A systematic approach to nD orientation representation Bernd Rieger * , Lucas J. van Vliet Pattern Recognition Group, Department of Applied Physics, Delft University of Technology, Lorentzweg 1, NL 2628, CJ Delft, The Netherlands Received 17 July 2002; received in revised form 14 November 2003; accepted 18 November 2003 Abstract In this paper, we present new insights in methods to solve the orientation representation problem in arbitrary dimensions. The gradient structure tensor is one of the most used descriptors for local structure in multi-dimensional images. We will relate its properties to the double angle method (2D) and the Knutsson mapping. We present a general scheme to reduce the dimensionality of the Knutsson mapping and derive some properties of these reduced mappings. q 2003 Elsevier B.V. All rights reserved. Keywords: Gradient structure tensor; Knutsson mapping; Continuous orientation representation; Local structure 1. Introduction We define simple neighborhoods in images as areas that are shift invariant in at least one direction and not shift invariant in at least one other direction. Such areas play a key role in the description of local structure. The aforementioned shift directions can be determined up to point inversion. Therefore, a pair of opposite directions is designated by a single orientation. The first order intensity variation is the gradient. A collection of local gradients is needed to compute a dominant orientation. The accompanying intensity vari- ations and that of orthogonal directions can be used to describe lines, surfaces and edges as well as texture. A characterization of simple neighborhoods is by the domi- nant orientation [1,2,7,10]. Orientation is direction up to point inversion, therefore, leaving room for ambiguity in representation. Direction is described by the full angle, which is in 2D characterized by q [ ½0; 2p; and in general by n 2 1 angles in nD. Direction can also be represented by vectors, for example v ¼ð1; 2Þ or w ¼ð21; 22Þ: These vectors point in opposite directions but have the same orientation. Representing orientation by vectors (direction infor- mation) leads to troublesome descriptions, in the sense that it is discontinuous. Representing a line in 2D by its angle with respect to a fixed coordinate axis and a plane in 3D by its normal vector is, therefore, not a suitable representation. In Fig. 1, a test and a real 2D image are shown on the left and the orientation fields on the right. We clearly see two jumps or discontinuities. They cannot be removed, for example, by phase unwrapping [5,14,20]. Phase unwrapping can only successfully be applied to discontinuities that form non- intersecting closed lines. A consistent definition of direction is only possible in a global frame work, whereas most image operators are bound to a local neighborhood. The heart of the problem is sketched in Fig. 2. The support of the operator may have a size as indicated by the circle. The scale is local, whereas the structure has a global connection. The content of the upper and lower window are the same, although the outward pointing normal vector changes continuously along the line. So we are left with two identical windows and an estimated normal orientation with a discontinuity. The gradient vectors in a local neighborhood need to be combined to obtain an estimate of the local orientation. A simple averaging of gradient vectors fails, on lines in 2D (planes in 3D, etc.) because vectors from opposite sides of the line point in opposite directions and will cancel each other. Thus, we need a suitable continuous representation of gradient vectors to average the structure inside a local window. Furthermore, a discontinuous representation is very often not suitable for further processing. Most image 0262-8856/$ - see front matter q 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2003.11.005 Image and Vision Computing 22 (2004) 453–459 www.elsevier.com/locate/imavis * Corresponding author. Tel.: þ31-015-278-60-54; fax: þ 31-015-278- 67-40. E-mail addresses: bernd@ph.tn.tudelft.nl (B. Rieger), lucas@ph.tn. tudelft.nl (L.J. van Vliet).