNR.

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Fig. 8 (a,b) Contrast, (c,d) CNR, and (e,f) SNR as a function of diameter with NSR = 0.5 and σ_N values (a,c,e) 0.1 and (b,d,f) 1.3.

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Fig. 9 (a) B-mode and (b) SLSC images from in vivo liver experiment. Representative coherence functions are displayed from the locations shown (c) inside and (d) outside of the target. The corresponding theoretical coherence functions were created from a target with a diameter of 11.97 mm, with noise values σ_N = 1.1, and NSR = 1.0 where indicated.

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Contrast as a function of target size in the presence of the noise model described by Eq. (7) is shown for both low (σ_N = 0.1) and high (σ_N = 1.3) values of 𝒩 in Figs. 8(a) and 8(b), respectively. Confirming our qualitative observations, the best contrast with the theoretical data is observed at smaller targets, as well as at lower lag values (e.g., M = 10) for the smaller targets. Contrast decreases as target size increases from 2 mm to 20 mm, and there is minimal contrast difference when varying the short-lag values (i.e., M) of the larger targets. Contrast values are as high as 24.66 dB with lower noise and the maximum contrast increases to 27 dB with increased noise. The contrast measurement is also more variable as the fluence-related noise (i.e., σ_N) increases, which indicates that this noise is responsible for the large standard deviations observed in previous measurements of contrast (e.g. Fig. 6 in Ref. [16]).

For larger target sizes, the similar contrast measured across multiple short-lag values seems to contradict our observations in Fig. 3, where the contrast appears to decrease as the short-lag value increases from M = 2 to M = 20 for a 5 mm-diameter target. However, the corresponding axial line plots in Fig. 3 demonstrate an increase in the coherence of the target boundaries with increasing M, while the target center coherence remains relatively constant as M increases, which explains this apparent discrepancy. The SLSC images in Fig. 3 were normalized to the brightest pixel in each image (which occurs at the target boundaries), and this normalization causes a perceived qualitative contrast decrease when comparing SLSC images created with increasing short-lag values. However, because the ROI for calculating contrast (indicated by the vertical lines in Fig. 7) is located at the target center and excludes the target boundaries, the calculated contrast remains relatively constant across the different M values.

CNR as a function of target size is shown for both low (σ_N = 0.1) and high (σ_N = 1.3) values of 𝒩 in Figs. 8(c) and 8(d), respectively. Unlike the contrast measurements in Fig. 8(b), there is a significant decrease in CNR when higher values of σ_N are included in the theoretical predictions. CNR values are as high as 33.18 with lower noise and the maximum CNR decreases to 4.5 with the addition of more noise. The CNR plots in Fig. 8(c) each experience a peak for target diameters ranging from 6–8 mm for lower σ_N values, but the measured CNR generally decreases as target size increases beyond 4 mm for the higher σ_N values, as shown in Fig. 8(d). The peaks in CNR measurements occur because of the location and size of the chosen ROI. Fig. 7 shows that for targets smaller than 6 mm in diameter, the ROI includes an increase in coherence caused by the narrower target width near the circular boundary, which results in a larger standard deviation within the target region and a lower CNR.

SNR as a function of target size is shown for both low (σ_N = 0.1) and high (σ_N = 1.3) values of 𝒩 in Figs. 8(e) and 8(f), respectively. Similar to the CNR measurements, there is an order of magnitude decrease in the measured SNR when the higher σ_N value is included in the theoretical predictions. SNR values are as high as 88.23 with lower noise values and the maximum value decreases to 7.0 with the increased noise. SNR also decreases as target diameter increases.

As shown in the axial line plots of Fig. 7, although the mean signal coherence remains relatively constant as noise increases, the standard deviation of signals within the target and background ROIs significantly increase as the noise level increases. Therefore, the trends of decreasing CNR and SNR with increasing noise levels are caused by the increased standard deviations of signals within the target and background regions of the SLSC images.

To demonstrate examples of required noise values to achieve similar performance to that observed in previous experimental data, Table 1 shows previously measured contrast, CNR, and SNR values reported in published papers [5,16,18,23] and lists the parameters that are needed for our model to match these values. In particular, we matched the reported lateral target size to a circular disk of the same size and adjusted the NSR and σ_N values in our model until the resulting measurements matched the reported measurements. We also matched the short lag value as a percentage of the receive array aperture rather than matching the number of elements. The corresponding laser energies are additionally listed for comparison. The σ_N values ranged from 0 – 1.3 and NSR values ranged from 0 – 8.5. The ROIs used to measure contrast, CNR, and SNR with our model were placed in the same locations described in Section 2, although the ROIs were not the same as previously reported ROIs. Nonetheless, we are more interested in the noise values required for our model to achieve similar contrast, SNR, and CNR to previous reports rather than using the noise models to recreate images that match the previously reported images. Note that it was difficult to match contrast with our model applied to the parameters in Ref. [23], likely because SLSC images were averaged to achieve higher contrast, and therefore a separate NSR value is reported to match contrast only. For this value of NSR (i.e., 4), the corresponding σ_N was 0.2.

4.2. Comparison of theory with in vivo liver data

Fig. 9(a) shows the ultrasound B-mode image of an in vivo blood vessel in the liver. Fig. 9(b) shows the corresponding photoacoustic SLSC image of the vessel created with M = 1. A theoretical SLSC image of a circular target with a diameter that matches the lateral width of the in vivo vessel in the B-mode image (i.e., 11.97 mm) was simulated. Although the axial dimensions of the theoretical and experimental images do not correspond (because we approximate the absorber function as circular), the results in the lateral dimension are assumed to be separable.

