1. Introduction

Diffuse optical tomography (DOT) is a non-invasive imaging tool that probes the functional structure of biological tissues and has been used in imaging of the brain and breast. The harmless near-infrared light used in DOT discriminates oxyhemoglobin, de-oxyhemoglobin, and water content due to different absorption coefficients in biological tissues [1,2]. Transmitted and/or reflected diffusive photons from a biological tissue are used tomographically to reconstruct maps of scattering and absorption coefficients of the tissue. Therefore, DOT has enormous potential to diagnose tissue abnormalities including cancerous tumors.

Data measured using a continuous wave (DC) source cannot, in general, be reconstructed to a unique solution due to the ill-posed nature of the forward scattering problem. One effective data-acquisition method to diminish this problem is to modulate the intensity of the incident beam source. In this frequency-domain measurement, modulation amplitude attenuation and phase delay of output beams are employed to reconstruct the absorption and scattering maps [3]. It could be mistakenly considered that higher modulation frequency f_m always leads to better signal-to-noise (SNR) of measurements due to larger amplitude and phase variations as measured at individual detectors. In reality, however, the performance is not always increasing as f_m increases due to noise; hence there exists an optimal f_m [4]. Generally, SNR of the modulation amplitude and/or phase under appropriate noise models is considered to estimate the system performance [4–7]. V. Toronov et al. [6] regarded a quantum shot noise as a main noise source in DOT and derived analytic expressions for amplitude and phase standard deviations as

Since f_m in DOT is usually hundreds of MHz or higher, current CCD or CMOS detectors cannot measure the modulation amplitude and phase of output beams. This is overcome by a heterodyne method, where the modulated gain of an image intensifier makes the amplified output modulate at the beat frequency that is the difference between f_m and the gain modulation frequency [9]. Because the beat frequency is typically much lower than the bandwidth of CCD or CMOS detectors, the heterodyne output can be measured precisely, thus obtaining modulation amplitude and phase images across the entire detector array. If the gain is modulated at the exact same frequency as f_m, but has a different phase from the source beam, the amplified output beam varies along the heterodyne output trajectory by changing the relative phase; this is known as a homodyne method [10]. With the homodyne method, noise can be easily reduced by increasing the detector exposure time or acquiring multiple images and averaging; an advantage not shared by the heterodyne method. Although it is technically possible to realize gain modulation frequencies up to 1GHz or more [7], the instrumentation needed to achieve this fast gain modulation is very expensive. Furthermore, the lifetime of the image intensifier’s cathode, where a modulated voltage is applied to achieve gain modulation, might be dramatically reduced under the circumstance of high frequency modulation. Therefore, it is beneficial to develop frequency-domain diffusive imaging methods that have high performance while operating at low modulation frequencies.

The use of out-of-phase modulation inputs, called a phased array, has been introduced as the effective method to increase the phase variation of modulated output beams in diffusive media [11,12]. A typical phased array system has two modulated inputs with π intensity phase difference and a single detector on the interface between two sources, where a null plane of abrupt amplitude and phase transitions is generated in homogeneous media. The amplitude and phase on a single detector are very sensitive to the variation of photon number difference from sources, which are strongly affected by inhomogeneity of the investigated phantom. Therefore, traditional phased array systems optically scan the phantom to detect a signal that perturbs the modulation amplitude and/or phase null plane. It has been reported that a phased array system is more robust to biological random noise than a single source system, and that the amplitude-phase crosstalk is almost canceled when both input beams are generated by one source [13,14]. Not confined to just scanning tissues, the idea of phased array is used in fluorescence imaging tomographic reconstructions under the assumption of diffuse approximation [14–16]. Recently, simulations have shown that the reconstructed images from a phased array show better resolution than a single source system, but it is hard to implement the system for reliable measurements due to the scanning [17].

Some groups studied the SNR of modulation phases in phased array systems to investigate signal detectability in diffusive imaging. In these studies, small objects are embedded in diffusive media to generate perturbed signals. Y.Chen et al. [18] showed that a phased array usually has higher signal detectability than a single source system. They also pointed out that the intended power imbalance between two input sources increases the detectability for perturbed signals in the inhomogeneous background. S. Morgan et al. [19] introduced the idea of varying the relative phase between two input modulation sources, which is effective at differentiating object sizes as well as locations. S. Morgan also investigated the detectability of various phased array configurations using a probabilistic detection theory in the presence of Gaussian noise [20]. He anticipated that the optimized phased array system would show better performance than a single source system when the power densities of input sources of these systems are the same. Even though there have been various SNR analyses for phased array systems, as described in this paragraph, to the author’s knowledge, all previous SNR analyses are with a single detector located on the transition null plane. This indicates that a phased array system has been attractive as a scanning tool to detect a perturbed signal in diffusive media.

