## 2 Continuous Wave Data and Uniqueness in Parameter Recovery  
Uniqueness/non-uniqueness problem in the parameter recovery methods of CW-based systems has been a long-discussed topic [13; 14] and has been shown that for a single wavelength multi-distance boundary intensity measurements there can be more than one solutions of absorption and reduced scattering coefficient distributions which will lead to the same boundary data. This non-uniqueness can be addressed by using multi-wavelength data and constraining the problem spectrally [14]. The reported findings to date discuss a very general scenario of image reconstruction where the recovered parameters are the three-dimensional distribution of absorption and reduced scattering coefficients. However, in the context of finding the absolute parameters of different regions in a layered medium such as the human head, this presented work brings an additional spatial constraint to the distribution of parameters. Here, by extending the arguments made in previous works [13; 14] it is shown that given such a regional constraint on parameters, each solution set is unique to a measured boundary intensity data, and also investigates the scenario when the measured data is either a spatial or 1 st spectral derivative of intensity (in log scale). Consider the following schematic of a two-layered medium as shown in Figure 1.  
Here, \(\mu_{a}\) is the absorption coefficient, \(\mu_{s}^{\prime}\) is the reduced scattering coefficient, and the diffusion coefficient (\(D\)) is defined as, \(D=1/3(\mu_{a}+\mu_{s}^{\prime})\approx 1/3\mu_{s}^{\prime}\). Re-writing the CW diffusion equation by change of variables to get the Helmholtz-type equation given as [13],  
\[-\
abla^{2}\Psi+\eta\Psi=\frac{q_{0}}{\gamma}. \tag{1}\]  
where \(\gamma^{2}=D\), \(q_{0}\) is the source distribution, \(\Psi=\gamma\Phi\), where \(\Phi\) is the photon density, and the variable \(\eta\) is given by,  
\[\eta=\frac{\
abla^{2}\gamma}{\gamma}+\frac{\mu_{a}}{\gamma^{2}}. \tag{2}\]  
Considering the light-source to be a point source at a location \(\mathbf{r}_{0}\) on the external medium boundary, and defining \(\Psi_{0}\) such that:  
\[-\
abla^{2}\Psi_{0}+\eta\Psi_{0}=\mathbf{q}_{0}. \tag{3}\]  
This implies that \(\Psi=\Psi_{0}/\gamma(\mathbf{r}_{0})=\Psi_{0}/\gamma_{1}\), as \(\gamma(\mathbf{r}_{0})=\gamma_{boundary}=\gamma_{1}\) in the given scenario and therefore \(\Phi_{\text{boundary}}=\Psi_{boundary}/\gamma_{1}^{2}\). For any measured boundary intensity data \(\Phi_{\text{boundary}}\), two non-unique solution sets \([\mu_{a},\ \mu_{s}^{\prime},L]\) and \([\overline{\mu}_{a},\ \overline{\mu}_{s}^{\prime},\ \overline{L}]\) (with their equivalent canonical parameters \(\eta\) and \(\overline{\eta}\)), can exist if they satisfy the following two conditions:  
Condition-1: \(\eta=\overline{\eta}\) everywhere inside the medium, this ensures \(\Psi_{0}\) to be the same for both solutions. Condition-2: \(\gamma_{boundary}=\overline{\gamma}_{boundary}\), together with condition 1 this ensures \(\Phi_{\text{boundary}}\) to be the same for both solutions. Figure 1: Schematic showing a two-layered medium. Applying condition 1, to this scenario, \(\eta=\overline{\eta}\) for region-1, region-2 and at the boundary between them. This requires the superficial layer thickness to be equal i.e., \(L=\overline{L}\), as without this the two solutions would have a different location of the interface between region-1 and region-2, and \(\eta\) can never be equal to \(\overline{\eta}\) at the regional interface.