# Matched-Filter Thermography
## 4 Matched Filtering in Thermography
### Thermal-Wave Radar (TWR)
#### 4.1.3 Select TWR Experimental Results  
One of the commonly used experimental models for characterization of photothermal radiometry systems is an opaque sample with blind holes as it mimics the presence of sharp subsurface defects at varying depths from the interrogation surface. Figure 5a,b show the results of a conventional frequency-domain photothermal radiometry (FD-PTR; analogous to lock-in thermography) carried out at 5 Hz along the centerline of holes with varying distances, \(l\), from the interrogation surface. It can be seen that the spatial scan can clearly detect the shallow subsurface holes/defects but not the deep ones. Figure 5c,d, on the other hand, depict the TWR spatial scan along the same lines as those of panels (a) and (b) using same experimental conditions used in the FD-PTR experiments (e.g., laser power, measurement time, etc.). The enhanced dynamic range of TWR can clearly be observed when comparing the performance of the two methodologies in detecting deep subsurface defects. Similar to FD-PTR, the TWR delay time can be fitted to the theoretical model, to estimate the subsurface depth of the holes (Figure 3d; solid line, and open circles, respectively).  
The theoretical case of subsurface absorber at depth \(l\) in a transparent medium can experimentally be modeled using back-painted glass sample of thickness \(l\). Experimental TWR results from such samples are presented in Figure 5e. For the front-surface painted glass sample, the presence of the TWR cross-correlation maximum at \(\tau\) = 0 indicates pure surface absorption (no delayed conductive contribution from underlayers). For the back-surface painted glass samples, as the thickness of the glass increases, the corresponding peak delay time also increases (as predicted by theory; Figure 3f). Furthermore, with an increase in glass thickness, the TWR signal is strongly attenuated as thermal-waves decay exponentially across the glass thickness. The TWR peak delay time, \(\tau_{p}\), is directly  
Figure 5: Experimental results obtained using (**a,b**) single-frequency photothermal radiometry and (**c,d**) TWR in an opaque sample with blind holes. Remaining thickness of holes are depicted in the insert; (**e**) experimental TWR responses from glass slides of various thicknesses painted on the black on the front or back surface [21].  
linked to the depth of the absorber and as such can be fitted to the diffusive equation \(\tau_{p}(t)=k_{1}l^{2}+k_{2}\) to approximate the depth of absorbers (Figure 3f; solid line and \(\
abla\)).  
Figure 6 shows the TWR images from a transparent sample with two iso-depth absorbers with different light absorption coefficients. Panels (a)-(c) are the images obtained using cross correlation amplitude, peak delay time, and phase, respectively, and panels (d)-(f) show their horizontal mean profiles, respectively. It can be seen that the amplitude channel is representative of the amount of energy absorption by the two equally deep absorbers (green and black paints), yielding significantly different amplitude values. Consequently, the amplitude channel is not a true measure of the depth of the absorber. However, the peak delay time and phase values are linked to the true depth of the absorbers as they maintain the same value over the two absorbers regardless of their absorption coefficients. In terms of SNR, the amplitude channel is significantly stronger than the peak delay time and phase channels, and therefore the amplitude images should always be used to complement the information obtained from the phase and peak delay time images.