In this paper, we investigate the performance of a phased array when the detector on the exit face of a phantom is an array, such as CCD or CMOS cameras – CCD-based phased array system. Unlike previous works, we will analyze systems using objective, task-based measures of image quality [21,22]. That is, we will not just analyze the conventional SNR of the measured data but will assess the ability of an observer to perform relevant tasks using the image data. This methodology, which is based on the fact that medical images are acquired for specific purposes or tasks, has been used extensively to optimize other medical imaging modalities [22]. We define image quality by how well these relevant tasks can be performed. Measures such as resolution and contrast, while quantitative, do not necessarily correlate with task performance.

The tasks we have chosen are the detection of a small abnormality (i.e., a signal) and the task of distinguishing between two adjacent abnormalities (a task related to the resolution of reconstructed images). This approach denotes that we expect that the CCD-based phased array can be tomographically used for diffusive imaging without scanning. There is a great deal of literature reporting that large data sets measured from detector arrays in diffusive imaging systems increase the system performance and the quality of reconstructed images [23–26]. However, to the author’s knowledge, phased array systems with detector arrays have not been deeply investigated yet. With a detector array, the detectability for an embedded small abnormality in a homogenous diffusive medium is calculated using the concept of Hotelling observer assessment [21,22,27,28], which is further explained in the next section. Usually, the surface of detector array is imaged onto the phantom exit face, but for simplicity without losing generality, the influence of imaging, such as the decreased number of detected photons and additional noise factors, is not considered for calculating the Hotelling observer performance. The well-known noise model of Eq. (1) is adapted in our study, where the means of modulation amplitude and phase are simulated by the tMCimg Monte-Carlo software [29]. tMCimg is the widely used Monte-Carlo simulation tool considering the Henyey-Greenstein phase function for tissue scattering, which is openly downloadable. The result shows that phased arrays with a large amount of detector pixels (1600 in this paper) have high performance in tasks of detecting single abnormalities and distinguishing between adjacent abnormalities. Furthermore, the use of detector arrays allows us to achieve this high performance at substantially low modulation frequency f_m, thus significantly reducing the equipment cost.

2. Model

Figures 1(a) and 1(b) show side and top views, respectively, of the phantom configuration for the tMCimg simulator. The phantom is uniform with absorption (μ_a) and scattering (μ_s) coefficients set to 0.005mm⁻¹ and 5mm⁻¹, respectively, and the anisotropy parameter (g) is 0.8, which are in the known range of breast tissue optical characteristics [30,31]. The small abnormality embedded in the uniform background medium perturbs the diffused output photons. Detecting the data perturbation is conceptually the same as detecting the embedded abnormality, which is related to SNR of the data. In frequency-domain diffusive imaging, the data is the modulation amplitude and/or phase. The ability to detect this abnormality (or signal) is determined by the differences and variances in the measured data when compared to just the normal background. Thus, the term signal detectability refers to the ability of an observer to distinguish data with and without the abnormality.

 

Fig. 1 Side (a) and top (b) views of the phantom for the Monte Carlo simulation. The detector array shown in Fig. 1(a) is D₂ that is also indicated as a dot square in Fig. 1(b). Although not shown here, D₁ has the size of 2 × 2mm, which is located on the center of the phantom. The red arrows and dots indicate sources and gray squares indicate possible abnormality locations.

