Analysis of fast full angle fluorescence diffuse optical tomography with beam-forming illumination

Daifa Wang; Xin Liu; Jing Bai

doi:10.1364/OE.17.021376

1. Introduction

Fluorescence diffuse optical tomography (FDOT) is an optical imaging method that provides quantitative, three-dimensional imaging of fluorescence distribution inside living small animals. This technique overcomes the limitations of the simple and widely used planar reflectance imaging [1,2], which provides semi-quantitative planar imaging without depth information [3]. At present, FDOT has been successfully applied in oncology [4], treatment evaluation [5], and inflammation monitoring [6], etc. Combined with the anatomical structure provided by other imaging modalities such as MRI and CT, FDOT extends its applications in investigating glioma [7,8] and alzheimer’s diseases [9]. With the development of more accurate mathematical model of photon propagation in tissues, better reconstruction algorithms and imaging systems for general or specific applications, and novel targeting fluorescent dyes [10,11], more applications would be expected in fundamental researches, drug development, and clinical experiments.

In past years, great developments have been made in fluorescence diffuse optical imaging systems. The improvements are mainly focused on (i) non-contact delivery of the photons into and out of small animals, (ii) high spatial sampling of photon fields by using charge coupled device (CCD), (iii) implementation of full angle ( $360^{°}$ ) projections. These three factors are considered important in improving image qualities and simplifying the experiment procedures. The early fiber-based imaging systems typically place optical fibers around and in-contact with the imaged object for photons delivery [12–14]. In these systems, cylindrical imaging chamber and matching fluid are typically used to handle the irregular small animal geometry and simplify the complex fibers-fixing. Although full angle projections are provided in these systems, they have low spatial sampling of photon fields due to the small number of optical fibers, and relative complex experiment procedure due to matching fluid and in-contact photons delivery. Limited projection angle imaging systems with [15,16] (slab geometry) or without matching fluid [17–19] enable non-contact collecting and dense sampling of photon fields by using CCD. However, the resolution along projection direction is bottlenecked due to the limited projection angles. At present, full angle non-contact imaging systems have been reported [20,21]. These systems integrate these three factors and are expected to provide the optimal image qualities [22].

However, challenges still remain in imaging fast and whole body biological activities such as pharmacokinetics, where whole body field of view, short data collection time, optimal image qualities within current computational capacities should be considered simultaneously. Patwardhan et al. [16] designed a slab geometry based imaging system for real time whole body imaging. In their reports, they decreased the switching time between illumination sources using galvanometer-controlled mirrors and increased the data acquisition rate using a high frame rate EMCCD. In this article, we propose and analyze another strategy, where a non-contact, full angle imaging system was designed based on continuous-wave beam-forming illumination (BF-FDOT). The beam-forming illumination includes line illumination (L-FDOT), area illumination (A-FDOT), and multiple-points illumination (MP-FDOT). A-FDOT and MP-FDOT can be consider the same as L-FDOT in some extreme cases when squeezing the area illumination to one line in A-FDOT or having sufficient illumination points along one line in MP-FDOT. These systems are expected to provide the potential in real time whole body imaging with high quality images. Systematic studies should be performed to analyze and compare the performances of different kinds of beam-forming illumination based systems, and demonstrate the advantages of the beam-forming illumination over the conventional point illumination. The systematic studies will help us in better understanding which kind of beam-forming strategies is most appropriate for real time whole body imaging and obtaining corresponding parameters for better hardware design.

It is obvious that more projections, denser sampling of photon fields and finer reconstruction mesh will improve the reconstructed image qualities. Generally, the FDOT reconstruction can be described by a linear system m = Wn, where the weight matrix W maps the unknown fluorescence distribution n to known measurements m. The weight matrix W takes account of the light propagation in highly scattering tissues. When considering the number of pixels on a 512x512 CCD, 72 projections, and 30x30x60 reconstruction mesh, the weight matrix will contain approximate $10^{12}$ elements (~ $10^{3}$ GB). The high memory cost and succeeding long computation time exceeds current computational capacities. In visualizing pharmacokinetics in a long time period, successive 3D FDOT images need to be reconstructed so that the reconstruction time is especially critical for quick data analysis. In addition, the imaging time (data collection time) is related to the number of projections, which is important in real time imaging. Therefore, optimizations of experimental parameters of data acquisition and reconstruction are important to obtain the best image qualities with the least computational burden and imaging time. For various kinds of diffuse optical tomography (DOT) and FDOT imaging systems based on point sources, optimizations of experimental parameters are available [22–27]. Analysis of optimal experimental parameters has also been performed for different scanning patterns including line illumination [28]. However, the imaging system [28] is reflection based and mainly used in imaging large volume such as human brain.

Singular value analysis (SVA) has been widely used in various DOT [24,25] and FDOT [22,26] systems for determining the optimal source/detector arrangement, and the optimal field of view (FOV). SVA efficiently condenses the information contained in the weight matrix of different experimental parameters and system implementations to a spectrum vector. Then, evaluation and comparison can be easily made among these spectra which represent different experimental situations.

In this article, three categories of experiments were designed, where SVA was used as the main tool to find corresponding conclusions and reconstruction experiments were used to verify and supplement the findings obtained. Firstly, for FDOT with line illumination, we employed SVA to optimize projections number, detector sampling density, mesh sampling density, detector field of view along horizontal and vertical directions, and line source length. Secondly, To demonstrate the advantages of the FDOT with beam-forming illumination over the conventional FDOT with point illumination (P-FDOT) [20,21] in real time whole body imaging, experiments were designed to compare the spatial and temporal performances between L-FDOT and P-FDOT. To simplify the experiments design, only L-FDOT is involved, since it represents the strategy of all BF-FDOT systems and can be even considered as the extreme instances of the other two. Thirdly, considering the similarities of the three beam-forming strategies, only the specific experimental parameters of FDOT systems with area illumination or multiple-points illumination were analyzed. The specific parameters are the area width of A-FDOT and the number and density of multiple points of MP-FDOT. In this category of experiments, the performances of these three beam-forming strategies were also compared.

The outline of this article is presented as follows. In section 2, the methods used are detailed. In section 3, analytical and experimental results are described. In section 4, we discuss and conclude the major findings of this study.

