# Wavelia Breast Imaging: The Optical Breast Contour Detection Subsystem
## 2 Materials and Methods
### Optical Breast Contour Detection (OBCD) Subsystem: The 3D Breast Surface Reconstruction Procedure
#### 2.3.2 Denoising Procedure
These are defined in \(P_{C}\) by using the _z limit_ of the nipple section (see Figure 18). To apply the point removing criterion, the following parameters are defined:  
* \(z_{min}\) and \(z_{max}\): these values correspond to the minimum and maximum z coordinates of the bounding box of \(P_{C}\). * \(z_{nipple}\): this parameter concerns the z limit of the nipple section (see Figure 18). Its value is computed by using Equation (16), where \(Dz_{nipple}\) is an empirically defined value within the range 0 to 1 based on tests performed on human and phantom breasts. \[z_{nipple}=\ z_{min}+\ (z_{max}-z_{min})\cdot Dz_{nipple}\] (16)
* _valid distance:_ this parameter is applied to the points outside the nipple section. It corresponds to the maximal signed vertical distance to the reference rectangular grid for the denoising operation to preserve points corresponding to the breast surface. * _valid distance for nipple section:_ this parameter is applied to the points inside the nipple section, where the nipple points are above the breast surface. It corresponds to the maximal signed vertical distance to the reference rectangular grid for the denoising operation to preserve points corresponding to the nipple and areola vicinity on the breast surface. Figure 18: Nipple section (magenta points) in the not stretched point cloud (left-hand images) and in the stretched point cloud \(P_{C}\) (right-hand images): (**a**) point cloud corresponding to Patient 002; (**b**) point cloud corresponding to Patient 004. The removing criterion is applied to \(Pc_{C}\) by using \(z_{\mathit{nipple}}\), _valid distance for nipple section_ and _valid distance_. The resulting point cloud, after denoising, is named \(Pc_{D}\). Figure 19a,b illustrates the denoising process applied to the data from two patient scans. In Figure 19, two sample zones (yellow delimited zones) are highlighted before the denoising procedure (\(Pc_{C}\)) and after the denoising procedure (\(Pc_{D}\)). Observing the impact of the denoising procedure on these zones permits to clearly illustrate the efficacy of the denoising procedure to significantly reduce the noise. When the first artifacts removal process is finalized, \(Pc_{D}\) is transformed into a full filled rectangular grid \(Pc_{A}\). This point could \(Pc_{A}\) can be considered as a 2D image which allows using 2D imaging processing. To generate the full filled rectangular grid \(Pc_{A}\) from \(Pc_{D}\), the size of the rectangular grid is defined in the x and y axes by using Equations (14) and (15). Then, a rectangular grid is generated on the xy plane. To improve the robustness against the noise along the z axis, each of the z values of the grid (green points in Figure 20a) is computed as the median value of all the z coordinates of the points located inside the associated small rectangular section (small blue rectangles in Figure 20a). Some points of the rectangular grid will remain with no defined z value. These points are named \"empty points\", and the sections of empty points are named \"holes\" (see red points in Figure 20b). The points with a defined z value are named \"valid points\" (see green points in Figure 20b). Figure 19: Denoising step by using the _reference rectangular grid for denoising_. Two yellow delimited zones are shown in \(Pc_{C}\) (before denoising) and \(Pc_{D}\) (after denoising): (**a**) point cloud corresponding to Patient 002; (**b**) point cloud corresponding to Patient 004. To define the z coordinate of the empty points in the rectangular grid (the process to fill the holes), the parameter _ring height_ defined in Appendix B.2 is required. By using this parameter, we divide the valid points of the grid (green points in Figure 21) in two vertical sections: the first one concerns the points the z coordinates of which are in range of 0 to _ring height_ (bright green points in Figure 21), and the second one concerns the points the z coordinates of which are outside the first interval (dark green points in Figure 21).