## 2 Methods and Materials
### Correction for Lens Distortion
#### 2.4.4 Fatigue Crack Growth Data from Infrared Thermography  
Finding the Crack Tip Position  
To determine the crack tip position, we combine the thermoelastic effect and the heat dissipation measurements. The idea is to take the data recorded just after peak load, where the load decreases while the crack closure effect is negligible. At this stage, both the thermoelastic effect and the heat dissipation (if any) contribute to the temperature rise, with the maximum reached at the crack tip. The region behind the crack tip should experience a lower temperature change, as the closure effects are not active, and the stresses are comparatively low. The images taken at the upper and lower limit (marked with red in Figure 10) were stabilised with respect to the cyclic reference image--the image taken immediately after the test interruption at the mean load. The procedure was essentially the same as that described in Section 2.4.2 with one correction: the transformation matrix was used to transform the images with an _affine_ transformation to the reference image to find the temperature change between the images. By applying convolution with a specific kernel, which added the data covered by a 90 degree angle in front of a point of interest, and subtracted an equivalent region behind it, the sharp changes are maximised, and the crack tip can be easily located at the brightest point of the convoluted image as illustrated in Figure 11, comparing the temperature change map and the corresponding result of the convolution operation. The image can suffer from noise caused by the uncoated metallic surfaces at the crack faces. Therefore, the local extreme temperatures in the area outside the high temperature region surrounding the crack tip were replaced with the mean temperature for a given image. Motion Compensation and IRT/DIC Algorithm  
To measure and minimise the effect caused by the camera motion between recordings sessions, the cyclic reference image in each session was used. The procedure explained in detail in Section 2.4.2 was used to find the displacement between the images induced by the camera motion. ROI was selected to be approximately equal to half of the image without the crack. Since there is no crack in ROI, all deformations are relatively small, and the rigid body assumption holds. Figure 7 represents an example of finding how far the IRT camera has moved between two successive recordings. The transformation matrices can be combined to express the position of each reference image with respect to the first one (in any coordinate system chosen). Marking the notch position in an image, and finding its transformation matrix with respect to the first reference image, allows determining the crack position with respect to the notch. It is also possible to create a specific user-preferred coordinate system. Tracking data points in a record can be performed in two ways. The first way is to use a geometric transform (Section 2.4.2) based on the rigid body motion assumption, which is justified for small deformations. Alternatively, one can perform the correlative analysis of images based on IRT data, which can be used to estimate local displacements in many practical situations, where the optical imaging is difficult. The quality of these estimates  
Figure 11: Determination of crack tip position from the image enhanced by the convolution procedure highlighting the crack tip. Figure 10: Load signal, and region used to determine crack tip position. depends heavily on the quality of IRT images. A very elegant approach has been proposed by [PERSON] et al. [118] who applied the DIC technique to the images captured with an infrared camera in order to obtain perfectly synchronised kinematic and thermal full-field maps solely from the calibrated infrared measurements. Inspired by this work, we propose a simplified algorithm giving rise to similar results, although the quantitative comparison of the accuracy achieved by these two methods has not been performed yet.