2. Methods

2.1. Experimental setup

In observing the impact of different experimental parameters for real time FDOT imaging system with large field of view, we focused on these three full angle imaging systems with different kinds of beam-forming strategies. Firstly, a sufficiently narrow light beam along Z axis was used for illumination [Fig. 1(a) ], which was considered as a line source. Secondly, a uniform area light beam was used, as shown in Fig. 1(b). Thirdly, multiple-points along Z axis were used, as shown in Fig. 1(c). For the three imaging systems, the excitation light transports into the imaged object and excites the fluorescent dye inside. On the opposite side, the excitation light and the emission light out of the imaged object are collected by a CCD camera coupled with different band pass filters respectively. The imaged object is placed on a rotation stage, which enables collecting projections at arbitrary angles.

Fig. 1 The schematics of full angle imaging systems and the experimental parameters. (a)-(d) The top views and side views of full angle imaging systems with different illumination strategies, including line illumination (a), area illumination (b), multiple-points illumination (c), and point illumination (d). The blue lines and points in (a)-(d) indicate the illumination sources. (e) Illustration of detector horizontal field of view. (f) Illustration of the detector vertical field of view. (g) Illustration of detector spacing. (h) Simplified illustration of the 3D reconstruction mesh using a 2D mesh in X-Y slice.

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In singular value analysis and simulation experiments, the imaged object was a homogeneous cylinder with diameter of 2.0cm and height of 6.0cm. The phantom size selected represented the average dimensions of the mouse torso imaged in our previous in-vivo experiments. The optical properties of the cylindrical phantom were assumed as $μ_{s}^{'} = 10.0 {cm}^{- 1}, μ_{a} = 0.58 {cm}^{- 1}$ [22], which were close to optical properties of the mid-torso of mice. In Fig. 1(a), the line source length along vertical direction (Z axis) is an important hardware parameter of L-FDOT imaging system. In Fig. 1(b), the width of the area along horizontal direction and the length of the area along vertical direction are important hardware parameters of A-FDOT imaging system. In Fig. 1(c), the points number and points density are important hardware parameters of MP-FDOT imaging system. The three systems relate to each other. A sufficient narrow area or sufficient points along one line can both be considered as a line. The diagram of the conventional P-FDOT is also shown in Fig. 1(d). How many cycles of projections are used and which height slices the point sources of each cycle are placed at influence the whole body imaging abilities of P-FDOT. In Figs. 1(e) to 1(h), we also describe the experimental parameters such as detector horizontal/vertical field of view, detector spacing, and mesh spacing.

2.2. Forward and inverse problems

In highly scattering tissue medium, the light transportation can be modeled using the diffusion equation coupled with the Robin-type boundary condition [29]. Then, the Green’s function $G (r)$ describing the light transportation field due to a continuous-wave source term $S (r)$ can be obtained as follows,

{\begin{cases} - \nabla \cdot [D (r) \nabla G (r)] + μ_{a} (r) G (r) = S (r) r \in Ω \\ 2 q D (r) \partial G (r) / \partial \vec{n} + G (r) = 0 r \in \partial Ω \end{cases},

where Ω is the domain of the imaged object and

\partial Ω

is the boundary.

D (r) = 1 / (3 μ_{s}^{'} (r))

is the diffusion coefficient with the reduced scattering coefficient

μ_{s}^{'} (r)

at position r.

μ_{a} (r)

is the absorption coefficient. q is a constant depending on the optical refractive index mismatch on the boundary and

\vec{n}

denotes the outward normal of the boundary

\partial Ω

. For FDOT with point illumination, a collimated laser spot is usually modeled as an isotropic point source

S (r) = δ (r - r_{s})

, where

r_{s}

is the point one transport mean free path

l t r = 1 / μ_{s}^{'} (r)

into the medium from the illumination spot [29]. In analogy, a narrow light beam (line source) is modeled as a line one transport mean free path into the medium from the central axis of the narrow light beam, and an area light beam is modeled as an area one transport mean free path into the medium. Then, for the three kinds of beam-forming illumination strategies, the source terms are described using

L (r)

(line illumination),

A (r)

(area illumination), and

M P (r)

(multiple-points illumination) respectively as follows,

\begin{array}{l} L (r) : = {\begin{cases} \int_{{r_{l}}} L (r) d r = 1 \\ L (r_{1}) = L (r_{2}) r_{1}, r_{2} \in {r_{l}} \\ L (r) = 0 r \notin {r_{l}} \end{cases} \\ A (r) : = {\begin{cases} \int_{{r_{a}}} A (r) d r = 1 \\ A (r_{1}) = A (r_{2}) r_{1}, r_{2} \in {r_{a}} \\ A (r) = 0 r \notin {r_{a}} \end{cases}, \\ M P (r) : = 1 / M \sum_{{r_{m p}}} δ (r - r_{i}) r_{i} \in {r_{m p}} \end{array}

where

r_{l}

is the point on the line

{r_{l}}

,

r_{a}

is the point on the area

{r_{a}}

, and

r_{m p}

is the point of the multiple (M) points

{r_{m p}}

. Then, Eq. (1) can be solved using finite element method to obtain the Green’s functions for different source terms in Eq. (2). After that, the forward problem can be generated based on Normalized Born approximation [30], which reduces the mouse tissue heterogeneity influences. The ratio of the measured emission

Φ_{m} (r_{d})

and the corresponding excitation

Φ_{x} (r_{d})

at detector point

r_{d}

is formulated as follows,

\frac{Φ_{m} (r_{d})}{Φ_{x} (r_{d})} = Θ \int_{V} \frac{G_{S (r)} (r_{p}) G_{δ (r - r_{d})} (r_{p}) n (r_{p})}{G_{S (r)} (r_{d})} d r_{p},

where

r_{p}

is the point inside the volume Vconsidered for reconstruction, and

n (r_{p})

denotes the distribution of fluorescent dye at point

r_{p}

.

G_{S (r)} (r_{p})

denotes the Green’s function value at

r_{p}

due to an excitation source term

S (r)

.