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The abnormality or signal to be detected is 2 × 2 × 2mm³ and is placed at various locations within the uniform phantom (see Fig. 1). The possible abnormality locations (${\text{L}}_{\text{Sig}}$) are indicated as dark gray squares in Fig. 1, which are numbered for convenience. The separation between each ${\text{L}}_{\text{Sig}}$ is 4mm. We will simulate two types of signals to be detected: absorbing and scattering signals. For absorbing signals, μ_s of the signal is the same as the background but μ_a of the signal is set to 0.01mm⁻¹. For scattering signals, μ_a is the same as the background and μ_s is set to 5.005mm⁻¹. These signals mimic abnormal tissues generally showing higher μ_a and μ_s than values in normal breast tissues due to highly concentrated blood vessels. 2mm-diameter sources of a phased array system are simulated with 10⁸ photons each, which are indicated as two red arrows and dots in Figs. 1(a) and 1(b), respectively. Output photons from each source are recorded on the detector array located on the center of the phantom exit face, as shown in Fig. 1(a). 2 × 2mm (D₁) and 40 × 40mm (D₂) detector arrays are considered to investigate the effect of a large detector array. A relative time difference τ is introduced between two sources to model the relative phase difference for a fixed f_m. The modulation amplitude and phase for the phased array system are calculated by Fourier transforming the combination of two output photons from each source, where one is shifted from the other by the amount of τ.

The observer we will focus on is an ideal linear observer called Hotelling observer, which is considered a reliable assessment tool in statistical decision problems [21,22,28]. The Hotelling observer SNR for the current type of study, where both signals and background are known exactly, is defined as

When data measured at different detector pixels are considered as independent, Κin Eq. (2) becomes a diagonal matrix having diagonal components of ${\sigma }_i^2$, where i incidates a detector pixel. Under this assumption, Eq. (2) is simplified to

It is known that Bayesian ideal observer calculates the most reliable performance but the computation of this observer is often impractical [22]. The efficiency of other observers, such as Hotelling observer, can be defined from the ideal observer’s performance, where higher observer efficiency indicates the performance from the observer is more reliable. Hotelling observer is equivalent to Bayesian ideal observer when data distributions are Gaussian [22]. Although statistics of modulation amplitude and phase are not Gaussian, the skewness of their probability density functions don’t deviate much from that of Gaussian distribution [32]. Therefore, it can be stated that Hotelling observer SNRs of Eqs. (4) and (5) are highly efficient, which validates the use of Hotelling observer assessment for this study.

3. Simulation result

Usually, the measurement of modulation amplitudes is less reliable than phase due to surface roughness of tissues and/or stray light [33]. Therefore, we mainly investigate $SN{R}_{\phi }^2$ for the Hotelling observer performance of the phased array systems. The detector pixel size $l_D$ is 1 × 1mm, so N in Eq. (5) is 4 and 1600 for D₁ and D₂, respectively. The term β in Eqs. (4) and (5) is not physically meaningful, because this quantity just scales the system’s Hotelling observer performance. Therefore, β is arbitrarily chosen, and held constant for all simulations. The characteristics of$SN{R}_A^2$ and the effect of different $l_D$ on $SN{R}_{\phi }^2$ are discussed later. Figures 2(a) –2(d) show $SN{R}_{\phi }^2$ from two different detectors D₁ and D₂ for absorbing signals located on ${\text{L}}_{\text{Sig}}$ = 1 and 3. For each result, $SN{R}_{\phi }^2$ with τ = 0, 2ns, and 4ns are shown as the function of f_m, where π of the relative phase differences between two input sources happen at f_m = 125MHz and 250HMz for τ = 2ns and 4ns, respectively. The system configuration for Figs. 2(a) and 2(b) is similar to that of a conventional scanning phased array system because of the point-like detector D₁, where $SN{R}_{\phi }^2$ values near π of the relative phase are relatively high, but show minimum at the exact π. S. Morgan et al. [19] pointed out this feature of a phased array system with the explanation that the steepest phase gradient is generated on the phase transition plane with the source phase difference of near π, which is not possible for the exact π. The detecting mechanism of the conventional phased array system can be understood from Figs. 2(a) and 2(b), where the movement of the absorbing abnormality around the phase transition plane caused by phased array scanning dramatically changes detectability, when the phase difference between two sources are near π. Although not shown in Fig. 2, $SN{R}_{\phi }^2$ for${\text{L}}_{\text{Sig}}$ = 2 shows a similar behavior with Fig. 2(b), which indicates the abrupt SNR improvement happens when the inhomogeneity is scanned around the phase transition plane.

 

Fig. 2 Hotelling observer SNRs for the task of detecting small absorbing signals at${\text{L}}_{\text{Sig}}$ = 1 and 3 in a phased array system with the detector D₁ ((a) and (b)) and D₂ ((c) and (d)) are shown. These SNRs are calculated with modulation phases. SNRs of all examined signal locations at fixed modulation frequencies with D₁ and D₂ are shown in (e) and (f), respectively.

