signal detectability in diffusive media using phased arrays in conjunction with detector arrays

Signal detectability in diffusive media using phased arrays in conjunction with detector

arrays

Dongyel Kang,* and Matthew A. Kupinski

College of Optical Sciences, University of Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA *[email protected]

Abstract: We investigate Hotelling observer performance (i.e., signal detectability) of a phased array system for tasks of detecting small inhomogeneities and distinguishing adjacent abnormalities in uniform diffusive media. Unlike conventional phased array systems where a single detector is located on the interface between two sources, we consider a detector array, such as a CCD, on a phantom exit surface for calculating the Hotelling observer detectability. The signal detectability for adjacent small abnormalities (2mm displacement) for the CCD-based phased array is related to the resolution of reconstructed images. Simulations show that acquiring high-dimensional data from a detector array in a phased array system dramatically improves the detectability for both tasks when compared to conventional single detector measurements, especially at low modulation frequencies. It is also observed in all studied cases that there exists the modulation frequency optimizing CCD-based phased array systems, where detectability for both tasks is consistently high. These results imply that the CCD-based phased array has the potential to achieve high resolution and signal detectability in tomographic diffusive imaging while operating at a very low modulation frequency. The effect of other configuration parameters, such as a detector pixel size, on the observer performance is also discussed.

©2011 Optical Society of America

OCIS codes: (000.5490) Probability theory, stochastic processes, and statistics; (030.6600) Statistical optics; (290.7050) Turbid media.

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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 fm 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 fm increases due to noise; hence there exists an optimal fm [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

and ,A DC A ACa a (1)

respectively, where a

DCand

a

AC are mean DC and modulation amplitude. β is a term affected

by the measurement system which is also proportional to the amount of quantum shot noise. It is experimentally verified that the standard deviations in Eq. (1) are reasonable for the frequency-domain DOT [6,7]. It has recently been reported that the optimal fm for a single source system considering the noise model of Eq. (1) is between 400MHz and 600MHz for small tissue volumes [4]. If the distance between source and detector is closer and/or a tissue has a lower scattering coefficient, the optimal fm increases up to 800MHz [8].

Since fm 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 fm 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 fm, 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 fm, 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 (μa) and scattering (μs)

coefficients set to 0.005mm1

and 5mm1

, 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.

(a) (b)

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.

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 ( SigL ) are

indicated as dark gray squares in Fig. 1, which are numbered for convenience. The separation

between each SigL is 4mm. We will simulate two types of signals to be detected: absorbing

and scattering signals. For absorbing signals, μs of the signal is the same as the background

but μa of the signal is set to 0.01mm1

. For scattering signals, μa is the same as the background

and μs is set to 5.005mm1

. These signals mimic abnormal tissues generally showing higher μa and μs than values in normal breast tissues due to highly concentrated blood vessels. 2mm-diameter sources of a phased array system are simulated with 10

8 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 fm. 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

2 1t

HSNR g g Κ (2)

7 8 9 4 5 6 1 2 3

Detector array (4040mm)

15mm

16mm

62mm


where 1 0g g g with 1g being the mean data vector when the signal is present and

0g being the mean data vector when the signal is absent. The term 0 11 2 K K K is the

average covariance matrix of the data without and with the signal - 0K and K1 covariance

matrices, respectively. The dimension of K is N × N, where N is the number of detector pixels. As shown in Eq. (2), the Hotelling observer SNR can be calculated from only first and second moments of data.

When data measured at different detector pixels are considered as independent, Κ in Eq.

(2) becomes a diagonal matrix having diagonal components of 2

i , where i incidates a

detector pixel. Under this assumption, Eq. (2) is simplified to

2

2

21

.N

i

H

i i

gSNR

(3)

Equation (3) is a measure of the squared mean data difference ( 2g ) as seen in each detector

element normalized by the variance in that element ( 2 ); that is, it is the squared difference

between the conventional SNRs from different system states. It is important to distinguish the Hotelling SNR values we are calculating as compared to conventional SNRs computed in previous work [4–7,18–20]. The Hotelling SNR addresses the detectability of a signal using all detector measurements accounting for the small signal (i.e., tumor) to be detected. Conventional SNR measurements are on a detector-by-detector basis and do not measure any sort of signal detectability since there is no signal of interest in their formulation. The Hotelling observer has been used extensively in the literature to characterize system performance in a number of imaging modalities [21,22,27,28]. Substituting Eq. (1) to Eq. (3) produces the Hotelling observer SNRs of modulation amplitude and phase as