G_{δ (r - r_{d})} (r_{p})

denotes the Green’s function value at

r_{p}

due to a point source

δ (r - r_{d})

at

r_{d}

. Θ is a unitless calibration constant which accounts for the excitation light power, the unknown gain and attenuation factors of the system. When the volume V is sampled in voxels, Eq. (3) can be discretized into vector form as

\frac{Φ_{m} (r_{d})}{Φ_{x} (r_{d})} = \frac{Θ Δ V}{G_{S (r)} (r_{d})} [G_{S (r)} (r_{p 1}) G_{δ (r - r_{d})} (r_{p 1}) \dots G_{S (r)} (r_{p N}) G_{δ (r - r_{d})} (r_{p N})] (\begin{array}{l} n (r_{p 1}) \\ ⋮ \\ n (r_{p N}) \end{array}),

where

Δ V

is the volume of the descretized voxel. For multiple source-detector pairs, a linear system

m = W n

could be generated based on Eq. (4),

(\begin{array}{l} Φ_{m} (r_{d 1}) / Φ_{x} (r_{d 1}) \\ ⋮ \\ Φ_{m} (r_{d M}) / Φ_{x} (r_{d M}) \end{array}) = W (\begin{array}{l} n (r_{p 1}) \\ ⋮ \\ n (r_{p N}) \end{array}) .

The unknown fluorescence distribution n was obtained by solving the linear system using algebraic reconstruction technique [31] with relaxation parameter $λ = 0.1$ and iteration number 1500.

In Eq. (5), the measured data are assumed linear to the dye concentrations. In fact, the dye concentrations also partly contribute to the optical properties including absorption and diffusion coefficients. Many reports use a nonlinear iterative model when considering this [32,33]. However, linear assumption is reasonable when we consider the optical properties such as absorption coefficient as one whole part. In other words, we don’t separate the absorptions of tissues and dye concentrations. This consideration is adopted in many diffuse optical tomography guided fluorescence diffuse optical tomography algorithms [19,34,35]. In addition, the Normalized Born method used herein can largely reduce the influences of inaccurate estimation of optical properties [30].

2.3. Singular value analysis and cost functions

Singular value analysis was used to study the generic characteristics of different experimental parameters, where the weight matrix was decomposed to $W = U S V^{T}$ . U and V are two orthonormal matrices and S is a diagonal matrix consisting of the singular values of W. The singular value spectrum was then normalized by its maximum element and truncated by a specific threshold. We will comment on the use of normalized rather than absolute singular values in Section 4. The normalized singular values above a specific threshold represent the useable image-space modes which can be detected in the experimental setup [22,24–26]. The threshold $10^{- 4}$ was empirically determined and used in [22,26] and could well account for the experimental noise level during FDOT experiments. Of course, the threshold is not universal because the noise level is complex and different in different practical experiments. However, it is representative and allows a generic criterion for obtaining the number of useable image-space modes (number of singular values above threshold, NSVAT) from measurements with different experimental parameters. A range of other thresholds from $10^{- 3}$ to $10^{- 5}$ were also repeated to deal with different levels of noises.

When improving the image qualities, the imaging time and computational burden including memory and computation time may increase. In addition, the complexity of system implementation was also considered. These cost functions combined with NSVAT were used to determine the optimal experimental parameters compromising among temporal resolution, spatial resolution, computational burden, and complexity of system implementation.

The imaging time of a full angle imaging system is the sum of rotation time and total exposure time of images. Typically, since the rotation speed is set to $6^{\circ} / s$ to prevent possible skew and internal mouse organs movement, one cycle of projections take approximate 1 minute rotation time. The total exposure time of images is the product of the projections number and the exposure time of one projection image. The exposure time of one image varies in different experiments but is typically 2 to 3 seconds in our previous experiments in pharmacokinetics. The computation memory is linear to the elements number of the weight matrix. When solving the inverse problem using 1500 ART iterations, the computation time is linear to the element number of W. The actual computation time on our personal computer (CoreTM2 Quad processor and 8GB RAM) for a 10000x10000 weight matrix was approximate 8.7 minutes.

2.4. Experimental sets for L-FDOT

The L-FDOT system is a typical FDOT system under study. A-FDOT and MP-FDOT can be considered as L-FDOT in some extreme cases. Therefore, we focused on the optimization of experimental parameters for L-FDOT in this section. As shown in Table 1 , we optimized six experimental parameters for L-FDOT system including projections number (A1), detector vertical FOV (A2), detector horizontal FOV (A3), detector spacing (A4), mesh spacing (A5), and line source length (A6). The phantom used was described in section 2.1. The reconstruction mesh was over 2.6cmx2.6cmx5.8cm 3D region and only the mesh inside the imaged object was considered for analysis, which was described in the following parts as the reconstruction mesh inside the imaged object and 5.8cm height range. Each parameter was optimized with itself varied while keeping other parameters constant. Each case was performed one to three times with different configurations, as shown in Table 1.

Table 1. The experimental sets for L-FDOT.

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2.5. Reconstructions of simulated data for L-FDOT

Simulation experiments were performed to further confirm the findings obtained by singular value analysis in section 2.4. As shown in Fig. 2(a) , 6 small cylindrical fluorescent tubes (0.3cm diameter and 0.3cm length) were embedded inside the cylinder phantom at different height slices, which simulated the whole body distribution of fluorescent dye. The 6 tubes were divided into 3 groups with their centers at different height slices (z = 1.7cm, 3.0cm, and 5.25cm). Each group consisted of two tubes with 0.3cm edge to edge distance ( $x = 0.0 cm, y = 0.3 cm; x = 0.0 cm, y = - 0.3 cm$ ). The concentration of fluorescent dye was assumed as 1 unit inside the tubes and 0 inside the background. The other optical properties of the tubes were assumed the same as those of the background. The excitation and emission data were synthesized using the finite element method based on Eq. (3), where $Φ_{x} (r_{d}) = G_{L (r)} (r_{d})$ and $Φ_{m} (r_{d}) = \int_{V} G_{L (r)} (r_{p}) G_{δ (r - r_{d})} (r_{p}) n (r_{p}) d r_{p}$ . Data were synthesized for experimental setups with different projections numbers and different line source lengths. The detectors were distributed over 1.8cm horizontal FOV and 5.8cm vertical FOV with 0.2cm spacing. For the purpose of numerical reconstructions using synthetic data, we have to choose a signal to noise level. Since this is tied to the number of CCD counts, not the flux that can be computed from the diffusion equation, we fix it using the following procedure: for excitation measurements, we determine a measurement time $S_{0}$ so that the CCD pixel that sees the highest light intensity would record 2000 CCD counts. Likewise, we determine a count time $S_{1}$ by requiring that the detector with the highest fluorescent light intensity registers 2000 fluorescent CCD counts. $S_{0}$ and $S_{1}$ were computed using an experiment with 24 projections and a 4cm long line source, and the same times $S_{0}$ , $S_{1}$ were also used for all other experiments. The noise with standard deviation $\sqrt{0.49 U + {2.8}^{2}}$ [36] was added to the synthetic data, where U was the signal. The noise model takes account of the Poisson probability model of experimental optical measurement and the readout noise of CCD. In the following parts of this article, this noise model was used for all synthetic data. Excitation measurements less than 40 counts were not considered in the reconstruction procedure. Of course, the different amplification factors between the excitation and fluorescence measurement channels were corrected before reconstructions. The reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm mesh spacing was used for reconstruction.

