# Measurement of Contrast and Spatial Resolution for the Photothermal Imaging Method
## 2 Image Quality Analysis Theory for Photothermal Imaging  
### Contrast  
Contrast is a measure of the difference between the magnitude of signals from an internal structure and those from the surrounding or background of the internal structure. In general, the purpose of an imaging method is to examine the inside of an object to be measured and to detect any structural  
Figure 1: Schematic diagram of photothermal imaging: (**a**) detection of photothermal signals; (**b**) detection of internal material through scanning. differences from the intended design or any abnormal defects. In order to easily detect such defects, imaging methods with high contrast are preferred [18]. This section focuses on the modulation index that quantifies contrast in reference to periodic signals of constant amplitude, as shown in Figure 2.  
The modulation index (\(m_{freq}\)) of a periodic signal \(f(x)\) whose maximum and minimum amplitudes of signal are \(A_{max}\) and \(A_{min}\), respectively, can be defined as Equation (1) [18; 19], where it is the ratio of the average value of this signal (\((A_{max}+A_{min})/2\)) to the size of the amplitude (\((A_{max}-A_{min})/2\)) of the periodic signal \(f(x)\). \[m_{freq}=\frac{A_{\max}-A_{\min}}{A_{\max}+A_{\min}}. \tag{1}\]  
In the case of an imaging method that utilizes the photothermal effect, as with other imaging methods, the contrast changes depending on the size of the object to be measured (in this study, the size of the internal structure). The size of an object to be measured can be quantified on the basis of the spatial frequency, which can be derived by applying the Fourier transform to a spatial structure in the space. In summary, an imaging method's contrast can be quantified from the modulation index, and the size of a structure to be measured can be quantified from the spatial frequency [15; 17]. The relation between the spatial frequency and modulation index is called the modulation transfer function (MTF). The MTF of an imaging method can be calculated by either of the following methods:  
* For line pairs of varied sizes, the modulation index of a spatial frequency corresponding to each line pair is calculated [20; 21]. * A Fourier transform is applied to the line spread function (LSF) derived from line pairs of the same size [20; 21]. The first method (Method A), which uses line pairs of varied sizes, is the most typical way to derive the MTF, but it is disadvantageous in that a large number of specimens is required in order to derive the entire MTF profile. For this reason, comparing the MTF in various conditions drastically increases the number of numerical analyses or experiments. The second method (Method B), which utilizes the LSF, can be implemented in a relatively simple manner [21]. On the assumption that there are infinite specimens with the internal structure shown in Figure 3a in the \(x\)- and \(y\)-directions, 1-D scanning on the specimen surface using the photothermal imaging method will produce the phase delay change results shown in Figure 3b. The sigmoidal phase difference signal derived near the boundary shown in this figure is called the edge response function (ERF). As the ERF is differentiated, the LSF as in Figure 3c can be obtained. \[MTF(f_{\rm s})=\mathcal{F}[LSF(x)]. \tag{2}\]  
Figure 2: Example of periodic signals with a constant amplitude. Physically and mathematically, the LSF and MTF are in the relation of a Fourier transform as in Equation (2) [18]. By normalizing the results of applying the Fourier transform to the LSF, the MTF of the imaging system can be derived as shown in Figure 3d. These two methods of deriving the MTF (Methods A and B) have been found in a number of studies to have the same mathematical and physical meanings [15; 20]. The present study derived the MTF by using the relatively simpler Method B and verified contrast changes depending on the spatial frequency.