In this section, we assume a point-like change in absorption, simulate the corresponding measurement vector, and reconstruct the 3D map of the absorption changes. This gives us the imaging point spread function of our reconstruction algorithm and enables us to assess the performance of the method and compare it to a CW reconstruction scheme.
3.1 Reconstruction examples
shows examples of reconstructions for a point-like inclusion located at the lateral position 1, between source and detector, as defined in , and with a depth z varying between 0.5 cm and 3 cm by steps of 0.5 cm. Both CW and TD reconstructions are presented. The rendered volume shows the contour of 80% of the maximum in the reconstructed absorption change. The following parameters were used: 5 gates starting at delays 0.6, 1.1, 1.6, 2.1, and 2.6 ns with a width wGate = 300 ps (gates 2 to 6 in ); β = 20; α = 10−3; signal to noise ratio = 100 at the peak of the TPSF.
CW and TD reconstructions of a point-like inclusion located at position 1 and at a depth z varying between 0.5 cm and 3 cm. The volumes show the contour of 80% of the maximum reconstructed change in absorption.
The following general observations can be made: the CW data always reconstruct the inclusion at the same depth. If a traditional Tikhonov regularization is used instead (i.e. if A
is used in place of B
), the CW reconstruction is pulled towards the surface (data not shown), where the sensitivity is maximum [35
]. The effect of the spatially varying regularization matrix L
is to force the reconstruction deeper under the surface. However, the reconstructed depth is unchanged with different actual depths for the CW data.
On the contrary, the TD method reconstructs the inclusion deeper as its actual depth increases. For the true inclusion at a depth of 1.5 and 2 cm, the reconstructed depth is actually slightly over-estimated, which results from the effect of the regularization matrix L. As the true inclusion gets deeper, the reconstructed depth becomes under-estimated (see inclusion at true depth 3 cm).
We also observe that the reconstructed volume at 80% of the maximum contrast is smaller for TD than for CW reconstructions, showing improved lateral resolution for this location of the inclusion.
More quantitative and systematic assessment of the effect of different parameters and of the improvement of TD over CW will be investigated in the following paragraphs.
3.2 Performance assessment
The reconstruction performances were assessed by a number of parameters. The location of the center of mass (COM) of the reconstructed inclusion was calculated by taking into account all voxels with a contrast above 80% of the maximum contrast in absorption: rCOM = (Σi, Vox≥80% Riri)/(Σi, Vox≥80% Ri), where ri and Ri are respectively the position and absorption contrast of the ith voxel. We define the localization error as the distance between the true inclusion and the COM, both in depth and laterally. We call lateral resolution the contrast-weighted sum of the lateral distances to the COM, over all voxels with a contrast above 80% of the maximum contrast: Res = (Σi, Vox≥80% Ri|ρi − ρCOM|)/(Σi, Vox≥80% Ri), where ρi and ρCOM are the lateral positions of the ith voxel and the COM respectively.
3.3 Optimal number of gates
We studied the influence of the number of gates included in the reconstruction. shows the evolution of the reconstructed depth (depth of the COM) as a function of the true depth of an inclusion located at lateral position 1, for different number of gates included (starting from gate 1 on ). The reconstruction improves, more strikingly for deep inclusions, as more late gates are included, up to 6 gates, after which the reconstruction does not improve anymore as we include more noisy data. The first gate only brings minor improvement for superficial inclusion (data not shown), and does not contribute to the data reconstruction for deep inclusions. We tried other combinations (data not shown), and found that the best one with our parameters was 5 gates every 500 ps from 0.6 ns to 2.6 ns.
Fig. 4 Reconstructed depth of the COM as a function of the inclusion true depth, for a point-like inclusion in position 1, for (a) different delay gate combinations (β = 20), and (b) different threshold coefficients in the spatially varying regularization (more ...)
With these parameters, the inclusion can be reconstructed with a depth error under 15 % down to approximately 2.5 cm. The reconstructed depths of deeper inclusions continue to increase, but with larger errors, e.g. an inclusion at 3 cm is reconstructed at 2.3 cm (an underestimation of almost 25%).
3.4 Influence of the spatially varying regularization
By penalizing more the voxels with higher sensitivity, the L
matrix enables us to reconstruct the inclusion better than a simple Tikhonov regularization [34
]. However this matrix has to be thresholded so that regions far from a source-detector pair do not inappropriately receive larger weighting. The effect of this threshold is presented in , where the depth of the reconstructed COM is plotted vs. the true depth of the inclusion, for CW data and TD data with the 5-gate combination discussed above, and for a β
factor of 10, 20, 50 and 100. With CW data, the absorption is reconstructed at a constant depth whatever its true depth, and this reconstructed depth simply increases as β
increases. For TD data, a smaller threshold (large β
) enables better reconstructions for deep inclusions (z
> 2.5 cm), but also leads to an overestimated reconstructed depth for medium-deep inclusions (z
around 1.5 to 2.5 cm). Moreover, for inclusions very close to the surface (under 1 cm) located just underneath a source or detector as in position 3 from , a large β
also leads to strong artifacts (data not shown). A β
= 20 was used in the following simulations, enabling a good compromise for reconstruction of different depths between 1 and 3 cm.