2

2

21

1 NAC

A

i DC i

aSNR

a

(4)

and

2 2

2

21

1 NAC

i DC i

aSNR

a

(5)

respectively, where the notation (α1/α2)i implies the ith component of α1 divided by the ith component of α2 like in Eq. (3). As explained in Eq. (1), is a measurement system related

parameter, and DCa ,

a

AC and are mean DC, modulation amplitude and phase calculated

from modulated diffusive output photons. Since K in Eq. (2) is the averaged covariance, DCa

and a

AC in Eqs. (4) and (5) are averaged means from different system states. As shown in

Eqs. (4) and (5), Hotelling observer SNR can be calculated by only mean data vectors that

could be simulated from tMCimg. Equations (4) and (5) also indicate that these 2

HSNR are

proportional to the number of photons because DCa and

ACa are linearly increased by the

number of photons. If 0g and 1g in Eq. (2) are the mean data generated from two adjacent

abnormalities or signals, Eq. (2) measures the ability of Hotelling observer to distinguish between two signals that are very close to one another - a measure of reconstructed resolution.

In order to differentiate this task from the signal-absent and present case, notations for 2

HSNR

of modulation amplitude and phase for this task are set to2SNR and 2

ASNR , respectively.


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 SNR2

for the

Hotelling observer performance of the phased array systems. The detector pixel size Dl 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 SNRA2 and the effect of different lD on

2SNR are discussed later. Figures

2(a)–2(d) show SNR2

from two different detectors D1 and D2 for absorbing signals located on

SigL = 1 and 3. For each result, SNR2

with τ = 0, 2ns, and 4ns are shown as the function of

fm, where π of the relative phase differences between two input sources happen at fm = 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 SNR2

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, 2SNR for SigL

= 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.

SNR2

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

SNR2

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 fm than Figs. 2(a) and 2(b). For the example of τ = 4ns and SigL =

1, peaks of SNR are observed around 80MHz for D2, and around 120MHz for D1. Another


(a) (b)

(c) (d)

(e) (f)

Fig. 2. Hotelling observer SNRs for the task of detecting small absorbing signals at SigL = 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.

important characteristic of 2SNR from D2 is that the performance for all studied absorbing

signal locations is relatively high at fixed fm ; this fm is called the optimal fm. This can be

partially observed in Figs. 2(c) and 2(d) that both2SNR s of τ = 4ns show peaks around fm =

80MHz, which is not the case for Figs. 2(a) and 2(b). The existence of the optimal fm is

obvious in Figs. 2(e) and 2(f), where SNR2

of all LSig at selected modulation frequencies are


shown for D1 and D2, respectively. Compared to Fig. 2(e), where the variation of 2SNR is

largely sensitive to LSig , the overall performance in Fig. 2(f) is consistently high for the

entire SigL .

(a) (b)

(c) (d)

(e) (f)

Fig. 3. Hotelling observer SNRs for two adjacent signals at for LSig = 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.

Hotelling observer SNRs for the task of distinguishing between adjacent signals are shown

in Fig. 3, of which notation is indicated as SNR

2. For

2SNR , two different system states in


Eq. (2) are two small absorbing abnormalities, where one is located on LSig and the other is

shifted 2mm in depth from the LSig . Different from 2SNR that indicates signal detectability

for the absorbing signal located on LSig , SNR

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,

only2SNR for signals shifted in depth is investigated in this paper. Figure 3 shows that

characteristics of the 2SNR variation from D1 to D2 are similar with the case of SNR

2. These

SNR

2 values are dramatically increased by the larger detector array, and an optimal fm,

where 2SNR 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

SNR2

and SNR

2 simultaneously at the same fixed modulation frequencies for absorbing

signals. Especially, for the phantom configuration of Fig. 1, the fm is ~80MHz for τ = 4ns, which is less than 100MHz, so the instrument cost could be dramatically reduced.

(a) (b)

Fig. 4. (a) Mean 2SNR calculated from all SigL are shown with different τ and μs, where the

optimal fm are almost invariant to μs. (b) the variation of 2SNR to SigL is shown for different

detector sizes at τ = 6ns and fm = 55MHz, where the increasing rate of2SNR is saturated.

Figure 4(a) shows averaged 2SNR from all LSig measured from D2 for different ms and τ,

where the optimal fm 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 SNR

2 is

similar and peaks of these values are observed at the almost same fm as SNR2

. As shown in

Fig. 4(a), the optimal fm 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 fm is ~40MHz and 55MHz for τ = 6ns and 8ns, respectively, which are much smaller than the previously reported optimal fm in a single source system. More importantly, these optimal fm for each τ are almost

invariant to the background ms 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 2SNR for s = 5mm

1 at fm = 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 SNR2

is saturated as the detector size increases. We confirmed that SNR

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 aAC2 aDC in Eq. (5) is dramatically decreased. Second, the

abrupt phase variation of a phased array is mainly concentrated on the central area, so 2 in

Eq. (5) measured from outer area is also decreased. Therefore, the overall contribution from

outer detector pixels to2SNR becomes very small. Additional study shows that this SNR

2

saturation by the increased detector array size happens for different source separations of

8mm and 24mm. Furthermore, amazingly, the optimal fm observed from the saturated SNR2

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 2SNR and

2SNR for absorption inhomogeneities within the entire sensitivity area

simultaneously show peak values at the same fm, where the fm is mainly determined by τ and almost invariant to other properties, such as μs and a source separation.

