In order to generate an accurate model of the anatomical region to be studied, it is necessary to be able to work on medical images obtained from either computed tomography or magnetic resonance, such that one can correct many aspects susceptible to occur during acquisition e. In this context, the open-source software InVesalius  is a proper option to help in this entire process through its edition tools. InVesalius is a free software under GNU General Public License, allowing researchers and developers from universities and research centers from all over the world to contribute with the improvement of it.
Until now, more than 10, people from countries are active users of the software. Written mainly in Python programming language, the input data to InVesalius consist of medical image files, either in Digital Imaging and Communications in Medicine DICOM or in Analyze file formats, originated from computed tomography and magnetic resonance imaging.
Many libraries are also implemented to give support to the features offered by the software, such as wxPython for the graphical interface and Visualization Toolkit VTK  for 2D and 3D visualization of the images and surfaces. Several features are offered to the users of InVesalius. The highlights among them include image segmentation, where the user can automatically segment an image by a threshold value and then refine the results manually; 3D surface reconstruction through the Marching Cubes algorithm  ; volumetric visualization generated by Ray Casting technique  ; and measurements, both linear and angular, on the 2D slices and 3D surfaces.
Figure 1 illustrates these main features of InVesalius tool respectively, top left, top right, second row and third row. These features allow the user to make analysis of virtual anatomical models, from which physical models can be printed with the aid of rapid prototyping, giving the medical community a reliable instrument to help planning surgeries. In order to offer the user more control over the model to be printed, the new feature described in this paper and available in a future version of InVesalius extracts the point cloud of the input surface, reconstructs it through NURBS parameterization   and exports the results to a STEP file, allowing further edition through any Computer Aided Design CAD software.
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NURBS is an acronym to Non-Uniform Rational B-Splines and was introduced to the Computer Graphics community after studies of several works on the physical properties of piecewise polynomials known as splines . As an extension of B-splines, NURBS functions share many of their relevant properties, mainly the continuity and local support ones, permitting the former a smoother representation of curves and surfaces and the latter, more control over changes on the forms resulting from alterations on the parameters values.
Parameterization by means of NURBS functions allows one to obtain a flexible and versatile modeling of curves and surfaces, being capable, therefore, to represent either simple curves and surfaces, as conics, or those more complex, such as free-form ones. Once modeled by NURBS, the curve or surface offers a malleability in the representation of its form via the rearrangement of the control points or reassignment of the weight values.
For a set of control points on the plane, with , a B-spline curve of order or, equivalently, of degree k is given by the formula. Figure 1. Some InVesalius features.
35 CURVE AND SURFACE RECONSTRUCTION
From Equation 2 , it is straightforward to see that the basis function is nonzero only on a finite number of subintervals, which are determined by knots belonging to A, a property known as local support. As the control point is multiplied by the basis function in Equation 1 , moving the former only affects the curve shape on the subintervals where the latter is nonzero.
Thus, knot values establish to what extent the control points affect the shape of the curve. A B-spline surface of order is defined by. By introducing weights to the control points, a intrinsically non-rational B-spline becomes rational and provides a way of adjusting the shape of the form without adding excessively new control points. Equations 1 and 3 change into.
Therefore, after reconstructing the surface as a NURBS one, the user is able to modify the shape of it to meet his needs, just altering the positions or weights of the control points. Furthermore, the local support property inherited from the B-splines ensures that any change to those parameters affects only a local region of the surface, and not the whole of it, giving the user more control over the adjustment of his region of interest.
Since one of the goals of our work was to allow the user to reopen the surface generated by InVesalius in any CAD software, AP was chosen as the file type to export the reconstruction results. Primarily, the algorithm accepts, as inputs, point clouds stored in Polygon File Format PLY or StereoLithography STL file types, the first for being most commonly used to store tridimensional data obtained from 3D scanners, the second because of its widely usage for rapid prototyping and 3D printing, and both for being supported by almost every software package known nowadays, including InVesalius.
After reading the input file, the algorithm extracts the data and opens a visualization window with the point cloud that was read. Once the visualization window is open with the input, the NURBS reconstruction can be executed to the entire surface or to a specific point set.
go to site In the latter case, a keyboard shortcut activates the selection cursor, allowing the user to draw a rectangle over the region of interest. Done this procedure, the script changes the content of the visualization window to show the set of points selected. In case the result is not as expected, the user can switch back to the original point cloud and make a new selection. Being the selection as anticipated, the algorithm saves it to an output file with the same format of the input.
The reconstruction step, either executed on the entire point cloud or on the selected region, is automatic. A new coordinate system, centered at the mean of the points of the cloud to be reconstructed, is created from the main eigenvectors obtained by PCA, pointing them into the directions of the maximum, middle and minimum extension of the point cloud. Also, a bounding box is estimated in the plane comprised by the maximum and middle eigenvectors and used to determine the extension of the NURBS domain. With all this, the NURBS surface is initialized with a minimal number of control points, according to the degree specified.
To improve the quality of the surface, refinements procedures are iteratively executed on it.
The strategy behind them is to add more control points to the initial surface by knot insertion, in order to increase flexibility in shape control. This can be done because, in any direction of the surface, the number of control points CP is tied to the number of knots K by the relation.
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So, in each iteration of the refinement, a knot is inserted in the middle of each knot direction and the NURBS surface with this new configuration is fitted to the point cloud, by minimizing, using least squares minimization, the system of equations derived from the difference between each point in the cloud and its associated point on the NURBS surface.
MPU implicit functions are fit to the input contours, defined as binary images, to produce smooth curves with controllable error bounds.
Full 2D Euclidean distance fields are then calculated from the implicit curves. A distance-dependent Gaussian filter is applied to the distance fields to smooth their medial axis discontinuities. Monotonicity-constraining cubic splines are used to construct smooth, blending slices between the input slices. A mesh that approximates the zero isosurface is then extracted from the resulting volume. The effectiveness of the approach is demonstrated on a number of complex, multi-component contour datasets. The table of contents of the conference proceedings is generated automatically, so it can be incomplete, although all articles are available in the TIB.
Surface Reconstruction From Non‐parallel Curve Networks
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