# Matched-Filter Thermography
## 4 Matched Filtering in Thermography
### LIT vs. TWR vs. TCT  
Although linear frequency modulation (chirp) and binary phase coding are both pulse compression techniques used for improving the axial/depth resolution of the matched filter, their implementation in thermal-wave fields yields different characteristics. Figure 10a plots the simulated responses of absorbers at various depths inside a turbid medium using TWR (0.1 Hz-4.9 Hz, 6.4 s). The TCT responses of the same simulated sample to an excitation at 2.5 Hz carrier frequency (i.e., the center frequency of TWR simulation) are depicted in Figure 10b. Comparison of these figures clearly shows that the narrow band nature of TCT results in more localized responses with minimal side lobes. These simulations suggest that, under similar experimental conditions, TCT is expected to outperform TWR in terms of depth resolution and SNR. Figure 10c,d lie at the heart of this review paper in which the simulated performances of LIT, TWR, and TCT are compared. The spectra of the simulated responses were superposed with white noise (SNR = 1) to obtain more realistic results and simulations were repeated 10 times at each absorber depth for measurement of error bars. All other simulation conditions were kept the same. Figure 10c,d clearly shows the improvement of axial/depth resolution (the slope of the lines) when changing the modulation scheme from single frequency to LFM and finally to BPC modulation. Moreover, due to the higher SNR of matched filtering, the size of the error bars are smaller in TWR and TCT compared to that of the LIT. Figure 9: (**a**) cross section of step wedge light-absorbing inclusion in a turbid medium; (**b**) conventional lock-in thermography (LIT) phase (unit: degree); (**c**) BPC peak delay time (unit: millisecond); and (**d**) BPC phase images (unit: degree) of the step wedge sample using thermal coherence tomography (TCT) (16-bit code). The curve in each image shows the mean horizontal profile of the corresponding contrast parameter; (**e**) pictures of the interrogated surface and cross section of a goat bone; TCT phase images of goat bone at (**f**) 10 Hz and (**g**) 1 Hz carrier frequencies [22]. Figure 11 shows experimental results obtained from a teeth matrix. The contacting surface of the teeth has been demineralized (thus absorbs light) to simulate the challenging clinical scenario of interproximal early caries in the field of Dentistry. Comparison of LIT and TCT phase images, panels (b) and (c), formed under identical experimental conditions, clearly demonstrates the enhanced axial resolution of the TCT over LIT. Quantitative analysis of these images suggests a 275% increase in contrast between interproximal caries and the surrounding intact areas in TCT over that of LIT. Moreover, the enhancement of depth resolution in TCT allows one to construct iso-delay images in which thermal contributions from over and under layers are effectively suppressed. Figure 11d-f show thermal coherence tomographic images obtained through mathematical delaying of the filter by 2.7 ms, 29.7 ms, and 45.9 ms, respectively, and registering signals coherent to such delayed filters. No heat source can be observed in the first (i.e., shallowest) slice, while the deepest slice shows healthy enamel areas. Defects and caries shift the thermal-wave centroid closer to the interrogated surface (compared to the surrounding healthy enamel) and, as a result, are revealed in the intermediate slice (Figure 11e). The results of Figure 11, experimentally, demonstrate that, by improving the axial resolution, TCT allows for deconvolution of thermal responses of subsurface defects from those of the non-defective regions. Figure 10: Simulated (**a**) TWR (0.1 Hz–4.9 Hz, 6.4 s) and (**b**) TCT (2.5 Hz, 16-bit coding) responses of turbid sample containing absorbers at several depths. Comparison of simulated phase responses using LIT, TWR, and TCT for absorbers in turbid medium.