For all previous simulation results, the detector pixel size Dl 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 Dl . Therefore, it is interesting to investigate the Hotelling observer SNR variation

to Dl when the whole size of a detector array is reasonably large. Figures 5(a) and 5(b) show

2SNR for absorbing signals and 2SNR for scattering signals, respectively, where cases of

Dl

= 1mm and 2mm in D2 are shown in each of them. As mentioned before, the scattering signals

have μs = 5.005mm1

and other characteristics are the same as background. Hotelling observer

SNRs for SigL = 2, 5, and 8 are not shown because these values are similar to SigL = 3, 6, and

9 cases, respectively. As shown in Fig. 5, decreasing Dl dramatically increases

2SNR and

2SNR . We confirmed that previously demonstrated benefits of a CCC-based phased array

including the optimal fm are remained with these enhanced values. However, it is observed

from other study that 2SNR for absorbing signals are almost invariant to

Dl .

The effect of changing Dl 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 ith 2

HSNR component is proportional to the number of photons measured on ith detector pixel.

Therefore, if Dl is decreased, the SNR value on each detector pixel is also decreased.

However, the whole 2

HSNR is calculated by summing all pixel values, and the number of

pixels is increased as Dl is decreased (assuming the size of the detector array is the same).

Resulting from this, 2

HSNR remains at least the same as long as the amounts of DCa and

ACa on a single detector pixel are proportionally changed to Dl . This implies that there should

be other factors to further increase 2

ACa and/or 2 of 2SNR and

2SNR in Figs. 5(a) and

5(b), respectively. Figures 5(c) and 5(d) show one-dimensional profiles of modulation phase

difference from absorbing and scattering signals, respectively. for absorbing signals

are mainly non-negative or non-positive, which is already observed elsewhere [19]. For this

situation, when Dl is reasonably small compared with the variation of , the summed value

of 2 from all detector pixels is almost invariant to Dl . Therefore, Dl doesn‟t affect 2SNR


for small absorbing abnormalities. The situation is, however, different for from scattering

signals, as shown in Fig. 5(d), where each 2 value could be dramatically increased as Dl

decreases by reducing the abrupt oscillating part of in each detector pixel area. Therefore,

smaller Dl achieves higher

2SNR as shown in Fig. 5(b). The increase of 2SNR in Fig. 5(a)

can be similarly understood. Generally, most biological tissues are inhomogeneous in terms of μa and μs. Therefore, it can be concluded that imaging smaller areas on the exit surface of

phantoms to each detector pixel always increases both 2SNR and 2SNR . There is, however,

the limitation to increase2SNR and

2SNR by reducing Dl , 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.

(a) (b)

(c) (d)

Fig. 5. Hotelling observer SNRs are increased as a detector pixel size ( lD ) 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.

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 SNRA2 in Eq.

(4) in a phased array system. Figures 6(a) and 6(b) show 2

ASNR from absorbing signals at SigL


= 1 and 3 with D1 and D2, respectively, where a detector array of D2 increases overall 2

ASNR

dramatically like2SNR . For both detector types, it is observed that 2

ASNR of τ = 0ns is

monotonically decreasing as fm increases, which is similar to the result in a single source

system [4,6]. For SigL = 1 with D1, introducing nonzero τ between two modulation sources

makes 2

ASNR oscillate to fm having minimums at π phase differences. The trend of 2

ASNR

variation forSigL = 3 does not differ much from the case of

SigL = 1 except there are small

peaks at π phase differences. Figure 6(b) shows that the detector array of D2 increases 2

ASNR

at SigL = 3 more than the case of

SigL = 1 and additionally, 2

ASNR 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 2

ASNR calculated from modulation amplitudes for two adjacent absorbing

signals, where basic characteristics are analogous 2

ASNR . The noticeable difference from

2

ASNR is that peaks at π phase differences between two sources are more remarkable.

(a) (b)

(c) (d)

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.

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, SNRA2 shows relatively high

values at fm = 250MHz, but this is not the case for SNR2

, 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 2

ASNR is usually much larger than SNR2

, 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 SNR2 rather than 2

ASNR , 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 SNR2

and

2SNR are simultaneously high at a fixed fm that is called as an optimal fm 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 fm 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 fm 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.

Acknowledgments

Authors acknowledge Stefano Young for his discussion about the tMCimg simulator.


signal detectability in diffusive media using phased arrays in conjunction with detector arrays

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