## 2 Model Specification of Data Processing
### Pre-Processing  
The pre-processing step is the reconstruction of the interferogram [9; 10; 11; 12; 13]. For this purpose, one needs the images obtained in the initial state and the deformation state. On the one hand, the initial state can be the time of the beginning of the experiment, denoted \(t_{0}\), and the deformation time is the set of other times of the DHSPI acquisitions, i.e., from time \(t_{1}\) to time \(t_{n}\) On the other hand, the initial state can be time \(t_{i}\) and the deformation time the next one, the time \(t_{i+1}\). The method is the same for both cases, but the second method is used in this experiment because if the technique were to be applied to a work of art in a museum, it would be logical to compare two images with the minimum time difference. During the acquisition of the DHSPI, five images of the surface of the object under study are recorded in order to optimise the signal-to-noise ratio. At time \(t_{i}\), the pixel intensity of each image is formulated as:  
\[I_{n}(x,y,t_{i})=I_{O}(x,y,t_{i})+I_{R}(x,y,t_{i})+2\sqrt{I_{O}(x,y,t_{i})*I_{R }(x,y,t_{i})}\cos\Bigl{[}\varphi_{R}(x,y,t_{i})-\varphi_{B}(x,y,t_{i})-n*\frac {\pi}{2}\Bigr{]}\ with\ n=0,\ldots,4 \tag{1}\]  
where \(I_{O}(x,y,t_{i})\) is the object beam intensity at pixel \((x,y)\), \(I_{R}(x,y,t_{i})\) is the reference beam intensity, \(\varphi_{R}(x,y,t_{i})\) is the reference speckle pattern phase and \(\varphi_{B}(x,y,t_{i})\) is the phase of the undeformed sampling light beam.  
\[I_{n}(x,y,t_{i+1})=I_{O}(x,y,t_{i+1})+I_{R}(x,y,t_{i+1})+2\sqrt{I_{O}(x,y,t_{i+ 1})*I_{R}(x,y,t_{i+1})}\cos\bigl{\{}\varphi_{R}(x,y,t_{i+1})-\varphi_{B}(x,y,t_ {i+1})-n*\frac{\pi}{2}+\Delta\varphi(x,y,t_{i+1})\bigr{\}}\ with\ n=0,\ldots,4 \tag{2}\]  
where \(\Delta\varphi(x,y,t_{i+1})\) is phase difference that represents the surface movement between time \(t_{i}\) and \(t_{i+1}\).  
In order to obtain the phase of time \(t_{i}\), the images are recombined and deduced from Equation (1):  
\[\tan\bigl{[}\varphi_{R}(x,y,t_{i})-\varphi_{B}(x,y,t_{i})\bigr{]}=\frac{2(I_{ 2}(x,y,t_{i})-I_{4}(x,y,t_{i}))}{(2I_{3}(x,y,t_{i})-I_{1}(x,y,t_{i})-I_{5}(x,y,t_{i}))} \tag{3}\]  
To obtain the phase of time \(t_{i+1}\), the images are recombined and deduced from Equation (2):  
\[\tan\bigl{[}\varphi_{R}(x,y,t_{i+1})-\varphi_{B}(x,y,t_{i+1})+\Delta\varphi( x,y,t_{i+1})\bigr{]}=\frac{2\bigl{(}I_{2}(x,y,t_{i+1})-I_{4}(x,y,t_{i+1}))}{(2I_{3} (x,y,t_{i+1})-I_{1}(x,y,t_{i+1})-I_{5}(x,y,t_{i+1}))} \tag{4}\]  
From Equations (3) and (4), the phase difference can be obtained:  
\[\Delta\varphi=\tan^{-1}\biggl{[}\frac{2(I_{2}(x,y,t_{i+1})-I_{4}(x,y,t_{i+1}) )}{(2I_{3}(x,y,t_{i+1})-I_{1}(x,y,t_{i+1})-I_{5}(x,y,t_{i+1}))}\biggr{]}-\tan ^{-1}\biggl{[}\frac{2(I_{2}(x,y,t_{i})-I_{4}(x,y,t_{i}))}{(2I_{3}(x,y,t_{i})-I_ {1}(x,y,t_{i})-I_{5}(x,y,t_{i}))}\biggr{]} \tag{5}\]  
Figure 1: Interference principle.  
The phase difference is obtained by construction in the interval \([-\pi,\ \pi]\) thanks to the MatLab function, 'atan2' [14]. Therefore, the phase is wrapped and it needs to be unwrapped to obtain the surface deformation of the artwork.  
Even if it is not always necessary, denoising the signal could enhance the quality of the data treatment.