Fig. 2 Phantom and tubes settings for simulated and experimental experiments. (a) Simulated experiment. 6 cylinder tubes (diameter of 0.3cm and height of 0.3cm) were placed at different heights (1.7cm, 3.0cm, and 5.25cm) inside a cylinder phantom (diameter of 2.0cm and height of 6.0cm). The edge to edge distance was 0.3cm for each two tubes at the same height. (b) Experimental experiment. Two tubes (glass tubes of 0.3cm diameter filled with 10uL, 1.3uM ICG) were placed inside a cylinder phantom (a glass cylinder of 3.0cm diameter filled with 1% intralipid). The center distance of the two tubes along Z axis was 2.4cm.

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2.6. Reconstructions of experimental data for L-FDOT

Similar to the simulation experiments, reconstructions of experimental data were performed. As shown in Fig. 2(b), two transparent glass tubes (0.3cm diameter) filled with 10 $μL$ , 1.3 $μM$ ICG were immerged in a cylinder phantom at different heights. The phantom was made of a glass cylinder (3.0cm diameter) filled with 1% intralipid ( $μ_{s}^{'} =10 {.0cm}^{-1}, μ_{a} = 0.02 {cm}^{-1}$ ). The two tubes were far away from each other with center distance 2.4cm along Z axis to mimic whole body distribution. The line source was a narrow light beam of which the length was 4.3cm and the width was maintained less than 0.16cm over 5cm depth of field. The total power of the line source was about 5mW. 72 excitation and emission images were collected every $5^{°}$ using a 14 bit EMCCD, part of which were used to verify the impact of different numbers of projections. The detectors were distributed over 2.2cmx5.4cm FOV with 0.2cm spacing. Excitation measurements less than 40 counts were not considered in the reconstructions. The reconstruction mesh was over 3.0cmx3.0cmx5.4cm 3D region with 0.1cm spacing and only the mesh inside the imaged object was considered for reconstruction.

2.7. Comparison experiments between L-FDOT and P-FDOT

To compare the temporal and spatial performances between L-FDOT and P-FDOT, singular value analysis and reconstruction experiments were both performed. For P-FDOT with only one cycle of projections, the fluorescent targets far away from the point illumination slice couldn’t or weakly illuminated. Then, fluorescent targets far away from the point illumination slice may not be accurately reconstructed due to information shortage and the ill-posed nature of FDOT.

In this section, the optimal vertical detector FOV was firstly determined for P-FDOT with one cycle of projections using SVA. After obtaining the optimal vertical FOV, the cycles of projections of P-FDOT were varied while keeping other parameters constant. SVA was performed to evaluate how many cycles will contain larger NSVAT than L-FDOT. Finally, analysis of reconstructions of simulated data was used to confirm the findings obtained by SVA. All the experimental sets were performed on the phantom described in section 2.1. In this section, the mesh inside the imaged object and 5.8cm height range with 0.1cm spacing was used for analysis. The details of the experimental sets are as follows.

2.7.1. B1. The optimal detector vertical FOV for P-FDOT

The detector vertical FOV was varied while keeping other parameters constant. The point source was at 3cm height slice. 24 projections, 1.8cm detector horizontal FOV and 0.2cm detector spacing were used.

2.7.2. B2. SVA analysis of the number of cycles for P-FDOT

To obtain the best performance with multiple cycles, the optimal distribution of point sources along Z axis was firstly determined using SVA. The optimal vertical FOV was used and the other experimental parameters were the same as those in study B1. The NSVAT of P-FDOT with different cycles of projections were then compared to that of L-FDOT. For L-FDOT with 4cm long line source, 24 projections, detectors over 1.8cmx5.8cm FOV with 0.2cm spacing, and reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing were used.

2.7.3. B3. Reconstructions of simulated data for P-FDOT

The reconstruction experiments were used to verify and supplement the findings in study B2. The phantom and tube settings were the same as those in section 2.5. For P-FDOT system, each cycle of projections consisted of 24 projections (evenly distributed over $360^{°}$ ). The detectors were distributed over the optimal vertical FOV and 1.8cm horizontal FOV with 0.2cm detector spacing. The number of cycles was from 1 to 4. The excitation and emission data were synthesized using finite element method based on Eq. (3), where $Φ_{x} (r_{d}) = G_{δ (r - r_{s})} (r_{d})$ and $Φ_{m} (r_{d}) = \int_{V} G_{δ (r - r_{s})} (r_{p}) G_{δ (r - r_{d})} (r_{p}) n (r_{p}) d r_{p}$ . For P-FDOT with one cycle of projections, the excitation measurements were normalized with their maximum value mapped to an experimental measurement $2 \times 10^{3}$ counts ( $S_{0}$ ). Similarly, the fluorescence measurements were normalized with their maximum value mapped to an experimental measurement $2 \times 10^{3}$ counts ( $S_{1}$ , $S_{1} / S_{0} = 8.8$ ). For other experimental setups, the same $S_{0}$ and $S_{1}$ were also used. Noises, which were defined in section 2.5, were added to the synthetic data. Excitation measurements less than 40 counts were not considered in the reconstruction procedure.

2.8. Experimental sets for beam-forming strategies

The three systems with different beam-forming strategies relate to each other, where a sufficient narrow area or sufficient points along one line direction can either be considered as a line. Hence, the trends of most experimental parameters optimization including projections number, detector FOV, detector spacing, mesh spacing for A-FDOT and MP-FDOT could be expected similar to those for L-FDOT. In addition, the area length of A-FDOT and the multiple points’ height range of MP-FDOT were expected to have similar trends as the line source length of L-FDOT. In this section, we focused on analyzing the specific experimental parameters for different BF-FDOT systems, such as the area width of A-FDOT, and the multiple points’ density of MP-FDOT. During these experiments, we also compared the performances of different beam-forming strategies. All studies in this section were performed on the cylinder phantom described in section 2.1. In the following studies (C1-C3), 24 projections, 1.8cm detector horizontal FOV, 0.2cm detector spacing, the mesh inside the imaged object and 5.8cm height range with 0.1cm spacing were used. The corresponding experimental sets are detailed as follows.