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$SN{R}_{\phi }^2$ calculated from modulation phases detected from D₂ is dramatically increased, as shown in Fig. 2(c) and 2(d). In addition to the dramatic improvement, it is observed that $SN{R}_{\phi }^2$ from D₂ still shows a minimum value at π phase difference, but the width of this dip is much wider than that of the D₁ detector. Resulting from this broadening, peak SNR values for each τ are shifted to lower f_m than Figs. 2(a) and 2(b). For the example of τ = 4ns and ${\text{L}}_{\text{Sig}}$ = 1, peaks of SNR are observed around 80MHz for D₂, and around 120MHz for D₁. Another important characteristic of $SN{R}_{\phi }^2$ from D₂ is that the performance for all studied absorbing signal locations is relatively high at fixed f_m ; this f_m is called the optimal f_m. This can be partially observed in Figs. 2(c) and 2(d) that both$SN{R}_{\phi }^2$s of τ = 4ns show peaks around f_m = 80MHz, which is not the case for Figs. 2(a) and 2(b). The existence of the optimal f_m is obvious in Figs. 2(e) and 2(f), where$SN{R}_{\phi }^2$ of all ${\text{L}}_{\text{Sig}}$at selected modulation frequencies are shown for D₁ and D₂, respectively. Compared to Fig. 2(e), where the variation of $SN{R}_{\phi }^2$ is largely sensitive to${\text{L}}_{\text{Sig}}$, the overall performance in Fig. 2(f) is consistently high for the entire${\text{L}}_{\text{Sig}}$.

Hotelling observer SNRs for the task of distinguishing between adjacent signals are shown in Fig. 3 , of which notation is indicated as$SN{R}_{\Delta \phi }^2$. For $SN{R}_{\Delta \phi }^2$, two different system states in Eq. (2) are two small absorbing abnormalities, where one is located on ${\text{L}}_{\text{Sig}}$ and the other is shifted 2mm in depth from the ${\text{L}}_{\text{Sig}}$. Different from $SN{R}_{\phi }^2$ that indicates signal detectability for the absorbing signal located on ${\text{L}}_{\text{Sig}}$,$SN{R}_{\Delta \phi }^2$ measures the resolution of reconstructed images for these two adjacent signals. It is known that the depth resolution of the reconstructed image in DOT is usually worse than the lateral resolution [34]. Therefore, only$SN{R}_{\Delta \phi }^2$ for signals shifted in depth is investigated in this paper. Figure 3 shows that characteristics of the $SN{R}_{\Delta \phi }^2$variation from D₁ to D₂ are similar with the case of$SN{R}_{\phi }^2$. These $SN{R}_{\Delta \phi }^2$ values are dramatically increased by the larger detector array, and an optimal f_m, where $SN{R}_{\Delta \phi }^2$of all signal locations are relatively high and stable, exists. It is estimated from Figs. 3 and 4 that considering a detector array in a phased array system can stabilize both $SN{R}_{\phi }^2$ and$SN{R}_{\Delta \phi }^2$ simultaneously at the same fixed modulation frequencies for absorbing signals. Especially, for the phantom configuration of Fig. 1, the f_m is ~80MHz for τ = 4ns, which is less than 100MHz, so the instrument cost could be dramatically reduced.

 

Fig. 3 Hotelling observer SNRs for two adjacent signals at for ${\text{L}}_{\text{Sig}}$ = 1 and 3 in a phased array system with the detector D₁ ((a) and (b)) and D₂ ((c) and (d)) are shown. These SNRs are calculated with modulation phases. SNRs of all examined signal locations at fixed modulation frequencies with D₁ and D₂are shown in (e) and (f), respectively.

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Fig. 4 (a) Mean $SN{R}_{\phi }^2$ calculated from all${\text{L}}_{\text{Sig}}$ are shown with different τ and μ_s, where the optimal f_m are almost invariant to μ_s. (b) the variation of $SN{R}_{\phi }^2$to${\text{L}}_{\text{Sig}}$ is shown for different detector sizes at τ = 6ns and f_m = 55MHz, where the increasing rate of$SN{R}_{\phi }^2$ is saturated.