Representative coherence functions inside and outside of the vessel are shown in Figs. 9(c) and 9(d), respectively. When comparing the in vivo coherence functions to the theoretical coherence functions created without noise, the random variations at higher lags are not well modeled and there is a negative spike at the lower lags outside of the vessel. These observations support the introduction of noise models to obtain spatial coherence functions that are consistent with experimental data. We added source-related random noise, 𝒩, with σ_N = 1.1 and system-related noise with NSR = 1.0 to achieve a closer match between the in vivo and theoretical spatial coherence functions obtained inside and outside of the vessel. At a short-lag value of M = 1, the contrast, SNR, and CNR of the in vivo SLSC image were 14.0 dB, 3.6, and 1.5, respectively. The corresponding contrast, SNR, and CNR for the theoretical SLSC image were 11.3, 3.7, and 3.0, which represents 2.7 dB, 0.1, and 1.5 differences in contrast, SNR, and CNR, respectively.

5. Discussion

We introduced and implemented two additive noise models to simulate source-related noise (e.g., fluctuations in the initial pressure distribution) and system related noise (e.g., thermal and electronic) in spatial coherence measurements of photoacoustic data. The random additive noise model closely mimics the randomness in coherence estimates that we observe in experimental data (e.g., Figs. 1(a), 9(c), and 9(d)), which was not present in our noiseless models of photoacoustic spatial coherence [19, 20]. The mathematical descriptions in Section 2.3 relate this random noise to variations in the incident fluence distribution, which is random and known to be highly dependent on the scattering medium. Table 1 indicates that the contribution from fluence variations is generally consistent for multiple scattering environments, with σ_N values that span 0 – 1.3. Similarly, modeling the system-related noise as a delta function with an amplitude that is based on the ratio between the lag 1 and lag 0 spatial coherence measurements provides a working theory to explain the trends seen by Bell et al. [5], where averaging signals acquired with a pulsed laser diode increased the measured coherence at lag 1. The results in Table 1 and Fig. 9 are consistent with this theory because the contribution from this type of noise has a greater impact when the laser energy is lower, with NSR values that span 0.2 – 1 for Nd:YAG lasers (no averaging included [16,18]) and 0 – 8.5 for the pulsed laser diode [5]. In comparison to the 0.2 – 1 NSR values needed when no averaging is involved for the Nd:YAG laser, the NSR values can be as high as 4 when averaging SLSC images produced with an Nd:YAG laser [23].

The inclusion of noise in the spatial coherence model lays the groundwork for estimation and predictions of expected photoacoustic SLSC image quality in experimental data and enables us to quantify expectations for a range of target sizes and short-lag values (i.e., M). For example, Fig. 8 shows that the smaller target lateral cross sections in the image plane can be displayed with larger M values in the presence of noise to achieve better contrast, which was also observed in previous experimental data, where the circular axis of brachytherapy seeds was displayed with higher M values than the long axis of the same seeds [16, 17]. This trend is also seen in Fig. 9, where the spatial coherence functions of the larger 1 cm target indicate that high-contrast SLSC images are limited to the lower short-lag values (e.g., M = 1). Another interesting observation is that the contrast measurements are more variable as σ_N increases, which provides a working theory to explain the large variations in contrast measurements that are typically seen in experimental data. These results indicate that these large fluctuations are caused by variations in the fluence distribution, which varies due to scattering and is also highly variable when the light source is sufficiently far from the photoacoustic target (e.g., > 5mm, as indicated by Fig. 6 in Ref. [17]).

When these noise models are included before measuring target contrast, CNR, and SNR, these measurements more closely represent the results observed in previous experimental data [5,16–18]. Specifically, SNR and CNR values significantly decrease with the addition of higher levels of fluence-related noise (i.e., 𝒩), as shown in Fig. 8. In these higher noise environments, variations in the contrast measurement also increase. In comparison, the high NSRs that can be addressed with signal averaging prior to calculating spatial coherence do not appear to affect the overall image contrast, because SLSC is a correlation-based technique that is not directly dependent on signal magnitudes. Although there is minimal impact on SLSC image quality, the coherence functions are still affected by the chosen NSR value. Based on these observations, it is evident that our new noise models enable us to identify the best trade-off among contrast, CNR, SNR, and M in photoacoustic SLSC images, and this choice is based on target size.

One difference between experimental and simulated data that is not included in the noise models presented in this paper is the presence of significantly large (and often negative) values in coherence estimates, particularly at higher lag values. This could be due to a lower number of measurements of spatial coherence at the larger lags. In general, our noise models do not address the larger positive or negative values at higher lags, and this will be the focus of future work. Our future work will additionally simulate 2D absorber functions that mimic experimental data.

6. Conclusion

We developed and tested two models to describe the observed noise in photoacoustic spatial coherence measurements. The development of these noise models enables more realistic simulation of trends observed in experimental photoacoustic SLSC image data. These trends include random variations in the spatial coherence functions at higher lag values and changes in the contrast, CNR, and SNR image quality metrics as functions of target size and the short-lag value M that is used to display images. We have additionally provided a working theory to explain the large variations in contrast measurements that are typically seen in experimental data and the effects of averaging signals acquired with a pulsed laser diode to increase the measured coherence at lag 1. These noise models are promising for future optimization of coherence-based photoacoustic image quality.