2.1.

Reconstruction Method

The conventional image reconstruction method involves finding the optical properties of the medium from a set of measurements made at the surface. The optical properties are then combined to calculate the chromophore concentrations at wavelength λ using the following equation:

The multispectral method involves reconstructing chromophore concentrations and scatter parameters using all measurements at all wavelengths simultaneously.¹⁶ The problem is nonlinear but can be linearized and reduced to the so-called difference method, based on the assumption that the true chromophore concentrations are close to an initial guess. For CW measurements, only intensity $A$ is measured, hence changes in the measured data $\Delta A$ at $n$ measurement wavelengths are related to changes in the chromophore concentrations $\Delta c_i$ and scatter amplitude $\Delta a$ through matrix $\tilde{J}$ ,

Chromophore concentrations and scatter parameters can be obtained by solving an ill-posed inverse problem at all wavelengths. In the work presented here, this is aided using the zero-order Tikhonov regularization,

2.2.

Identification of the Optimal Wavelengths in Optical Topography

3.1.

Adult Head Model

3.1.2.

Three-layer model

Figure 1 shows the second adult head model, which is a slab with the same dimensions as the homogeneous model, consisting of three layers that imitate the scalp, skull, and intracerebral tissue. Scalp is represented by a $4\text{-}\mathrm{mm}$ thickness layer, and the skull is $7\mathrm{mm}$ thick. Absorption and scattering spectra of skull are described in Firbank ³⁶

Fig. 1

Three-layer adult head model and probe placement.

Skin absorption spectrum

Three main chromophores were considered for scalp: hemoglobin, water, and melanin. Their estimated concentrations in skin are summarized in Table 1. The melanin absorption coefficient can be approximated by the expression³⁰

Eq. 12

$${\mu }_{a,\text{melanin}}=(6.6\times {10}^{10}){\lambda }^{-3.33},$$

in ${\mathrm{mm}}^{-1}$ , where $\lambda$ is the wavelength represented in units of nanometers. Considering the contributions from all skin chromophores, the total absorption coefficient of the skin layer can be represented as

Eq. 13

$${\mu }_{a,\text{skin}}\left(\lambda \right)={ϵ}_{\mathrm{Hb}{\mathrm{O}}_2}\left[\mathrm{HbT}\right]\mathrm{S}{\mathrm{O}}_2+{ϵ}_{\mathrm{HHb}}\left[\mathrm{HbT}\right](1-\mathrm{S}{\mathrm{O}}_2)+{\mu }_{a,{\mathrm{H}}_2\mathrm{O}}W+{\mu }_{a,\text{melanin}}M,$$

where $M$ is the fraction of melanin content. It was not necessary to consider a background absorption for the skin layer, because the absorption coefficient values already show a good agreement with the values presented by Simpson ³⁷

Skin scattering spectrum

The wavelength dependence of the reduced scattering coefficient of skin has been modeled by Bashkatov ³⁸ as a combination of Mie and Rayleigh scattering given by

Eq. 14

$${\mu }_{\mathrm{s},\text{skin}}^{\prime }=7.37{\lambda }^{-0.22}+1.1\times {10}^{11}{\lambda }^{-4},$$

in ${\mathrm{mm}}^{-1}$ , where $\lambda$ is the wavelength in nm. The first term corresponds to Mie scattering due to large collagen fibers and the second to Rayleigh scattering mainly due to small collagen fibrils.

3.2.

Similarity Measures

The sum of squared differences (SSD) can be used to measure the similarity between two sensitivity profiles. For PMDFs $X$ and $Y$ with voxels $i$ , the SSD is defined as

3.3.

Optimal Wavelengths

Sensitivity profile maps (PMDFs) were obtained for the wavelength range of $650–990\mathrm{nm}$ in steps of $10\mathrm{nm}$ , making a total of 35 wavelengths. Initially, the PMDFs were combined in sets of three wavelengths $\left(C_3^{35}\right)$ , making a total of 6545 wavelength sets. Thereafter, wavelength sets of four wavelengths were investigated, which results in 52,360 sets. For a medium with $i$ absorption chromophores, at least $n=i+1$ wavelengths are required when the scattering parameter $a$ is allowed to vary.¹⁶ Therefore, in order to find the solution for both absorption and scattering coefficients for a medium with two chromophores, it is necessary to employ at least three wavelengths. It is common for optical topography systems to acquire data at just two wavelengths. Therefore, all possible wavelength pairs are compared (total of 595), to provide an insight into what can be achieved using only two measurement wavelengths.

