Line Geometry for 3D Shape Understanding and Reconstruction Helmut Pottmann, Michael Hofer, Boris Odehnal, and Johannes Wallner Technische UniversitÄat Wien, A 1040 Wien, Austria. fpottmann,hofer,odehnal,[email protected] Abstract. We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on estimated surface normals we use approximation techniques in line space to recognize and reconstruct rotational, helical, developable and other surfaces, which are character- ized by the con¯guration of locally intersecting surface normals. For the computational solution we use a modi¯ed version of the Klein model of line space. Obvious applications of these methods lie in Reverse Engi- neering. We have tested our algorithms on real world data obtained from objects as antique pottery, gear wheels, and a surface of the ankle joint. Introduction The geometric viewpoint turned out to be highly successful in dealing with a variety of problems in Computer Vision (see, e.g., [3, 6, 9, 15]). So far mainly methods of analytic geometry (projective, a±ne and Euclidean) and di®erential geometry have been used. The present paper suggests to employ line geometry as a tool which is both interesting and applicable to a number of problems in Computer Vision. Relations between vision and line geometry are not entirely new. Recent research on generalized cameras involves sets of projection rays which are more general than just bundles [1, 7, 18, 22]. A beautiful exposition of the close connections of this research area with line geometry has recently been given by T. Pajdla [17]. The present paper deals with the problem of understanding and reconstruct- ing 3D shapes from 3D data. The data are assumed to be of a surface-like nature | either a cloud of measurement points, or another 3D shape representation such as a triangular mesh or a surface representation in parametric or implicit form | and we assume that we are able to obtain a discrete number of points on the surface and to estimate surface normals there. We are interested in classes of surfaces with special properties: planar, spherical and cylindrical surfaces, sur- faces of revolution and helical surfaces; and the more general surface classes of canal, pipe, and developable surfaces. For applications in CAD/CAM it is essen- tial that such special shapes are not represented by freeform surfaces without regard to their special properties, but treated in a way more appropriate to their `simple' nature. Line geometry enters the problem of object reconstruction from point clouds via the surface normals estimated at the data points. In fact, modern 3D pho- tography and the corresponding software delivers such normals together with 2 Helmut Pottmann et al. the data points. It turns out that the surface classes mentioned above can be characterized in terms of their surface normals in an elegant, simple and compu- tationally e±cient way. Appropriate coordinates for lines (which yield a model of line space as a certain 4-dimensional manifold embedded in R6) will allow to classify point clouds and their normals (i.e., the so-called `normal congruence') by means of tools such as principal component analysis. Previous work There is a vast body of literature on surface reconstruction. Since we are interested in the reconstruction of special surfaces, we do not review the part of literature which deals with the reconstruction of triangular meshes or general freeform representations. In Computer Vision, recognition and recon- struction of special shapes is often performed by methods related to the Hough transform. Originally designed for the detection of straight lines in 2D, it re- ceived much attention and has been generalized so as to be able to detect and reconstruct many other shapes (see, e.g., [11, 14]). Pure Hough transform meth- ods work in `spaces of shapes' and quickly lead to high dimensions and reduced e±ciency. In order to avoid these problems, such tools are sometimes augmented by methods from constructive geometry. This approach is already close to tech- niques invented by the CAD community, which use geometric characterizations of surfaces (e.g. by means of the Gaussian image) for data segmentation and the extraction of special shapes (see the survey [24]). Many papers deal with axis estimation of rotational surfaces, like [25]. See [8] for an overview on the Hough transform, the RANSAC principle, and the least squares approach. In the present paper, however, rotational surfaces occur only as a special case. The use of line geometry for surface reconstruction has been introduced by [20]. There cylinders, surfaces of revolution and helical surfaces are recognized by the fact that their surface normals are contained in a so-called linear line complex. In particular surfaces which can be moved