# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 2 Methods
### Noise Correlations  
Since DTOF signals are transformed to windows space, the noise will also suffer changes. In this part, noise transformations and correlations are analysed to understand noise influence. The DTOF signal measured by a detector is theoretically described as  
\[u(t)=x(t)+\varepsilon(x(t)), \tag{21}\]  
where \(x(t)\) is the noiseless signal and \(\varepsilon\) is the noisy term which depends on \(x(t)\) magnitude, uncorrelated (that is, \(\text{Cov}(\varepsilon(x(t)),\varepsilon(x(t^{\prime})))=0\) for \(t\
eq t^{\prime}\)) and follows a Poisson distribution. Applying the Fourier transform to \(u(t)\) yields,  
\[ \begin{split} U(f)&=X(f)+\text{Y}(f)\\ &=\int_{-\infty}^{\infty}x(t)e^{-i2\pi ft}\,\text{d}t+\int_{- \infty}^{\infty}\varepsilon(x(t))e^{-i2\pi ft}\,\text{d}t,\end{split}  \tag{22}\]  
where the second term in the right part is the noisy contribution and \(U(f)\) can be considered a random variable. As stated before, in time domain, the noise at each time bin is independent, however after  
Figure 7: (**Left**) Several Tukey functions with different \(\alpha\) values. (**Right**) Spectrum magnitude of previous Tukey functions. applying the Fourier transform this property does not hold anymore. For example, the covariance between two different frequencies is,  
\[E\left[U(f_{1})\cdot\overline{U(f_{2})}\right]-E\left[U(f_{1}) \right]\cdot E\left[\overline{U(f_{2})}\right] =E\left[\int_{-\infty}^{\infty}u(t)e^{-i2\pi f_{1}t}\,\mathrm{d}t \cdot\overline{\int_{-\infty}^{\infty}u(t)e^{-i2\pi f_{2}t}\,\mathrm{d}t}\right] \tag{23}\] \[\qquad-E\left[\int_{-\infty}^{\infty}u(t)e^{-i2\pi f_{1}t}\, \mathrm{d}t\right]\cdot E\left[\int_{-\infty}^{\infty}u(t)e^{-i2\pi f_{1}t}\, \mathrm{d}t\right]\] \[=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left(E\left[u(t) \cdot u(t^{\prime})\right]-E\left[u(t)\right]\cdot E\left[u(t^{\prime})\right] \right)e^{-i2\pi f_{1}t}e^{2\pi f_{2}t^{\prime}}\,\mathrm{d}t\,\mathrm{d}t^{\prime}\] (24) \[\left(\text{Due to TR signal noise}\right) =\int_{-\infty}^{\infty}E\left[u(t)e^{-i2\pi(f_{1}-f_{2})t}\, \mathrm{d}t\right.\] \[\left(E\left[u(t)\right]=x(t),\text{due to shot}/\text{Poisson noise}\right) =\int_{-\infty}^{\infty}x(t)\,e^{-i2\pi(f_{1}-f_{2})t}\,\mathrm{d}t\] \[=X(f_{1}-f_{2}),\]  
so the covariance of Fourier transform signal at two different frequencies is equal to the Fourier coefficient of the noiseless signal at frequency \(f_{1}-f_{2}\). See Appendix B for an alternative proof. From this result, the covariance between any temporal window datatypes can be determined. Let us define a window datatype shifted \(kt_{0}\) in time as,  
\[M_{k,t_{0}}=\int_{0}^{\infty}u(t)\,w(t-kt_{0})\,\mathrm{d}t=\int_{-\infty}^{ \infty}U(f)\,\overline{W(f)}e^{i2\pi ft_{0}}\,\mathrm{d}f, \tag{25}\]  
where \(t_{0}\) is the shift unit length and \(k\) is the number of shifts steps that are taken. Then, the covariance between shifted window datatypes is,  
\[ \begin{split}\text{Cov}(M_{k},M_{l})&=E\left[\int_{- \infty}^{\infty}U(f)\,\overline{W(f)}e^{i2\pi f_{k_{0}}}\,\mathrm{d}f\cdot \overline{\int_{-\infty}^{\infty}U(f)\,\overline{W(f)}e^{i2\pi ft_{0}}\, \mathrm{d}f}\right]\\ &\qquad-E\left[\int_{-\infty}^{\infty}U(f)\,\overline{W(f)}e^{2 \pi ft_{0}}\,\mathrm{d}f\right]\cdot E\left[\overline{\int_{-\infty}^{\infty} U(f)\,\overline{W(f)}e^{i2\pi ft_{0}}\,\mathrm{d}f}\right]\\ &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left(E\left[U(f) \cdot\overline{U(f^{\prime})}\right]-E\left[U(f)\right]\cdot E\left[\overline{ U(f^{\prime})}\right]\,\right)\overline{W(f)}\,e^{2\pi ft_{0}}\,W(f^{\prime})\,e^{-i2\pi f ^{\prime}ft_{0}}\,\mathrm{d}f\,\mathrm{d}f^{\prime}\\ &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}X(f-f^{\prime}) \,\overline{W(f)}\,W(f^{\prime})\,e^{i2\pi(tk_{0}f-lt_{0}f^{\prime})}\, \mathrm{d}f\,\mathrm{d}f^{\prime}.\end{split}  \tag{26}\]  
From Equation (26), the covariance and correlation matrices of any window can be obtained. Note that for standard (and Mellin-Laplace) moments the shift is already fixed by \(n\) (and \(p\)) value.