The aim of this study is to determine which wavelength set gives the best combination of: (i) good separation between absorption and scattering, (ii) good separation between $\left[\mathrm{Hb}{\mathrm{O}}_2\right]$ and [HHb], and (iii) excellent overlap between sensitivity profiles. Quantitatively, we aim to find the wavelength sets that maximize the residual norm $R$ [Eq. 7], minimize the condition number $\mathbf{k}$ of the matrix ${\epsilon }_i\left(\lambda \right)$ [Eq. 9], and minimize the SSD [Eq. 15]. The matrix $E$ in Eq. 8 for three wavelengths is reduced to

3.4.

PMDF Comparison

The PMDFs for wavelengths from $650\text{to}990\mathrm{nm}$ are compared by quantifying the differences in their sensitivity in terms of penetration depth, volume, and full width at half maximum (FWHM), in order to determine if these differences are significant. Absolute sensitivity values are used (because the absorption PMDF has negative sensitivity), and the threshold is set to 0.01. An analysis is performed of how these differences affect the measurements and image reconstructions.

3.5.

Image Reconstruction

Simulations were performed to compare the cross-talk effects in images reconstructed from measurements at wavelengths selected using our method against those from measurements at wavelengths selected using Corlu ’s method.¹⁶ The simulated medium consisted of a slab with 59,202 nodes, resulting in 38,572 elements, and dimensions $85\times 85\times 30\mathrm{mm}$ . An array with eight detectors and eight sources (described previously by Correia ³⁹) is centrally placed at $z=0\mathrm{mm}$ . However, here only source-detector separations from $14\text{to}41\mathrm{mm}$ were considered. TOAST was used to generate CW model data at each wavelength, to which 5% random Gaussian noise was added. The background had brain optical properties, which were described in Section 3.1.1. Hence, the deoxy-hemoglobin concentration [HHb] is $24\mu \mathrm{M}$ , the oxy-hemoglobin concentration $\left[\mathrm{Hb}{\mathrm{O}}_2\right]$ is $56\mu \mathrm{M}$ , and the scattering amplitude $a$ is $32{\mathrm{mm}}^{-1}{\mathrm{nm}}^{-1}$ . Three perturbations with dimensions $10.6\times 10.6\times 10.6\mathrm{mm}$ were placed within the medium, as shown in Fig. 2 . The center of each perturbation was at $11.7\mathrm{mm}$ below the imaging surface. A single parameter was changed for each perturbation, which consisted of either an increase in [HHb] of $1\mu \mathrm{M}$ , in $\left[\mathrm{Hb}{\mathrm{O}}_2\right]$ of $4\mu \mathrm{M}$ , or in $a$ of $1{\mathrm{mm}}^{-1}{\mathrm{nm}}^{-1}$ . These perturbations are sufficiently small to ensure that the concentrations and scattering amplitude can be retrieved by linear reconstruction, and are roughly consistent with the magnitude of cerebral hemodynamic changes. Images were reconstructed using the multispectral method on a $24\times 24\times 9$ regular pixel basis.

Fig. 2

Cross section $xy$ at $z=11.7\mathrm{mm}$ of the medium used in the simulation.

4.1.