2.8.1. C1. The influence of area width

The area width was varied while keeping other parameters constant. The area length used was 4cm and the bottom of the area was at z = 1cm height. The detector vertical FOV of A-FDOT was also dominated by the excitation signal intensity threshold, similar as that of L-FDOT. The optimal detector vertical FOV was analyzed before calculating the NSVAT of different area widths.

2.8.2. C2. The influence of multiple points’ density

The multiple points’ density was varied while keeping other parameters constant. The multiple points were evenly distributed inside height ranges from 1cm to 5cm.

2.8.3. C3. Reconstructions of simulated data for A-FDOT and MP-FDOT

Reconstructions of simulated data were performed to further confirm the findings in C1 and C2 by SVA. The phantom and tubes settings were the same as those in section 2.5. The line source was 4cm long with its bottom at z = 1cm for L-FDOT. The area length was 4cm with its bottom at z = 1cm height for A-FDOT. The area width was varied from 0.2cm to 1.6cm. The multiple points were evenly distributed inside height range from 1cm to 5cm. The total number of points was varied from 3 to 6.

For A-FDOT, the excitation and emission data were synthesized using finite element method based on Eq. (3), where $Φ_{x} (r_{d}) = G_{A (r)} (r_{d})$ and $Φ_{m} (r_{d}) = \int_{V} G_{A (r)} (r_{p}) G_{δ (r - r_{d})} (r_{p}) n (r_{p}) d r_{p}$ . For A-FDOT with 0.2cm area width, the excitation measurements were normalized with their maximum value mapped to an experimental measurement $2 \times 10^{3}$ counts ( $S_{0}$ ). Similarly, the fluorescence measurements were normalized with their maximum value mapped to an experimental measurement $2 \times 10^{3}$ counts ( $S_{1}$ , $S_{1} / S_{0} = 12.0$ ). For other area widths, the same $S_{0}$ and $S_{1}$ were used.

For MP-FDOT, the excitation and emission data were synthesized using the finite element method based on Eq. (3), where $Φ_{x} (r_{d}) = G_{M P (r)} (r_{d})$ and $Φ_{m} (r_{d}) = \int_{V} G_{M P (r)} (r_{p}) G_{δ (r - r_{d})} (r_{p}) n (r_{p}) d r_{p}$ For MP-FDOT with 6 points, the excitation measurements were normalized with their maximum value mapped to an experimental measurement $2 \times 10^{3}$ counts ( $S_{0}$ ). Similarly, the fluorescence measurements were normalized with their maximum value mapped to an experimental measurement $2 \times 10^{3}$ counts ( $S_{1}, S_{1} / S_{0} = 11.5$ ). For other points number, the same $S_{0}$ and $S_{1}$ were used.

Noises, which were defined in section 2.5, were added to the synthetic data. Excitation measurements less than 40 counts were not considered in the reconstruction procedure.

3. Results

3.1. Singular value analysis for L-FDOT

SVA was used to assess the effects of different experimental parameters for the L-FDOT system. As shown in Fig. 3(a) , the singular value spectra associated with study A1 are normalized to their maximum elements and plotted on a logarithmic scale. The noise threshold $10^{- 4}$ was chosen as the cut-off singular value for SVA and is plotted as a line in Fig. 3(a). The studies shown were also repeated with thresholds from $10^{- 3}$ to $10^{- 5}$ . Although the absolute value of NSVAT changed with different thresholds, the general trends in imaging performances were independent of the thresholds chosen.

Fig. 3 Singular value analysis of experimental parameters for L-FDOT. (a) The singular value spectra for different projections numbers. The experimental parameters used are detailed in study A1. (b) Study A1, singular value analysis of the number of projections (evenly distributed over the full angle) for L-FDOT. Plots are for setups with 0.1cm (circles) and 0.15cm (pluses) mesh spacings. (c) Study A2, singular value analysis of detector vertical FOV. Plots are for setups with 3cm (pluses), 4cm (diamonds), and 5cm (circles) long line sources. (d) Study A3, singular value analysis of detector horizontal FOV. Plots are for setups with 3cm (pluses), 4cm (diamonds) and 5cm (circles) long line sources. (e) Study A4, singular value analysis of detector spacing. Plots are for setups with 0.1cm (circles) and 0.15cm (pluses) mesh spacings. (f) Study A5, singular value analysis of mesh spacing. Plots are for setups with 0.2cm (pluses), 0.15cm (diamonds) and 0.1cm (circles) detector spacings. (g) Study A6, singular value analysis of line source length.

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3.1.1. Study A1

As shown in Fig. 3(b), the NSVAT is plotted as a function with respect to projections number for two different mesh spacings (0.1cm and 0.15cm). An initial sharp increase of NSVAT is observed up to 12 projections for these two mesh spacings. For the two mesh spacings, increasing projections number increases NSVAT. The increase speed of NSVAT, however, becomes low and low as the increase of projections number, which is especially apparent for 0.15cm mesh spacing. The NSVAT flattens out after 18 projections for 0.15cm mesh spacing. For 0.1cm mesh spacing, small increase is observed when the projections number is over 18. The increase in NSVAT corresponds to more useable image-space modes (better spatial resolution). However, increasing the projections number increases the imaging time (lower temporal resolution) and linearly increases the computational burden. When compromising between the small improvement and the significantly increased burden, 18 projections are considered optimal for the L-FDOT system.

3.1.2. Study A2

Detector points with ultra low excitation light signals, which were less than 2% of the maximum signal intensity, weren’t considered in reconstructions. In the existence of noises, these measurements don’t contain information but induce artifacts into the reconstructed images. As shown in Fig. 3(c), the NSVAT is plotted as a function with respect to the detector vertical FOV for three line source lengths (3cm, 4cm, and 5cm). It is seen that increasing the vertical FOV linearly increases NSVAT. For 3cm long line source, the optimal vertical FOV is dominated by the excitation signal intensity threshold. For line source with length longer than 4cm, the optimal vertical FOV is close to the length of the imaged object.