within themselves in more than one way (right circular cylinders, spheres, planes) are detected. The tech- nique is extendable to surfaces which may be locally well approximated by the surface types mentioned above [2, 13, 21]. Contributions of the present paper Inspired by the line geometric work on reverse engineering of special shapes, our paper presents a broader line geometric framework for the solution of problems in 3D shape understanding, segmentation and reconstruction: We discuss a point model for line space, namely a certain 4-dimensional al- ² 4 6 gebraic manifold M of order 4 in R . It is better suited for line geometric approximation problems than the classical Klein model or the model used in [20], which is limited to linear line complexes. This point model makes it possible to perform the basic shape recognition tasks² via principal component analysis (PCA) of a point cloud (contained in M 4), which represents the estimated surface normals of the input shape. This procedure is further improved here and unlike [20] is stable in all special cases. The idea of looking for surface normals which intersect makes it possible to apply² line-geometric methods to the problem of recognition and reconstruction of canal surfaces (which are the envelope of a one-parameter family of spheres), Line Geometry for 3D Shape Understanding and Reconstruction 3 and of moulding surfaces. The latter denotes a certain class of sweep surfaces which contains the developable surfaces and the pipe surfaces. The segmentation of composite surfaces into their `simple' parts is addressed here² in so far as recognition of surface type is essential for segmentation. Our algorithms may be included as a `black box' into a segmentation algorithm. 1 The 4-dimensional manifold of lines in R6 This paragraph discusses a computationally attractive point model of (i.e., co- ordinates in) the 4-dimensional manifold of straight lines in space. It is closely related to the classical Klein quadric (i.e., PlucÄ ker coordinates). We think of an oriented line L as one equipped with a unit vector l indicating the direction of the line | so that there are two oriented lines for each line of space. Then L is determined by l and the moment vector ¹l, which is computed by means of an arbitrary point x on L as ¹l = x l. ¹l is independent of the choice of x on L. The six numbers (l;¹l) are called normalize£ d PluckerÄ coordinates of L. `Normalized' means that l = 1; further, they satisfy the orthogonality condition l ¹l = 0. k k 3 ¢ Conversely, any two vectors l;¹l R which satisfy these two conditions deter- 2 3 mine a unique oriented straight line L in R , which has (l;¹l) as its normalized PluckÄ er coordinates. If we do not distinguish between the two opposite orientations of the same line, we may use all multiples of the pair (l;¹l) as coordinates of a line. Of course, we still have the condition l ¹l = 0. Such homogeneous coordinate vectors of lines represent those points of ¯v¢e-dimensional projective space which are contained 4 4 in the Klein quadric M2 given by the equation (x; x¹) M2 x x¹ = 0. This interpretation of lines is well studied in classical geometry2 , see,[21].¢ The present paper pursues the following approach, which is closely related to the Klein quadric. We use only normalized coordinate vectors, and so we identify an oriented line with the point (l;¹l) in six-dimensional Euclidean space R6. In this way we obtain a mapping ® of oriented lines to points of a 4-dimensional manifold M 4 R6. M 4 is algebraic of degree 4, and is the intersection of the cylinder Z5 and½ the cone ¡ 5 de¯ned by Z5 : x2 = 1; ¡ 5 : x x¹ = 0: ¢ We use the Euclidean distance of points in R6 in order to measure distances between oriented lines G; H: If G® = (g; g¹) and H® = (h; h¹), then d(G; H)2 = (g h)2 + (g¹ h¹)2: (1) ¡ ¡ The reasons why we prefer this distance function are the following: On the one hand, it is quadratic and thus lends itself to minimization. On the other hand, in a neighbourhood of the origin of the coordinate system, (1) models a distance of lines which is in accordance with visualization. The slight drawback that this is no longer the case in regions far away from the origin is in fact not important in most applications, as such applications very often de¯ne a natural region of interest, where we can put the origin of our coordinate system. 4 Helmut Pottmann et al. Remark 1. Another method for introducing coordinates and measuring distances for lines, which has been used in the past, is to ¯x two parallel planes and describe a line by the two intersection points with that plane (cf. [21]). This leads to simpler formulae and a region of interest which is bounded by the two ¯xed planes (in our case, this region is a sphere centered in the origin).