Optimal Wavelengths

For simplicity, the absorption and diffusion sensitivity profiles are dealt with separately. The first results are shown in Fig. 3 for the homogeneous model, considering absorption sensitivity profiles only and three wavelength sets. Note that the SSD values in Fig. 3 represent the average for each wavelength set. Because of the large number of points, further analysis of the wavelength sets was limited to those for which $R⩾0.2$ , $\mathbf{k}⩽100$ , and (because the SSD for two similar PMDFs is $\sim 0.001$ ) $\mathrm{SSD}⩽0.002$ (dark circles in Fig. 3). Each criterion has a different order of magnitude; hence, the results contained in this new range were normalized. The distance between the criteria values of a certain set of wavelengths in this range and the point $(\mathrm{SSD},\mathbf{k},R)=(0,0,1)$ was also calculated, which is the point with the smallest SSD, smallest condition number criteria, and highest residual. The wavelength distribution of the ten sets of three wavelengths with the shortest distance are displayed in the histogram in Fig. 4 . Histograms are obtained by counting the number of times a certain wavelength occurs, which is done separately for the three wavelengths, and normalized to the most frequently occurring wavelength. For comparison, Fig. 5 shows the histogram for the ten wavelength sets within the limit $R⩾0.2$ and $\mathbf{k}⩽100$ , whose distance is closest to the point $(\mathbf{k},R)=(0,1)$ . The means and standard deviations $\sigma$ of each wavelength distribution were calculated in order to find the optimal wavelengths and measure their spread. A $5\text{-}\mathrm{nm}$ uncertainty error is combined with $\sigma$ $({\sigma }_{\text{total}}=\sqrt{{\sigma }^2+5^2})$ , due to the bin size. Using the three-wavelength selection criteria, the wavelengths are $680\pm 5$ , $726\pm 7$ , and $835\pm 17\mathrm{nm}$ . When only the high residual and low condition number criteria are used the optimal wavelengths are $650\pm 5$ , $724\pm 7$ , and $915\pm 17\mathrm{nm}$ . The results suggest that when the new PMDF overlap condition is included the smallest wavelength becomes larger, and the largest wavelength becomes smaller.

Fig. 3

Plot of the sum of squared differences SSD (in millimeters), condition number $\mathbf{k}$ and residual norm $R$ , where each point represents a set of three wavelengths.

Fig. 4

Histogram of the optimal wavelength distributions that satisfy the three wavelength selection conditions, for the homogeneous model, three wavelengths, and two chromophores ( $\mathrm{Hb}{\mathrm{O}}_2$ and HHb).

Fig. 5

Histogram of the optimal wavelength distributions that satisfy the high residual and low condition number criteria, for the homogeneous model, three wavelengths, and two chromophores ( $\mathrm{Hb}{\mathrm{O}}_2$ and HHb).

The optimal four wavelengths that satisfy the three criteria are $680\pm 5$ , $715\pm 14$ , $733\pm 7$ , and $828\pm 9\mathrm{nm}$ . Adding a fourth wavelength broadened the central wavelength range and shifted the highest wavelength to a smaller value. The wavelength set obtained without including the third condition is $650\pm 5$ , $715\pm 7$ , $727\pm 7$ , and $919\pm 13\mathrm{nm}$ . The differences between these results and those presented by Corlu ¹⁶ are inevitably due to differences in the wavelength range and spacing, chromophore extinction coefficients, and criteria limits imposed.

The SSD for diffusion PMDFs is always slightly smaller than for absorption PMDFs. This is probably because scattering does not have a strong wavelength dependence. Thus, the optimal wavelength sets found using the absorption SSDs also return small diffusion SSDs.

The histograms for the three-layer adult head model for three and four wavelengths are shown in Figs. 6 and 7 , respectively. For the three wavelength distributions, the optimal wavelength values are $680\pm 5$ , $725\pm 10$ , and $876\pm 12\mathrm{nm}$ , and for four wavelengths these values are $685\pm 7$ , $719\pm 9$ , $731\pm 8$ , and $873\pm 9\mathrm{nm}$ . The main difference between the optimal wavelengths found for the homogeneous model and the three-layer model occurs for the largest wavelength, which becomes larger for the latter. The other wavelengths are approximately the same and, similar to the homogeneous case, adding a fourth wavelength resulted in the merger of the two central wavelengths (i.e., the mean value is approximately the same but the range is larger). Also for this model, the diffusion SSDs are smaller than the corresponding values for absorption; hence, the previous results also apply for diffusion-sensitivity profiles.

Fig. 6

Histogram of the optimal wavelength distributions that satisfy the three wavelength selection conditions, for the three-layer model, three wavelengths and two chromophores ( $\mathrm{Hb}{\mathrm{O}}_2$ and HHb).

Fig. 7

Histogram of the optimal wavelength distributions that satisfy the three-wavelength selection conditions, for the three model, four wavelengths, and two chromophores ( $\mathrm{Hb}{\mathrm{O}}_2$ and HHb).

The wavelength distributions obtained using our method were similar when the scattering power $b$ was varied from 0 to 2.8, whereas for the two-criteria method similar wavelength distributions were obtained for a range of $b$ from 0 to 1.2.

4.2.

PMDF Comparison

4.2.1.