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Figure 4(a) shows averaged $SN{R}_{\phi }^2$ from all ${\text{L}}_{\text{Sig}}$ measured from D₂ for different${\mu }_s$and τ, where the optimal f_m are at around 80MHz, 55MHz, and 40MHz for τ = 4ns, 6ns and 8ns, respectively. Although not shown here, it is observed that the behavior of averaged $SN{R}_{\Delta \phi }^2$ is similar and peaks of these values are observed at the almost same f_m as $SN{R}_{\phi }^2$. As shown in Fig. 4(a), the optimal f_m is decreased and the absolute peak value is slight increased, as τ increases. Therefore, it can be concluded that the higher τ is the better the Hotelling observer performance is in the CCD-based phased array. Notice that the optimal f_m is ~40MHz and 55MHz for τ = 6ns and 8ns, respectively, which are much smaller than the previously reported optimal f_m in a single source system. More importantly, these optimal f_m for each τ are almost invariant to the background ${\mu }_s$as determined from Fig. 4(a). This characteristic of the CCD-based phased array is highly desirable to DOT because it implies that the system can be optimized for all types of tissue properties.

Figure 4(b) shows $SN{R}_{\phi }^2$ for${\mu }_s$ = 5mm⁻¹ at f_m = 55MHz with τ = 6ns for various sizes of the detector array for the fixed detector pixel size of 1 × 1mm. It is reasonable to consider that Hotelling observer performance in Eq. (2) is increased more for the higher number of detector pixels, because of more terms added. It is observed in Fig. 4(b), however, that the increasing rate of $SN{R}_{\phi }^2$ is saturated as the detector size increases. We confirmed that$SN{R}_{\Delta \phi }^2$ shows the similar trend, even though the results are not shown here. There are two reasons for this phenomenon. First, photon numbers detected by the pixels far from the center are much smaller than the central area, so $a_{AC}^2/a_{DC}$in Eq. (5) is dramatically decreased. Second, the abrupt phase variation of a phased array is mainly concentrated on the central area, so $\Delta {\overline{\phi }}^2$in Eq. (5) measured from outer area is also decreased. Therefore, the overall contribution from outer detector pixels to$SN{R}_{\phi }^2$ becomes very small. Additional study shows that this $SN{R}_{\phi }^2$ saturation by the increased detector array size happens for different source separations of 8mm and 24mm. Furthermore, amazingly, the optimal f_m observed from the saturated $SN{R}_{\phi }^2$ for these different source separations are almost the same. Overall, Fig. 4 indicates that applying a CCD-based phased array system generates a sensitivity area inside diffusive media. Both $SN{R}_{\phi }^2$ and $SN{R}_{\Delta \phi }^2$ for absorption inhomogeneities within the entire sensitivity area simultaneously show peak values at the same f_m, where the f_m is mainly determined by τ and almost invariant to other properties, such as μ_s and a source separation.

For all previous simulation results, the detector pixel size $l_D$ is fixed to 1mm, which can be changed in a real situation. Experimentally, an imaging system images a CCD or CMOS detector plane to the exit surface of a phantom and the transverse magnification of the system can change $l_D$. Therefore, it is interesting to investigate the Hotelling observer SNR variation to $l_D$when the whole size of a detector array is reasonably large. Figures 5(a) and 5(b) show $SN{R}_{\Delta \phi }^2$ for absorbing signals and $SN{R}_{\phi }^2$ for scattering signals, respectively, where cases of $l_D$ = 1mm and 2mm in D₂ are shown in each of them. As mentioned before, the scattering signals have μ_s = 5.005mm⁻¹ and other characteristics are the same as background. Hotelling observer SNRs for ${\text{L}}_{\text{Sig}}$ = 2, 5, and 8 are not shown because these values are similar to ${\text{L}}_{\text{Sig}}$ = 3, 6, and 9 cases, respectively. As shown in Fig. 5, decreasing $l_D$dramatically increases $SN{R}_{\Delta \phi }^2$and $SN{R}_{\phi }^2$. We confirmed that previously demonstrated benefits of a CCC-based phased array including the optimal f_m are remained with these enhanced values. However, it is observed from other study that $SN{R}_{\phi }^2$ for absorbing signals are almost invariant to $l_D$.

