## 2 Materials and Methods
### Estimation of Deformation Parameters of the Tested Paper Sample  
The measurement data obtained as a result of the pre-processing and analysis of the images were further analyzed in order to determine the value of so-called the deflection arrow of the paper sample and the parameters of the sinusoidal function modelling the shape of the sample edges. This is preceded by a rotation of the coordinate system of the machine to position the edge of the sample at the beginning of the test in a vertical direction. This operation is performed on the basis of the coordinates of the sample mounting in the machine holders. After this correction, the deflection arrow is defined as the difference in the horizontal coordinate of the places of attachment of the sample in the handles and the place of the greatest deflection of the sample. The shape of the sample is then modelled using a sinusoidal function of image coordinates \(x=f(y)\):  
\[x=C+A\cdot sin\left(\frac{2\cdot\pi\cdot y}{Pr}+\varphi\right), \tag{1}\]  
where:  
\(C\)-constant--the shift relative to the beginning of the coordinate system (it can change in subsequent photos as a result of, for example, camera vibrations, it is not important in measuring the deflection of the sample),  
\(A\)--amplitude of the sine wave function modelling the deflection,  
\(Pr\)--the period of the sine wave modelling the deflection (at the initial stage of measurement, before the buckling of the sample, it cannot be determined accurately),  
\(\varphi\)--the initial phase of the sine wave modelling the deflection (not relevant from the point of view of deflection analysis, but must be a variable in order to match the model well with the measurement data). The desired values of the parameters of the sample shape modelling function are searched in the optimization procedure that minimizes the \(SSE_{Sh}\) error between the measurement data and the modelling result. The error is described by Equation (2):  
\[SSE_{Sh}=\sum_{i}\left(x_{i}-C-A\cdot\sin(\frac{2\cdot\pi\cdot y_{i}}{Pr}+ \varphi)\right)^{2}, \tag{2}\]  
where:  
\(i\)--index of the point at the edge of the test sample,  
\(x_{i}\)--the horizontal coordinate of the \(i\)-th point at the edge of the sample,  
\(y_{i}\)--the vertical coordinate of the \(i\)-th point on the edge of the sample. Figure 8: Displacement of the handle of the testing machine estimated from a series of images before and after synchronization with data from the machine. Meaning of the individual charts in each drawing:  
(a)--an image of the sample with the edge detected,  
(b)--the shape of the detected edge in the coordinate system determined on the basis of the indicated points of attachment of the sample,  
(c)--the shape of the force curve as a function of displacement of the handle with the marked place of recording of the shown image,  
(d)--the value of the deflection arrow as a function of displacement of the movable handle,  
(e)--the value of the deflection arrow as a function of force,  
(f)--work on deformation of the sample as a function of displacement of the handle,  
Figure 10: The results of the compression analysis of the paper sample at a clamping length of 4 mm at the maximum compressive force. (g)--the amplitude value of the sine wave function modelling the shape of the sample as a function of displacement of the handle,  
(h)--the period of the sinusoidal function modelling the shape of the sample as a function of displacement of the handle,  
(i)--the value of the sample shape modelling error as a function of the handle displacement. A round, red marker on the background of curves representing individual functional dependencies shows the obtained values in the discussed phase of the compression process. Figure 9 shows the state of the sample at the time of transition from the pure compression phase to the buckling phase of the sample. In the moments preceding this state, modelling the shape of the sample with a sinusoidal function does not make sense.