Volume

The relative volume was calculated for the absorption PMDFs $\left({\mathrm{PMDF}}_{{\mu }_a}\right)$ and diffusion PMDFs $\left({\mathrm{PMDF}}_{\kappa }\right)$ , for both head models. Figure 8 shows the relative volumes for the different wavelengths and the value of $N_{650}$ for each case. The volume of the ${\mathrm{PMDF}}_{{\mu }_a}$ is larger than that of the ${\mathrm{PMDF}}_{\kappa }$ . For the homogeneous model, the ${\mathrm{PMDF}}_{{\mu }_a}$ volume resembles an inverted ${\mu }_a$ spectrum of brain tissue, because for large ${\mu }_a$ the PMDF volume becomes smaller and for small ${\mu }_a$ the volume becomes larger. For the ${\mathrm{PMDF}}_{\kappa }$ , the volume decreases as expected, given that the volume decreases with decreasing ${\mu }_s^{\prime }$ . However, the volume for ${\mathrm{PMDF}}_{\kappa }$ does not vary as much as for ${\mathrm{PMDF}}_{{\mu }_a}$ . The relative volumes for the layered model are similar to those obtained for the homogeneous case.

Fig. 8

Relative volume of (a) the ${\mathrm{PMDF}}_{{\mu }_a}$ and (b) ${\mathrm{PMDF}}_{\kappa }$ for the homogeneous head model.

4.2.2.

FWHM

Figure 9 shows the difference between the depth FWHMs and that corresponding to the wavelength $650\mathrm{nm}$ , ${\mathrm{FWHM}}_{650}$ . In general, the difference is $<0.6\mathrm{mm}$ , which indicates that the spread of regions of higher sensitivity does not vary much with wavelength. The FWHM differences are smaller for ${\mathrm{PMDF}}_{\kappa }$ , in particular for the homogeneous model, which shows a relatively flat spectrum for wavelengths of $<970\mathrm{nm}$ .

Fig. 9

FWHM for the depth direction of the (a, c) ${\mathrm{PMDF}}_{{\mu }_a}$ and (b, d) ${\mathrm{PMDF}}_{\kappa }$ profiles for the homogeneous (a, b) head model and (c, d) layered model.

4.2.3.

PD

PD is larger for the ${\mathrm{PMDF}}_{{\mu }_a}$ , as shown in Fig. 10 . The graphs show the differences with respect to the PMDF PD at $650\mathrm{nm}$ , ${\mathrm{PD}}_{650}$ . For the homogeneous model, the maximum depth is $15.6\mathrm{mm}$ and occurs for a wavelength of $720\mathrm{nm}$ . The lowest PD is $13.2\mathrm{mm}$ and occurs for a wavelength of $970\mathrm{nm}$ . This means that the maximum PD difference is $2.4\mathrm{mm}$ . For the layered model, the maximum penetration depth is $14.4\mathrm{mm}$ for a wavelength of $720\mathrm{nm}$ . For a wavelength of $970\mathrm{nm}$ , the PD attains a minimum value of $12.0\mathrm{mm}$ . The largest difference between PD is $2.4\mathrm{mm}$ . As a consequence of the voxel size, which only allows discrete spatial steps, the relative PD spectra exhibit a steplike behavior.

Fig. 10

Penetration depth of the (a, c) ${\mathrm{PMDF}}_{{\mu }_a}$ and (b, d) ${\mathrm{PMDF}}_{\kappa }$ profiles for the homogeneous (a, b) head model and (c, d) layered model.

The ${\mathrm{PMDF}}_{\kappa }$ exhibits a small variation in the PD for the different wavelengths (Fig. 10). For the homogeneous model, the combination of low ${\mu }_a$ and high ${\mu }_s^{\prime }$ for wavelengths from $670\text{to}840\mathrm{nm}$ results in a broader ${\mathrm{PMDF}}_{\kappa }$ with higher PD. The maximum PD is $12.6\mathrm{mm}$ , and the largest difference between PD is $0.6\mathrm{mm}$ . The ${\mu }_s^{\prime }$ for skull decays faster than for brain tissue, and at wavelengths of $>750\mathrm{nm}$ their ${\mu }_s^{\prime }$ values diverge significantly. When the external layers have lower ${\mu }_s^{\prime }$ and ${\mu }_a$ than the inner layer, the sensitivity profiles shift toward the surface. The high absorption of the brain tissue at $970\mathrm{nm}$ reduces the interrogation depth to $9\mathrm{mm}$ , compared to the maximum at $11.4\mathrm{mm}$ .

4.3.

