# Matched-Filter Thermography
## 4 Matched Filtering in Thermography
### Thermal Coherence Tomography (TCT)
#### 4.2.1 Theory and Modeling  
Detailed derivations for analytical expression of a BPC signal and TCT responses of absorbers in turbid media are provided by [PERSON] and [PERSON] in their 2011 publication in the journal of Physical Review Letters [22]. Briefly, the TCT response of absorbers in turbid media can mathematically be modelled as:  
\[\left\{\begin{array}{l}\frac{\partial^{2}\theta(zz\omega)}{\partial z^{2}}- \sigma^{2}\theta(z;\omega)\ =\ -\frac{\mu_{g}}{\kappa}I(z)F(\omega)\\ -\kappa\frac{\partial\theta(zz\omega)}{\partial z}\Big{|}_{z=0}\ =\ 0\\ -\kappa\frac{\partial\theta(zz\omega)}{\partial z}\Big{|}_{z=\frac{I}{\kappa} =\ I(I)F(\omega)\\ \theta(z;\omega)\ =\ \Im\{T(z,t)-T_{\infty}\}\end{array}\right., \tag{10}\]  
where \(I(z)\) is the local fluence of optical excitation at depth \(z\). Note the similarity of the thermal-wave field equation in the frequency-domain to the Helmholtz equation of hyperbolic wave-fields. Equation (10) is an ordinary non-homogenous differential equation system whose solution can be found as superposition of the homogeneous and particular solutions [22]. Figure 8a plots the theoretical CC phase of two extreme \(\overline{\mu}_{IR}\) cases for a 7-bit binary code as a function of the subsurface absorber depth, \(l\), at several carrier frequencies. In both \(\overline{\mu}_{IR}\) cases, at a given carrier frequency, the deeper the absorber, the larger the amount of phase shift, showing a more delayed contribution from the deeper absorber. Moreover, an increase in the carrier frequency reduces the maximum probing depth due to the reduced thermal diffusion length, as expected. The key point in this figure is the effect of \(\overline{\mu}_{IR}\). The infrared radiation captured by the detector is composed of a delayed conductive thermal-wave portion and an instantaneous direct Planck emission. While an increase in the direct emission improves the signal-to-noise ratio of the amplitude channel, it deteriorates the maximum probing depth of the phase channel. This occurs because the direct emissions from absorbers at different depths are instantaneous and therefore all in phase, dominating the depth-dependent phase information of the conductive portion and consequently limiting the maximum probing depth, as observed in Figure 8a.  
Figure 8b shows how the height (peak amplitude channel) and location (peak delay time channel) of the CC signal peak behave as a function of the absorber depth at various carrier frequencies. Generally, a shallower absorber results in a higher amplitude and shifts the CC peak location to shorter delay times to manifest a less attenuated thermal-wave source closer to the interrogated surface. In other words, matched-filtering localizes the energy of the long-duty BPC excitation under a narrow peak whose location on the delay time axis is linked to the depth of the source and allows one to construct iso-delay images. As such, TCT can be considered as the diffusion equivalent of optical coherence tomography (OCT). Moreover, while the amplitude channel has by far the highest SNR, its reliability depends on the illumination and sample surface conditions because, unlike phase and peak delay time, it is not an emissivity normalized quantity. Another interesting feature of TCT  
Figure 7: Schematic procedure of Binary Phase Coding (BPC) signal construction in time (**top**) and frequency (**bottom**) domains for a 7-bit code and 5 Hz carrier [22].  
imaging is the effect of code length. While increasing the code length does not alter the CC phase and peak delay time curves of Figure 8, it increases the pulse compression ratio, i.e., increases the amplitude of the CC peaks and SNR.