NURBS procedures in reverse engineering.

Ivanov, Marian ; Pupaza, Cristina

1. INTRODUCTION

Reverse Engineering (RE) methodologies are fairly mature. However, one of the most important issues is still how to adequately define and manipulate surface details, in order to accurate represent and easily modify the original shape.

NonUniform Rational Spline (NURBS) are the ultimate RE output for CAD-CAM-CAE applications. These surfaces can be constructed based on the CAD entities from the curve phase or by using polygon meshes for surface fitting. The main objective of the NURBS surface definition phase in RE is to prepare a patch structure of quadrangular shapes for surface construction and for further utilization of the model. The patches can be drawn on the polygon model manually, semi-automatically, or automatically based on a target patch count and the curvature of the model. Although most RE software packages provide an automatic method to generate patch structures quickly, manual organization of the patch structure is always required to obtain the optimal layout, because, in many cases features such as edges and primitives cannot be recognized and patches do not adequately define the surface curvature (Raja & Fernandes, 2008). Examples of multilevel approximation have been reported (Moller, 2004), focusing on virtual reality simulations. Recent papers present an algorithm of NURBS surface fitting for reverse engineering (Dan & Lancheng, 2006), but the algorithm is computational intensive and the model has to be divided. Each patch has to be digitized separately, which is not the best choice. Many RE challenges are in fact modeling capabilities needed to process the 3D captured data. At present these functionalities are already implemented in advanced CAD systems (Lombard, 2008) and the designer has to use them in a rapid and efficient manner. The present paper deals with new procedures implemented in CAD systems used in processing RE models. A human face model was recovered and details were improved using NURBS technology in SolidWorks 2008.

2. SHAPE REPREZENTATION USING NURBS

NURBS are mathematical representations of 2D or 3D objects, which can be standard shapes or free-form shapes. Advantages of using NURBS representation are: * flexible design to a wide range of shapes; * compact expressions that can be evaluated and displayed quickly; * useful especially in 3D modelling, allowing the designer to easily manipulate control vertices and control curvature and the smoothness of contours; * defined from little data; * used to represent exactly the conic curves, with the added advantage of local control; * easy and stable evaluators, breaking and joining, degree elevation, and affine invariance.

NURBS enables local modification. Changing a control point affects the shape of the curve only in the neighbourhood of that point. The actual range of the change depends on the degree used. NURBS lets the CAD system to interpolate a large number of points while maintaining a low-degree basis function. This helps eliminate twists and spikes that often result from using high-degree polynomials (Piegl & Tiller, 1997). NURBS are generally characterized by four features: control points, weights, degree, and knot vector (Fig. 1). NURBS curve C(u), which is a vector-valued piecewise rational polynomial function, is defined as (Altmann, 2005)

C(u) = [n.summation over (i=0)] [w.sub.i] [P.sub.i] [N.sub.i,k] (u)/[n.summation over (i=0)] [w.sub.i] [N.sub.i,k] (u) (1)

where [w.sub.i] are weights, Pi--control points (vector), [N.sub.i,k]--normalized B-spline basis functions of degree k. These B-splines are defined recursively as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [t.sub.i] are the nodes forming a knot vector U{[t.sub.0], [t.sub.1], [t.sub.m]}. The definition of the NURBS curve (1) can be rewritten using rational basis functions [R.sub.i,k] (u) (4).

C(u) = [n.summation over (i=0)] [P.sub.i] [R.sub.i,k] (u) (4)

The NURBS surface is defined in a similar way. NURBS are actually regarded as a standard way to create and represent complex objects and free form shapes.

[FIGURE 1 OMITTED]

3. MODEL RECOVERING AND IMPROVEMENT

The point cloud model of a scanned human face was transformed into a 3D image file created in Virtual Reality Modeling Language and saved in a WRL--ASCII text format. This file contains only the coordinates of the measured points and viewpoint coordinates for the initial view of the model. The most important operations for optimizing the file were noise reduction and gap cleaning, abnormal surface cleaning, polygon and mesh refinement and polygon mesh decimation.

3.1 Mesh recovering and cleanup

A visual inspection of the model was proceeded before mesh preparation and cleanup. Figure 2 shows the WRL format, which was transformed into a polygon model.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

All holes were automatically detected and filled. In order to reduce the dimensions of the model and to emphasize the modelling procedure some simplifications were done. This helped speeding up surface generation of the model, which contains no symmetrical features. Individual submeshes were identified by colors (Fig. 3). The sensitivity was adjusted to obtain a desired submesh separation, which was improved in a later stage.

3.2 Surface extraction and solid model generation

After geometry recovering and mesh refinement the NURBS patches were obtained. When generating patches guided creation of the faces was chosen to better control the shapes. Editing of the patches was needed because the nose of the model proved to be distorted.

The geometry and topology of the model were checked with 'zebra stripes' option, in order to find the face imperfections, difficult to detect otherwise. This time no irregularities or breaks in the model were found. Figure 4 shows stripe checking options and editing of the patches. New curves and control points were added on the NURBS surfaces in order to allow improving local details. The geometry of the model was updated by means of new parameter definition and interactively changing control points. When changing the local shape of the model tangency constraints were imposed (Fig. 5). Surface extraction converted all the regions into faces. When the base model was completed and the surfaces were turned into a solid, the model was saved in an STEP format.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

4. CONCLUSION

Most RE applications need at least two software packages to complete the data processing chain: one for RE data manipulation and the other for the geometric modeling work, which is based on CAD entities or NURBS surfaces. This attempt proved that advanced modeling technologies help the designer to reduce the processing time and close the gap between traditional RE packages and CAD systems. The procedure was tested on free form shapes, but it is useful for mechanical components with complex geometries as well. The designer has to carefully check that the geometry of the model remained unchanged, controlling the number of control points of the grids when constructing NURBS patches, although automatic model recovering and cleanup functionalities are user friendly. Further work is in progress for retrieving and editing the parameter definition of the surfaces in order to integrate the procedure in an optimization loop.

5. REFERENCES

Altmann, M. (2005). About Nonuniform Rational B-Splines NURBS. Available from: http://www.cs.wpi.edu/~matt/ courses/cs563/talks/nurbs.html. Accessed: 2005.05.05

Dan, J.; Lancheng, W. (2006). An Algorithm of NURBS Surface Fitting for Reverse Engineering, Int. Journal of Advanced Manufacturing Tech., Springer-Verlag, DOI 10.1007/ s00170-005-0161-3, No. 31, pp. 92-97, London

Lombard, M. (2008). SolidWorks. Surfacing and Complex Shape Modelling. Wiley Publishing Inc., ISBN 978-0-470-25823-1, Indianapolis, Indiana

Moller, D.P.F. (2004). Computational Modeling an Simulation for Industry. Proceedings of the 16th European Simulation Symposium, Gyorgy Lipovszki, Istvan Molnar (Ed.), SCS Press, ISBN 1--56555-286-5.

Piegl, R.; Tiller, W. (1997). The NURBS Book, Springer-Verlag, ISBN 3-540-61545-8, Berlin Heidelberg

Raja, V.; Fernandes K.J. (2008). Reverse Engineering. An Industrial Approach, Springer-Verlag, ISBN 978-1-84628855-5, London

IVANOV, Marian & PUPAZA, Cristina *