Image Reconstruction

Images were reconstructed from simulated measurements at wavelengths selected using our new proposed method, and from measurements at wavelengths selected using the method of Corlu ¹⁶ Figure 11 shows the $xy$ cross-sectional images at $z=11.7\mathrm{mm}$ [Fig. 11] and $z=8.4\mathrm{mm}$ [Fig. 11], using measurement wavelengths at 680, 720, and $830\mathrm{nm}$ , simultaneously. Figure 12 shows the $xy$ cross-sectional images at $z=11.7\mathrm{mm}$ [Fig. 12] and $z=8.4\mathrm{mm}$ [Fig. 12], using measurement wavelengths at 650, 720, and $910\mathrm{nm}$ , simultaneously. Note that the color scales are not the same to enhance the differences between the images. For both sets of wavelengths, the images exhibited cross-talk between chromophores and also scattering. Cross-talk is particularly significant between oxy- and deoxy-hemoglobin concentrations in the deoxy-hemoglobin reconstruction and more evident in the images reconstructed using the wavelengths selected by our method. However, the deoxy-hemoglobin perturbation appears closer to the real position in the images reconstructed from wavelengths selected by our method than in the images obtained from wavelengths selected using the two-condition criteria. The $\left[\mathrm{Hb}{\mathrm{O}}_2\right]$ is underestimated in all the cases. The [HHb] is overestimated at $z=8.4\mathrm{mm}$ and underestimated at $z=11.7\mathrm{mm}$ . For comparison, images were reconstructed from simulated measurements at two wavelengths that are commonly used, 780 and $850\mathrm{nm}$ , and a third wavelength located between these two at $820\mathrm{nm}$ . Figure 13 shows the $xy$ cross-sectional images for this wavelengths set at $z=11.7\mathrm{mm}$ and Fig. 13 the reconstructed images at $z=8.4\mathrm{mm}$ . Cross-talk is clearly visible in the oxy-hemoglobin concentration and scattering amplitude images. The calculated concentration and scattering amplitude values are underestimated, except for [HHb] calculated at $z=8.4\mathrm{mm}$ . However, the [HHb] does not appear in the images in the correct position.

Fig. 11

Reconstructed images at (a–c) $z=11.7\mathrm{mm}$ and (d–f) $z=8.4\mathrm{mm}$ of (a, d) oxy-hemoglobin, (b, e) deoxy-hemoglobin, and (c, f) scattering amplitude from measurements at wavelengths 680, 720, and $830\mathrm{nm}$ , for the homogeneous model.

Fig. 12

Reconstructed images at (a–c) $z=11.7\mathrm{mm}$ and (d–f) $z=8.4\mathrm{mm}$ of (a, d) oxy-hemoglobin, (b, e) deoxy-hemoglobin, and (c, f) scattering amplitude from measurements at wavelengths 650, 720, and $910\mathrm{nm}$ , for the homogeneous model.

Fig. 13

Reconstructed images at (a–c) $z=11.7\mathrm{mm}$ and (d–f) $z=8.4\mathrm{mm}$ of (a, d) oxy-hemoglobin, (b, e) deoxy-hemoglobin, and (c, f) scattering amplitude from measurements at wavelengths 780, 820, and $850\mathrm{nm}$ , for the homogeneous model.

The same simulation study was performed using four wavelengths, and the corresponding $xy$ cross-sectional images at $z=11.7\mathrm{mm}$ and $z=8.4\mathrm{mm}$ obtained using the wavelengths found using our method are shown in Figs. 14 and 14 , respectively. Because of the limited sensitivity of the measurement to scattering to regions deeper than $12\mathrm{mm}$ below the surface, the reconstructions were found to have less cross-talk at $z=8.4\mathrm{mm}$ than at the true target depth $z=11.7\mathrm{mm}$ . Figure 15 shows the $xy$ cross-sectional images at $z=11.7\mathrm{mm}$ [Fig. 15] and $z=8.4\mathrm{mm}$ [Fig. 15], using the wavelengths selected by the two-criteria method. Reconstructions at $z=8.4\mathrm{mm}$ for both sets of wavelengths were similar and showed very small cross-talk. Nevertheless, the scattering amplitude and deoxy-hemoglobin concentration calculated at $z=8.4\mathrm{mm}$ were overestimated. The slightly better reconstructions at $z=11.7\mathrm{mm}$ obtained using our wavelength set could be due to the higher sensitivity of these wavelengths to the depth at which the targets are located. Furthermore, even though the sensitivity to scattering is quite limited in deeper regions, the PD of the ${\mathrm{PMDF}}_{\kappa }$ has its maximum at these wavelengths. For comparison, images were reconstructed using the wavelengths 750, 780, 820, and $850\mathrm{nm}$ . Figure 16 shows the $xy$ cross-sectional images at $z=11.7\mathrm{mm}$ , and Fig. 16 shows the images at $z=8.4\mathrm{mm}$ , obtained using the latter measurement set. Cross-talk is visible in the images; the calculated chromophore concentrations and scattering amplitude are underestimated; and the deoxy-hemoglobin perturbation appears misplaced in the image.