3.1.3. Study A3

As shown in Fig. 3(d), the NSVAT is plotted as a function with respect to the detector horizontal FOV for three line source lengths (3cm, 4cm, and 5cm). The 4.8cm vertical FOV used for 3cm long line source is the optimal vertical FOV determined in study A2. For all cases, increasing the horizontal FOV linearly increases NSVAT. Therefore, the detector horizontal FOV close to the width of the object silhouette is considered optimal.

3.1.4. Study A4

As shown in Fig. 3(e), the NSVAT is plotted as a function with respect to the detector density for two mesh spacings (0.1cm and 0.15cm). A sharp increase of NSVAT is observed for detector densities up to 5cm⁻¹. The NSVAT flattens out when the detector density becomes larger than 5cm⁻¹. In addition, increasing the detector density quadratically increases the computation time and memory. Therefore, the optimal detector spacing is determined as 0.2cm.

3.1.5. Study A5

As shown in Fig. 3(f), the NSVAT is plotted as a function with respect to the mesh density for three detector spacings (0.1cm, 0.15cm and 0.2cm). The mesh spacing was analyzed from 0.07cm to 0.2cm, while analyzing denser mesh sampling exceeded the capacity of our personal computer. Increasing the mesh density yields improvements in NSVAT, even when the mesh spacing is less than the detector spacing. However, the increase speed of NSVAT becomes much smaller for mesh densities larger than 10cm⁻¹. In addition, the increase in computation time and memory is the third power of that in mesh density. Considering that, the optimal mesh spacing is determined as 0.1cm.

3.1.6. Study A6

The study was performed for line source length varied from 2.0cm to 5.5cm. The vertical FOV used were the optimal vertical FOV determined in study A2. As shown in Fig. 3(g), the NSVAT is plotted as a function with respect to the line source length. A sharp increase of NSVAT is observed up to 4cm line source length. Further increasing the line source length still yields small improvement in NSVAT. The improvement in NSVAT is reasonable, because longer line source provides more uniform illumination along Z axis. Although increasing the line source length doesn’t yield additional computational burden and imaging time, the selection of line source length should be balanced between effectively illuminating the imaged object and preventing direct transportation to CCD without interacting with mouse tissues. Considering this, 4cm long line source is selected in our experiments.

3.2. Analysis of reconstructions of simulated data for L-FDOT

Verifications of the key findings obtained in the SVA study are depicted in Fig. 4 , where the reconstructed images of simulated data with different projections and different line source lengths are shown. The optimal values of other experimental parameters were used in these reconstructions. The tubes are more clearly resolved using 18 projections than using 9 and 4 projections. Further increasing the projections number to 36 yields small improvements compared with 18 projections. The results are consistent with the optimal projections number obtained using SVA in study A1. For 3cm long line source, the image contrasts of the tubes at 5.25cm height slice are much worse than those at the other two height slices. The non-uniform image contrasts are consistent with the lower optimal vertical FOV and the smaller NSAVT in studies A2 and A6. In contrast, nearly consistent image contrasts of the tubes inside whole body are demonstrated for 4cm long line source.

Fig. 4 Reconstructions of simulated data with 4-36 projections and 3-4 cm long line sources for L-FDOT. The images are at slice X = 0. The black squares represent the actual tubes. For each reconstruction, we used detectors distributed over 1.8cmx5.8cm FOV with 0.2cm spacing, and a reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing.

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3.3. Analysis of reconstructions of experimental data for L-FDOT

The reconstructed images of experimental data with different number of projections are shown in Fig. 5 . The maximum value of the reconstructed image for 36 projections was normalized to 1, resulting in a calibration factor used in other cases with different projections numbers. The image for 4 projections is underperforming, where the tubes are poorly located and many artifacts exist. From the four images, the impact of the projections number on the image qualities obtained by SVA is further confirmed.

Fig. 5 Reconstructions of experimental data with 4-36 projections for L-FDOT. The cross section images are at the height slices of the tubes centers, which are depicted using the red circles in the 3D view. The red curves on the cross images represent the phantom boundary, and the black circles represent the actual tubes. For each reconstruction, we used detectors distributed over 2.2cmx5.4cm FOV with 0.2cm spacing, and a reconstruction mesh over 3.0cmx3.0cmx5.4cm with 0.1cm spacing.

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3.4. Comparison experiments between L-FDOT and P-FDOT

3.4.1. Study B1

Detector points with ultra low excitation signals, which were less than 2% of the maximum signal intensity, weren’t considered in reconstructions. Then, the maximum detector vertical FOV is determined as 2.8cm by the excitation signal intensity threshold. Figure 6(a) plots the NSVAT with respect to the detector vertical FOV. Up to the maximum vertical FOV, increasing the vertical FOV linearly increases NSVAT. Therefore, similar as in study A2, the optimal vertical FOV is also dominated by the signal intensity threshold and therefore determined as 2.8cm.

Fig. 6 Singular value analysis of specific experimental parameters for P-FDOT. We used 24 projections, 1.8cm detector horizontal FOV, 0.2cm detector spacing for each cycle. The reconstruction mesh was inside the imaged object and 5.8cm height range with 0.1cm spacing. (a) Singular value analysis of the detector vertical FOV for P-FDOT with one cycle of projections. The optimal vertical FOV (2.8cm) was used in the following singular value studies (d-f). (b) Illustration of the symmetric distributions of point sources along Z axis for P-FDOT with two cycles of projections. (c) Illustration of the symmetric distributions of point sources along Z axis for P-FDOT with three cycles of projections. (d) Singular value analysis of the point sources distribtution along Z axis for P-FDOT with two cycles of projections. (e) Singular value analysis of the point sources distribtution along Z axis for P-FDOT with three cycles of projections. (f) Singular value analysis of the number of cycles when using the optimal point sources distributions obtained in (d) and (e). The NSVAT for L-FDOT with 4cm long line source is also plotted as a line in (f), where 24 projections, detectors over 1.8cmx5.8cm with 0.2cm spacing, and a reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing were used.

