## 2 Materials and Methods
### Image Processing Algorithm
#### 2.2.1 Geometrical Transformation  
The geometrical transformation is introduced in order to compensate the deformation of the discharge representation, which is caused by the cylindrical shape of the reactor, captured on the flat sensor of the camera. For that purpose, the Abel transformation [21; 22; 23] initially considered has been finally withdrawn, because the plasma distribution around the gas gap does not present the cylindrical invariance which is mandatory. A new method for calculation is proposed below.  
On the taken image, the reactor cylindrical shape is flattened by the camera, as shown on Figure 6. The reactor has length \(L\) and is placed along the Z axis; the camera's optical axis is on the X axis. The plane (X, Y) is an orthogonal cross-section of the reactor (Figure 6b). The points \(m(y_{p})\) cover the external surface of the reactor, at outer radius r, for \(\theta\) ranging from \(-90^{\circ}\) to \(90^{\circ}\). The point M(y), with y ranging from \(-\)R to \(+\)R, is the orthogonal projection, captured by the camera, of the points \(m(y_{p})\). Only the outer electrode is shown. Several geometrical assumptions are made:  
1. The streamer distribution is symmetrical to the plane X = 0 (the hidden face of the reactor is assumed to present the same aspect as the captured one). Therefore, only the front face is analyzed.
2. The camera plane (Y, Z) is far enough from the reactor that the light rays reach it in parallel. This ensures that the points M(y) and \(m(y_{p})\) are aligned parallel to the optical axis of the camera (X axis). Therefore, the height of the image captured by the camera (Figure 6c) is equal to 2.R.
3. Only the external surface of the reactor is considered--no depth of the plasma volume is considered (it means that possible streamers are seen as spots on the outer surface).  
Figure 4: Initial image, acquired by the camera.  
Figure 5: Initial cropped image (**a**), and grayscale transformed (**b**).  
Starting from the captured image, the goal of the geometrical transformation is to build the projected output image (Figure 6d): it is the rollout of the DBD reactor external surface. The height ratio between the captured image (Figure 6c) and the projected one (Figure 6d) is therefore equal to \(\pi/2\). The width is the same for both images and equal to the length L of the reactor.  
In order to build Figure 6d, each M(y) point of the acquired image is used to define the \(m(y_{p})\) point at vertical position \(y_{p}\); if a streamer appears on the original picture as a spot with \(\Delta z\Delta y\) size, on the transformed image it is spanned into \(\Delta z\Delta y_{p}\) with:  
\[y_{p}\ =\ R\times\theta\ =\ R\times\sin^{-1}\Bigl{(}\frac{y}{R}\Bigr{)} \tag{2}\]  
\[\Delta y_{p}\ =\ \frac{\Delta y\times R}{\sqrt{R^{2}-y^{2}}}\ =\ \frac{\Delta y}{\cos( \theta)} \tag{3}\]  
Equation (2) is obtained by a basic trigonometric manipulation: the length \(y_{p}\), which is also a circle segment, is equal to the radius \(R\) multiplied by the \(\theta\) angle. The ratio \(\Delta y_{p}/\Delta y\) from Equation (3) is a first-order approximation obtained with first y-derivative of Equation (2). Using these equations, the image is projected to cylindrical coordinates and rolled out. This process is repeated along the Z axis in all planes (X, Y) of the reactor to obtain the full picture. Figure 7b shows the obtained image in cylindrical coordinates.  
Figure 6: Principle of the geometrical transformation (projection): (**a**) 3D view of the reactor and the camera. (**b**) Orthogonal cross-section of the reactor. (**c**) Acquired picture. (**d**) Transformed picture.  
Figure 7: Projected unrolled image of the outer surface of the reactor: (**a**) Acquired grayscale image—Cylindrical coordinates output image (**b**). Red rectangle highlights the inner tube of the reactor.