Fig. 14

Reconstructed images at (a–c) $z=11.7\mathrm{mm}$ and (d–f) $z=8.4\mathrm{mm}$ of (a, d) oxy-hemoglobin, (b, e) deoxy-hemoglobin, and (c, f) scattering amplitude from measurements at wavelengths $680\mathrm{nm}$ , 710, 730, and $830\mathrm{nm}$ , for the homogeneous model.

Fig. 15

Reconstructed images at (a–c) $z=11.7\mathrm{mm}$ and (d–f) $z=8.4\mathrm{mm}$ of (a, d) oxy-hemoglobin, (b, e) deoxy-hemoglobin, and (c, f) scattering amplitude from measurements at wavelengths 650, 710, 730, and $910\mathrm{nm}$ , for the homogeneous model.

Fig. 16

Reconstructed images at (a–c) $z=11.7\mathrm{mm}$ and (d–f) $z=8.4\mathrm{mm}$ of (a, d) oxy-hemoglobin, (b, e) deoxy-hemoglobin, and (c, f) scattering amplitude from measurements at wavelengths 750, 780, 820, and $850\mathrm{nm}$ , for the homogeneous model.

The simulation results indicate that four measurement wavelengths are required, in order to reconstruct images with negligible cross-talk, from reflection measurements and using linear multispectral reconstruction.

For the three-layer adult head model presented here, the sensitivity to scattering perturbation occurring in the brain region is very low; hence, it is not possible to separate the two chromophores and scattering amplitude simultaneously.

Finally, simulations were performed using the same medium as before, but without the scattering perturbation and for two wavelengths only. Figure 17 shows the reconstructions at $z=11.7\mathrm{mm}$ and $z=8.4\mathrm{mm}$ obtained from measurements at wavelengths found using our method (690 and $880\mathrm{nm}$ ). The images show a good separation between chromophores, but at $z=8.4\mathrm{mm}$ , the calculated chromophore concentrations are overestimated and, at $z=11.7\mathrm{mm}$ , are slightly underestimated. The images reconstructed from measurements at wavelengths found using the two criteria method (710 and $970\mathrm{nm}$ ) exhibit a small amount of cross-talk (Fig. 18 ). The presence of cross-talk could be due to sensitivity differences between the two wavelengths. At $970\mathrm{nm}$ , only the top of the target is interrogated, whereas the light at $710\mathrm{nm}$ is likely to travel through the whole target volume.

Fig. 17

Reconstructed images at (a, b) $z=11.7\mathrm{mm}$ and (c, d) $z=8.4\mathrm{mm}$ of (a, c) oxy-hemoglobin, and (b, d) deoxy-hemoglobin from measurements at wavelengths 690 and $880\mathrm{nm}$ , for the homogeneous model.

Fig. 18

Reconstructed images at (a, b) $z=11.7\mathrm{mm}$ and (c, d) $z=8.4\mathrm{mm}$ of (a, c) oxy-hemoglobin, and (b, d) deoxy-hemoglobin from measurements at wavelengths 710 and $970\mathrm{nm}$ , for the homogeneous model.

All the simulations were repeated for up to three times larger concentration and scattering amplitude changes, and the results showed a similar tendency.

In applications of optical topography, the true optical properties of the medium are usually unknown; hence, the sensitivity matrix is usually calculated using an estimate of the average optical properties, and subsequently used to reconstruct images from measurements at different wavelengths. Given that the optical properties of the medium are wavelength dependent, the reconstructed images are affected by using sensitivity matrices generated for optical properties different from the true properties. If the estimate of the optical properties is significantly different from the true values, then the absorption perturbation is likely to appear misplaced in the image, particularly in the $z$ direction.