3.5 Influence of background optical properties
The evaluation of the baseline optical properties of a medium is subject to uncertainty [36
]. Therefore, we tested the influence of an error in the background optical properties on the reconstruction. We do not present data for this study, but enunciate general results we observed. An overestimation of the scattering coefficient (μ′s,recon
) by 20% led to an inclusion being reconstructed closer to the surface by approximately 1 to 3 mm, and vice versa for an underestimation of the scattering coefficient. We did not observe an effect of an over- or under-estimated absorption coefficient (by up to 50%) on the reconstructed depth. For these results, the true depth of the inclusion was varied between 0.5 cm and 3 cm.
3.6 Variation of depth localization error with lateral position
One major advantage of TD data is the depth information provided by the time of flight of photons. In , we showed the evolution of the reconstructed depth as a function of the true depth for one particular lateral position. In , we present the evolution of the reconstructed depth as a function of the lateral position of the inclusion, for a 1 cm and 2 cm deep inclusion. In both cases, the depth error is small (within 20% at 1 cm, and about 5% away from the edges of the medium at 2 cm).
Depth of the COM for (a) a 1 cm deep and (b) a 2 cm deep inclusion reconstructed with TD data (5 gates) as a function of the inclusion lateral position.
For the inclusion located at 2cm below the surface, it also worth noting that the reconstructed depth varies very little with the lateral position of the inclusion. This means that the reconstruction method has no “blind zone”, and the depth reconstruction performance remains good for any position of the inclusion. Note that we do not compare the depth error with that obtained by CW measurements as no depth information is provided without overlapping or multi-distance CW measurement.
3.7 Lateral localization and resolution
In this section, we study the performance of the reconstruction method in terms of lateral localization and lateral resolution, and compare it with a CW reconstruction method. and show the evolution of, respectively, the lateral localization error and the lateral resolution, for CW (left) and TD (right) reconstructions, at two different depths of the inclusion (1 cm, top, and 2 cm, bottom) as a function of the inclusion lateral position.
Lateral error (cm) for a 1 cm deep inclusion (top) and a 2 cm deep inclusion (bottom) reconstructed with CW (left) and TD (right) data, as a function of the inclusion lateral position.
Lateral resolution (cm) for a 1cm deep inclusion (top) and a 2 cm deep inclusion (bottom) reconstructed with CW (left) and TD (right) data, as a function of the inclusion lateral position.
For both depths, TD shows reduced error in the lateral localization and better lateral resolution. Importantly we also note that both the lateral resolution and the lateral localization for TD reconstructions are more uniform with the lateral position of the inclusion relative to the probe than with a CW reconstruction.
These improvements of TD over CW for the lateral localization and resolution can be explained intuitively based on the sensitivity profiles presented in . For later delay gates, the sensitivity profile of a given source-detector probes deeper into the medium giving us depth resolution, but also probes a larger region laterally providing additional information to improve lateral localization and resolution.
3.8 Contrast to noise ratio improvement
We have shown in a previous study [8
] that CW and TD systems yield similar contrast to noise ratio (CNR) for typical depths of cerebral activation (1 to 2 cm) and source-detector separations (2 to 3 cm) used in functional brain imaging, with CW even giving slightly better CNR. The intuitive explanation of this result is the following: even though TD data yield better contrast to deep inclusion by selecting photons which have traveled deep inside the medium, these measurements are also impeded by a much higher noise due to the low level of light at late delays. However the CNR of an image has to be evaluated in regards to other metrics of the image, in particular its resolution.
In this section, we varied the regularization parameter α between 10−4 and 10, both for CW and TD reconstructions, and studied the evolution of CNR and lateral resolution. is a parametric curve of the image CNR as a function of the lateral resolution. The true inclusion is located at position 1, and at a depth z = 1.5 cm. We compute the CNR as the average contrast to noise ratio for all voxels of contrast above 80% of the maximum contrast. The image noise was obtained by propagation of the measurement noise by: σx2= pAinvσy2pAinvT, where σx2 is the image covariance matrix.
Fig. 8 CNR versus lateral resolution for a regularization parameter α varying between 10−4 and 10. CNR is given per unit volume (cm3) and unit change in absorption (cm−1) of the inclusion. Regularization parameters of 1, 10−2 (more ...)
The plot illustrates the trade-off between CNR and resolution: as the regularization is increased, the CNR of the reconstructed inclusion increases, but is counter-balanced by a worsening of the lateral resolution. We observe that for an identical CNR of the image, the lateral resolution is improved by TD reconstructions compared to CW. Similarly, at identical lateral resolution, TD reconstruction enables a much higher CNR of the image.