# Matched-Filter Thermography
## 4 Matched Filtering in Thermography
### Thermal-Wave Radar (TWR)  
#### 4.1.1 Theory and Modeling  
The one-dimensional thermal-wave field induced in an opaque medium of thickness \(l\), Figure 2a, by a linearly frequency modulated (LFM) optical excitation on sample surface can be mathematically modeled in frequency-domain as:  
\[\frac{\partial^{2}\theta(z,\omega)}{\partial^{2}z}-\frac{i\omega}{\alpha}\theta (z,\omega)\ =\ 0, \tag{4}\]  
\[-\kappa\frac{\partial\theta(z,\omega)}{\partial z}\bigg{|}_{z\ =\ 0}\ =\ \Im\{f(t)\}\ =\ F(\omega),\]  
\[-\kappa\frac{\partial\theta(z,\omega)}{\partial z}\bigg{|}_{z\ =\ l}\ =\ 0,\]  
\[\theta(z,\omega)\ =\ T(z,\omega)-T_{\infty},\]  
where \(\alpha,\ \kappa,F(\omega),\ \text{and}\ \text{T}_{\infty}\) are thermal diffusivity, thermal conductivity, the incident LFM laser beam spectrum, and sample equilibrium temperature, respectively. Equation (4) is a homogeneous second-order ordinary differential equation that can be solved by finding the solutions of the corresponding characteristic equation and application of boundary conditions [21]. Therefore, using [PERSON]'s law of radiation, the radiometric signals registered by the infrared camera from the surface of an opaque sample with thickness \(l\), can be found as:  
\[S_{l}^{Opaque}(t)\ =\ \Im^{-1}[A(\omega,l)+B(\omega,l)], \tag{5}\]  
\[A(\omega,l)\ =\ \frac{F(\omega)}{\kappa\,\sigma}\,\frac{\exp(-2l\sigma)}{1- \exp(-2l\sigma)},\]  
\[B(\omega,l)\ =\ \frac{F(\omega)}{\kappa\,\sigma}\,\frac{1}{1-\exp(-2l\sigma)},\]  
where \(\sigma\ =\ \sqrt{i\omega/\alpha}\) is the complex wavenumber. Similarly, the TWR response of a semi-infinite opaque material can be calculated as:  
\[S_{\infty}^{Opaque}(t)\ =\ \Im^{-1}\bigg{[}\frac{F(\omega)}{\kappa\,\sigma} \bigg{]}. \tag{6}\]  
Figure 2: Schematic models of (**a**) an opaque sample with blind holes of various depths and (**b**) a turbid sample with absorber at various depths [21]. Moreover, using Equations (5) and (6), the delayed thermal contributions of subsurface defects can be isolated from the total TWR response by subtracting the response of the semi-infinite sample from that of the finite thickness sample (i.e., TWR subtraction mode):  
\[\Delta S_{Sub}^{Opaque}(t)\ =\ \Big{[}S_{l}^{Opaque}-S_{\infty}^{ Opaque}\Big{]}\ =\ \Im^{-1}\bigg{[}\frac{F(\omega)}{\kappa\,\sigma}\cdot\frac{2\exp(-2l\sigma)}{1- \exp(-2l\sigma)}\bigg{]}. \tag{7}\]  
Following a similar procedure, the TWR response of a subsurface absorber at depth \(l\) in a transparent medium, Figure 2b, can be modeled and found as:  
\[-\kappa\frac{\partial\theta(z,\omega)}{\partial z}\bigg{|}_{z\ =\ 0}\ =\ \Im\{f(t)\}\ =\ F(\omega),\]0  
\[-\kappa\frac{\partial\theta(z,\omega)}{\partial z}\bigg{|}_{z\ =\ 0}\ =\ \Im\{f(t)\}\ =\ F(\omega),\]1  
\[-\kappa\frac{\partial\theta(z,\omega)}{\partial z}\bigg{|}_{z\ =\ 0}\ =\ \Im\{f(t)\}\ =\ F(\omega),\]2  
where \(\overline{\mu}_{IR}\) is the average infrared absorption coefficient of transparent medium over the detection wavelength range. Therefore, by applying the frequency-domain interpretation of cross-correlation to Equations (5)-(8), the TWR/matched-filter output can be simulated for finite opaque and transparent materials as well as for semi-infinite opaque media. Figure 3a depicts the energy spectral density of simulated radiometric signals, Equation (5), from steel samples with various thicknesses along with the energy spectral density of the simulated radiometric signal of semi-infinite steel, Equation (6). For the purpose of comparison, the energy spectral density (ESD) of the optical excitation waveform (i.e., \(F(\omega)\) of the filter) is also included.