 

Fig. 5 Hotelling observer SNRs are increased as a detector pixel size ($l_D$) decreases for tasks of detecting (a) two adjacent absorbing and (b) scattering abnormalities. These SNR enhancements can be explained with mean phase differences for (c) absorbing and (d) scattering signals.

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The effect of changing $l_D$ in a CCD-based detector array on the Hotelling observer SNR can be understood by further investigating Eq. (5), where it is easily found out that the i_th $SN{R}_H^2$component is proportional to the number of photons measured on i_th detector pixel. Therefore, if $l_D$ is decreased, the SNR value on each detector pixel is also decreased. However, the whole$SN{R}_H^2$ is calculated by summing all pixel values, and the number of pixels is increased as $l_D$ is decreased (assuming the size of the detector array is the same). Resulting from this, $SN{R}_H^2$ remains at least the same as long as the amounts of $a_{DC}$ and $a_{AC}$on a single detector pixel are proportionally changed to $l_D$. This implies that there should be other factors to further increase $\Delta a_{AC}^2$ and/or $\Delta {\overline{\phi }}^2$of $SN{R}_{\Delta \phi }^2$ and $SN{R}_{\phi }^2$ in Figs. 5(a) and 5(b), respectively. Figures 5(c) and 5(d) show one-dimensional profiles of modulation phase difference $\Delta \overline{\phi }$ from absorbing and scattering signals, respectively. $\Delta \overline{\phi }$ for absorbing signals are mainly non-negative or non-positive, which is already observed elsewhere [19]. For this situation, when $l_D$is reasonably small compared with the variation of $\Delta \overline{\phi }$, the summed value of $\Delta {\overline{\phi }}^2$from all detector pixels is almost invariant to $l_D$. Therefore, $l_D$doesn’t affect $SN{R}_{\phi }^2$ for small absorbing abnormalities. The situation is, however, different for $\Delta \overline{\phi }$ from scattering signals, as shown in Fig. 5(d), where each $\Delta {\overline{\phi }}^2$ value could be dramatically increased as $l_D$ decreases by reducing the abrupt oscillating part of $\Delta \overline{\phi }$ in each detector pixel area. Therefore, smaller $l_D$ achieves higher $SN{R}_{\phi }^2$ as shown in Fig. 5(b). The increase of $SN{R}_{\Delta \phi }^2$ in Fig. 5(a) can be similarly understood. Generally, most biological tissues are inhomogeneous in terms of μ_a and μ_s. Therefore, it can be concluded that imaging smaller areas on the exit surface of phantoms to each detector pixel always increases both $SN{R}_{\phi }^2$ and $SN{R}_{\Delta \phi }^2$. There is, however, the limitation to increase$SN{R}_{\phi }^2$ and $SN{R}_{\Delta \phi }^2$by reducing $l_D$, because the noise model of Eq. (1) is not valid when the number of photons on a single detector pixel is below a certain level [32]. Increasing the measurement time (detector exposure time) could solve this problem, but this is prohibited or limited in some applications of diffusive imaging.

4. Discussion and conclusion

We observed that using a detector array substantially improves detectability in phased array systems where the detectability is the Hotelling observer performance calculated from modulation phases. It is interesting to investigate the effect of a detector array on$SN{R}_A^2$ in Eq. (4) in a phased array system. Figures 6(a) and 6(b) show$SN{R}_A^2$ from absorbing signals at ${\text{L}}_{\text{Sig}}$ = 1 and 3 with D₁ and D₂, respectively, where a detector array of D₂ increases overall $SN{R}_A^2$ dramatically like$SN{R}_{\phi }^2$. For both detector types, it is observed that $SN{R}_A^2$ of τ = 0ns is monotonically decreasing as f_m increases, which is similar to the result in a single source system [4,6]. For ${\text{L}}_{\text{Sig}}$ = 1 with D₁, introducing nonzero τ between two modulation sources makes $SN{R}_A^2$ oscillate to f_m having minimums at π phase differences. The trend of $SN{R}_A^2$ variation for${\text{L}}_{\text{Sig}}$ = 3 does not differ much from the case of ${\text{L}}_{\text{Sig}}$ = 1 except there are small peaks at π phase differences. Figure 6(b) shows that the detector array of D₂ increases $SN{R}_A^2$ at ${\text{L}}_{\text{Sig}}$ = 3 more than the case of ${\text{L}}_{\text{Sig}}$ = 1 and additionally, $SN{R}_A^2$of τ = 2ns and 4ns approaches to values of τ = 0ns, which are the clear improvement caused by D₂. Figures 6(c) and 6(d) show $SN{R}_{\Delta A}^2$ calculated from modulation amplitudes for two adjacent absorbing signals, where basic characteristics are analogous $SN{R}_A^2$. The noticeable difference from $SN{R}_A^2$ is that peaks at π phase differences between two sources are more remarkable.