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3.4.2. Study B2

The optimal distribution of point sources along Z axis is firstly analyzed for P-FDOT with two or three cycles of projections. Compared with asymmetric distribution, symmetric distribution of point sources along Z axis will lead to more uniform image qualities inside the whole body. The symmetric distribution of point sources along Z axis is depicted in Figs. 6(b) and 6(c), where p_dis represents the distance between neighboring point sources along Z axis. The optimal p_dis for two cycles of projections is 3cm, which is determined by the maximum NSVAT in Fig. 6(d). Similarly, it can be seen from Fig. 6(e) that the optimal p_dis for three cycles is 1.5cm. As shown in Fig. 6(f), we summarized the optimal NSVAT for P-FDOT with one to three cycles of projections and compared them with that for L-FDOT with 4cm long line source. It can be inferred that (i) two cycles of projections may be enough for P-FDOT to achieve comparable whole body performances as L-FDOT, and (ii) three cycles of projections are needed for P-FDOT to achieve better whole body performances than L-FDOT.

3.4.3. Study B3. Analysis of reconstructions of simulated data for P-FDOT

Figure 7 depicts the reconstructed images for P-FDOT with one to four cycles of projections. The point sources were placed at 3cm height for P-FDOT with one cycle of projections. The point sources were distributed following the optimal distributions in study B2 for P-FDOT with two (z = 1.5cm, 4.5cm) and three (z = 1.5cm, 3cm, 4.5cm) cycles of projections. Reconstruction was also performed for P-FDOT with 4 cycles of projections to further investigate the influence of cycles, where the point sources were placed at z = 1cm, 2cm, 3cm, 4cm height slices respectively. When comparing the images to that in L-FDOT with 4cm long line source in Fig. 4, it gives further understanding than in study B2. That is, at least three cycles of projections are needed for P-FDOT to reach comparable whole body performances as L-FDOT. P-FDOT with one cycle of projections has poor whole body performance, where the tubes far away the point illumination height slice were reconstructed inaccurately. However, tubes near the point illumination slice are more clearly resolved in P-FDOT with one cycle of projections than in L-FDOT. It confirms that P-FDOT with one cycle of projections is only suitable for observing local biological activities. Although the NSVAT of P-FDOT with two cycles of projections is only slightly smaller than that of L-FDOT, poor image contrasts are observed for the tubes at 3cm height slice. It is because better image qualities will be expected near the two point sources illumination height slices than those of L-FDOT.

Fig. 7 Reconstructions of simulated data for P-FDOT with 1-4 cyles of projections. The images are at slice X = 0. The black squares represent the actual tubes. 24 projections, detectors distributed over 1.8cmx2.8cm FOV with 0.2cm spacing were used for each cycle. The reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing was used for each reconstruction.

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3.5. Experimental sets for beam-forming strategies

3.5.1. Study C1

Figure 8(a) depicts the optimal vertical FOV with respect to different area widths. Similar as in study A2, the optimal vertical FOV is dominated by the excitation signal intensity threshold. The NSVAT for A-FDOT with different area widths are close to that for L-FDOT, as shown in Fig. 8(b). We can infer that increase in area width won’t lead to better whole body performance. In contrast, increase in area width will increase the possibility of direct light transportation to the CCD, especially for irregular mouse shapes. Therefore, A-FDOT doesn’t have advantages over L-FDOT.

Fig. 8 Singular value analysis of the specific experimental parameters for A-FDOT and MP-FDOT. 24 projections, detectors distributed over 1.8cm horizontal FOV with 0.2cm spacing, and reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing were used. (a) Plot of the detector vertical FOV versus the area width for A-FDOT with 4cm area length. (b) Singular value analysis of the effects of the area width for A-FDOT with 4cm area length. The optimal detector vertical FOV in (a) was used. (c) Singular value analysis of the effects of the points density for MP-FDOT with 4cm points distribution range. 5.8cm vertical FOV was used. The NSVAT for L-FDOT with 4cm long line source is also plotted as a line in (b)-(c), where 24 projections, detectors over 1.8cmx5.8cm FOV with 0.2cm spacing, and a reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing were used.

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3.5.2. Study C2

Figure 8(c) plots the NSVAT with respect to the points number inside height ranges from 1cm to 5cm. The NSVAT for MP-FDOT is slightly larger than that for L-FDOT. The NSVAT decreases as the points number increases. When the points number reaches 8, the NSVAT is nearly the same as that for L-FDOT. For small points number such as 3, although large NSVAT is demonstrated, there are regions up to 1.0cm way from each illumination height slices. Possible poor image qualities will be expected in these regions. This could be verified in study C3.

3.5.3. Study C3. Analysis of reconstructions of simulated data for A-FDOT and MP-FDOT

Figure 9 depicts the reconstructed images for A-FDOT and MP-FDOT. The reconstructed images for MP-FDOT with 3 points demonstrated relatively poor image contrast for tubes at 1.7cm height slice, which confirms the expectation in study C2. Therefore, although slightly larger NSVAT is demonstrated for 3 points than those for 6 or 8 points, the whole body performance for 3 points is worse than those for 6 or 8 points. The seemly uncommon phenomenon is because of the non-uniform image qualities inside different height regions. That is, the relative poor image qualities in some height regions compensate the relative good image qualities in some other height regions.

Fig. 9 Reconstructions of simulated data for MP-FDOT with different points densities, and A-FDOT with different area widths. The images are at slice X = 0. The black squares represent the actual tubes. 24 projections, detectors distributed over 1.8cm horizontal FOV with 0.2cm spacing, and a reconstruction mesh inside the imaged object and 5.8cm height range with 0.1cm spacing were used for each reconstruction. For MP-FDOT with 4cm points distribution range, 5.8cm vertical FOV was used. For A-FDOT with 4cm area length, the optimal vertical FOV in Fig. 8(a) were used.

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For A-FDOT, the reconstructed images further confirm that the increase in area width doesn’t yield improvements in image qualities.

4. Discussion and conclusion

As challenges remain in real time FDOT imaging fast biological activities through whole body, it is important to design imaging systems with real time abilities and whole body field of view. It is also important to optimize the experimental parameters including system implementation, data acquisition and utilization to obtain optimal image qualities while maintaining temporal resolution and computational efficiency. In this work, we proposed and analyzed three non-contact, full angle imaging systems with different beam-forming illuminations. We studied several of the most relevant parameters of the imaging systems. We also quantitatively compared the performances between the proposed BF-FDOT systems and the conventional P-FDOT systems.