 

Fig. 6 Hotelling observer SNRs for the task of detecting small absorbing abnormalities at = 1 and 3 in phased array systems with detectors (a) D₁ and (b) D₂ are shown. Hotelling observer SNRs for the task of detecting two adjacent abnormalities are shown in (c) and (d) for D₁ and D₂, respectively. These quantities of Hotelling observer SNR are calculated with modulation amplitudes.

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Previously, conventional SNRs of modulation amplitudes and phases have been separately considered to estimate the performance of frequency-domain diffusive imaging [4,20]. It is reasonable that observer performance is larger (smaller) when both quantities are simultaneously increased (decreased). For the most cases, though, the variations of these two quantities are not correspondent, so it is complicated to determine the system’s optimization condition. For example of the phased array with D₂ for τ = 4ns, $SN{R}_A^2$ shows relatively high values at f_m = 250MHz, but this is not the case for$SN{R}_{\phi }^2$, as shown in Figs. 6(b) and 2(d), respectively. One ad hoc method for compromising this issue is to simply consider the modulation amplitude and phase together in Eq. (2). For this case, dimensions of the data vectors and covariance matrix are 2N where N is the number of detector pixels. Since the modulation amplitude and phase are uncorrelated in frequency-domain measurement [32], the covariance matrix doesn’t have any non-zero off-diagonal terms. Therefore, the Hotelling observer SNR for this approach is just the summation of Eqs. (4) and (5). However, the absolute value of$SN{R}_A^2$ is usually much larger than $SN{R}_{\phi }^2$, as shown in Figs. 2 and 6, which means that it is not reasonable to just simply consider the modulation amplitude and phase together for Hotelling observer performance. Due to less reliability of measured modulation amplitudes, the current study mainly focuses on $SN{R}_{\phi }^2$ rather than $SN{R}_A^2$, but it is worthwhile investigating the method of effectively combining modulation amplitude and phase in frequency-domain diffusive imaging for more appropriate assessment.

The sensitivity area is observed in a CCD-based phased array, where both $SN{R}_{\phi }^2$ and $SN{R}_{\Delta \phi }^2$ are simultaneously high at a fixed f_m that is called as an optimal f_m in this paper. Additional simulations showed that performances for signals located outside boundaries determined by two sources are very poor, which means the sensitivity area width in the direction of connecting two sources is usually limited by the source separation, unless the distance between two sources is very small. Increasing the separation of two sources could extend the sensitivity area, but the overall performance is decreased. It is also observed for the case of small absorbing abnormalities that there is no clear boundary of the sensitivity area in the direction vertical to the line connecting sources. The sensitivity area’s vertical boundary is not strongly varied to the source separation as long as phantom properties are remained as the same. B. Chance et al. [35] contrived multiple sources in a phased array system, where the relative phase difference between sources is π and point detectors are located on multiple phase transition planes. We expect that the system of multiple modulation sources with a detector array could generate much large sensitivity area functioning at some fixed f_m that could be much less than 1/(2ω) by introducing some phase difference between sources, where ω is the modulation frequency of sources.

In conclusion, we investigated the effect of CCD or CMOS detectors on Hotelling observer detectability in phased array systems for the tasks of detecting small abnormalities (or signals) and for distinguishing between two adjacent signals in homogeneous diffusive media. Simulations show that the CCD-based phased array system could dramatically improve signal detectability and resolution of reconstructed images. Furthermore, we demonstrated that the modulation frequency optimizing the phased array system is generated by the detector array, where both signal detection and reconstruction resolution for small inhomogeneity located within some phantom area (i.e., sensitivity area) are simultaneously high. The value of the optimal f_m can be much smaller than 100MHz by increasing the time difference between two modulated input sources, which drives down the equipment cost for these systems. These results verify that a CCD-based phased array system is a high performance and cost-effective tomographic tool for diffusive imaging of tissue, which could be developed into new imaging systems for diffusive imaging applications.