For the L-FDOT system, the whole body performance relies on the line source length. The 4cm long line source is selected to compromise between effectively illuminating the imaged object and preventing direct excitation light transportation to the CCD. The temporal resolution relies on the projections number. 18 projections are considered optimal. Further increasing the projections number doesn’t yield significant improvements. It is because correlation is expected between adjacent measurements and projections due to the diffuse light transportation. The optimal spatial spacing is 0.2cm for detector spacing and 0.1cm for mesh spacing. Reducing the spatial sampling will lead to noticeable deteriorations in image qualities, while increasing the spatial sampling won’t yield significant improvements and result in nonlinearly increased computational burden. That’s why all the reconstructions in this article used 0.2cm detector spacing and 0.1cm mesh spacing. The optimal detector horizontal field of view of L-FDOT is close to the width of the object silhouette. The optimal detector vertical field of view of L-FDOT depends on the line source length used. It is dominated by the excitation light signal intensity threshold and is close to the length of the imaged object for line source length longer than 4cm.

Comparison experiments were made between L-FDOT and P-FDOT. L-FDOT demonstrates much better whole body image qualities than P-FDOT when temporal resolutions are the same. At least three cycles of projections are needed for P-FDOT to achieve comparable whole body performance as L-FDOT. Therefore, L-FDOT is more suitable than P-FDOT for observing fast biological activities through whole body. Of course, for cases where temporal resolution is not important, P-FDOT with sufficient cycles of projections are more appropriate, since more source-detector combination modes will be obtained and will lead to some extent improvements in spatial performances.

Specific characteristics of the A-FDOT system were analyzed. For A-FDOT with 24 projections, no better image qualities are observed than for L-FDOT with 24 projections. In addition, area illumination is more complex to be implemented than line illumination. At the same time, the probability of direct excitation light transportation to CCD is increased for wider area illumination, especially for irregular mouse shapes. Therefore, A-FDOT doesn’t have advantages over L-FDOT.

Specific characteristics of MP-FDOT were also analyzed. Results demonstrate that MP-FDOT with 4 to 6 points distributed inside 4cm height ranges gives comparable whole body performances as L-FDOT with 4cm long line source.

Singular value analysis reveals the bulk characteristics that lead to generic optimal parameters, which makes it easy to compare the performances among different imaging systems. It answers well most of the questions in this work. However, it still has some limitations in quantifying the consistency of the whole body performance, such as in studies B2 and C2. When highly non-uniform image qualities exist inside the whole body, the relative good image qualities in some height regions may compensate the poor image qualities in some other height regions. However, the differences cannot be noticed from the sum characteristics obtained by SVA. Although SVA has these limitations, it is still an essential tool in studies B2 and C2, where it guided the simulation experiments and assisted making correct conclusions.

In this article, we compared different experiments by counting the number of singular values that are larger than a certain factor times the largest singular values, rather than counting the number of singular values larger than a fixed threshold. Because all singular values can be multiplied by any factor by changing the integration time or the incident light intensity by the same factor, comparing relative sizes of singular values essentially amounts to the assumption that all experiments have the same signal-to-noise ratio with respect to the dominant mode in the singular value decomposition. While this normalization prevents realistic comparisons between line and point sources, for example, it is clear that such comparisons would also be affected innumerable other factors. There is a clear need for more systematic research in this direction, though this is beyond the scope of this article.

Overall, we have shown that full angle imaging systems with beam-forming illumination strategies offer significant improvements in fast whole body imaging. The analysis and optimization of experimental parameters give us a guideline in selecting and developing different BF-FDOT systems. It also tells us how to optimize the image qualities within acceptable computational burden and imaging time. Of course, improved image qualities are expected by improving the computational efficiency of FDOT inversion. Future works will be focused on using the analysis methods in this article to develop possible better FDOT systems.

Acknowledgments

The authors thank Dr. Tobias Lasser for useful discussions. The authors thank the reviewers and the editor for their helpful suggestions. This work was supported by the National Nature Science Foundation of China (No. 30670577, 60831003), the Tsinghua-Yue-Yuen Medical Science Foundation, the National Basic Research Program of China (No. 2006CB705700), the National High-Tech Research and Development Program of China (No. 2006AA020803), and the China Postdoctoral Science Foundation Funded Project.

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cases	projections number	detector vertical FOV	detector horizontal FOV	detector spacing	mesh spacing	line source length
A1	optimization	5.8cm	1.8cm	0.2cm	0.15cm/0.1cm	4cm
A2	24	optimization	1.8cm	0.2cm	0.1cm	3cm/4cm/5cm
A3	24	4.8cm/5.8cm/5.8cm	optimization	0.2cm	0.1cm	3cm/4cm/5cm
A4	24	5.8cm	1.8cm	optimization	0.15cm/0.1cm	4cm
A5	24	5.8cm	1.8cm	0.2cm/0.15cm/0.1cm	optimization	4cm
A6	24	optimal	1.8cm	0.2cm	0.1cm	optimization

Abstract

1. Introduction

2. Methods

2.1. Experimental setup

2.2. Forward and inverse problems

2.3. Singular value analysis and cost functions

2.4. Experimental sets for L-FDOT

2.5. Reconstructions of simulated data for L-FDOT

2.6. Reconstructions of experimental data for L-FDOT

2.7. Comparison experiments between L-FDOT and P-FDOT

2.7.1. B1. The optimal detector vertical FOV for P-FDOT

2.7.2. B2. SVA analysis of the number of cycles for P-FDOT

2.7.3. B3. Reconstructions of simulated data for P-FDOT

2.8. Experimental sets for beam-forming strategies

2.8.1. C1. The influence of area width

2.8.2. C2. The influence of multiple points’ density

2.8.3. C3. Reconstructions of simulated data for A-FDOT and MP-FDOT

3. Results

3.1. Singular value analysis for L-FDOT

3.1.1. Study A1

3.1.2. Study A2

3.1.3. Study A3

3.1.4. Study A4

3.1.5. Study A5

3.1.6. Study A6

3.2. Analysis of reconstructions of simulated data for L-FDOT

3.3. Analysis of reconstructions of experimental data for L-FDOT

3.4. Comparison experiments between L-FDOT and P-FDOT

3.4.1. Study B1

3.4.2. Study B2

3.4.3. Study B3. Analysis of reconstructions of simulated data for P-FDOT

3.5. Experimental sets for beam-forming strategies

3.5.1. Study C1

3.5.2. Study C2

3.5.3. Study C3. Analysis of reconstructions of simulated data for A-FDOT and MP-FDOT

4. Discussion and conclusion

Acknowledgments

References and links

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Figures (9)

Tables (1)

Equations (5)

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