Eurographics/ IEEE-VGTC Symposium on Visualization 2010 Volume 29 (2010), Number 3 G. Melançon, T. Munzner, and D. Weiskopf (Guest Editors) A Maximum Enhancing Higher-Order Tensor Glyph T. Schultz and G. Kindlmann Computer Science Department and Computation Institute, University of Chicago, USA — {t.schultz,glk}@uchicago.edu Abstract Glyphs are a fundamental tool in tensor visualization, since they provide an intuitive geometric representation of the full tensor information. The Higher-Order Maximum Enhancing (HOME) glyph, a generalization of the second-order tensor ellipsoid, was recently shown to emphasize the orientational information in the tensor through a pointed shape around maxima. This paper states and formally proves several important properties of this novel glyph, presents its first three-dimensional implementation, and proposes a new coloring scheme that reflects peak direction and sharpness. Application to data from High Angular Resolution Diffusion Imaging (HARDI) shows that the method allows for interactive data exploration and confirms that the HOME glyph conveys fiber spread and crossings more effectively than the conventional polar plot. Categories and Subject Descriptors (according to ACM CCS): Computer Graphics [I.3.5]: Curve, surface, solid, and object representation—Computer Graphics [I.3.8]: Applications— 1. Introduction Due to their multivariate nature, tensors cannot be fully rep- resented as grayscale or color values. Therefore, glyphs are the standard tool to inspect the full information of individual tensors. There exists a considerable choice of second-order tensor glyphs for various applications: Specialized glyphs emphasize different aspects of stress and strain tensors [HYW03]. In Diffusion Tensor Magnetic Resonance Imag- ing (DT-MRI), ellipsoids are the most traditional type of glyph [BMLB94], but more advanced glyphs based on com- ∗ Figure 1: Polar plots (a) make it difficult to see the shape posite shapes [WMK 99] or superquadrics [Kin04] have and orientation of peaks, which is readily revealed by our been constructed to overcome visual ambiguities that oc- HOME glyphs (b). In applications like HARDI, algebraic cur when projecting ellipsoids to the two-dimensional image surfaces (c) are counter-intuitive. space. Similar glyphs have been used to illustrate the tensor voting process [TTMM04] or selected properties of the Hes- sian near crease features [KSJESW09], and to visualize the nematic liquid crystal alignment tensor [JKM06]. Angular Resolution Diffusion Imaging (HARDI), it has been referred to as a “parametrized surface” [ÖM03], the In contrast, a single glyph dominates the visualization “higher-order tensor glyph” [HS05], or the “HARDI glyph” of higher-order tensors: namely, the surface whose distance [PPvA∗09]. To differentiate it from our novel glyph, we from the origin in each direction equals the value of the adopt the name “spherical polar plot” [Tuc04]. homogeneous form in that same direction (Figure 1 (a)). Despite its widespread use, this glyph lacks an established As shown in Figure 1, the polar plot makes it difficult name. It can be considered a generalization of the Reynolds to examine maxima of the homogeneous form: A perfectly glyph [HYW03], but that name refers to the particular ap- symmetric peak (left column) looks similar to an anisotropic plication to Reynolds stress tensors. In the context of High one, which is sharper in some directions than in others, and c 2010 The Author(s) Journal compilation c 2010 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. T. Schultz and G. Kindlmann / A Maximum Enhancing Higher-Order Tensor Glyph different orientations of anisotropic peaks are difficult to dis- and vice versa (Figure 1 (c)), they are counter-intuitive in cern (middle and right column). However, such differences applications like HARDI. play an important role in applications like HARDI, where The commonly used tensor ellipsoid, whose half-axes are they indicate anisotropic fiber spread. aligned with the eigenvectors and scaled with the eigenval- To overcome such problems, Schultz et al. [SWS09] ues, is obtained by replacing T in Eq. (1) with its squared − recently proposed the Higher-Order Maximum Enhancing inverse T 2. Özarslan and Mareci [ÖM03] point out that (HOME) glyph, which generalizes the second-order ten- due to the lack of a suitable inverse, this formulation cannot sor ellipsoid. When applied to two-dimensional higher-order be transferred to higher-order tensors. Therefore, this paper tensors that arise in image processing, it has been shown to takes a different route to generalizing the tensor ellipsoid, depict maxima more clearly. In this paper, we turn this def- which was first proposed by Schultz et al. [SWS09]. inition into a more general visualization tool by clarifying its mathematical properties and the conditions under which 3. Generalizing the Second-Order Tensor Ellipsoid it can be used. We also present an optimized implemen- tation that provides an efficient framework for interactive 3.1. Notation exploration of three-dimensional higher-order tensors (Fig- The coefficients Ti1i2...il of an order-l tensor T with respect ure 1 (b)). Finally, we support identification and localization to a given basis are indexed by l numbers. In this paper, we of peaks by a new coloring scheme and present novel appli- are concerned with totally symmetric tensors, whose coeffi- cation examples on HARDI data. cients are invariant under arbitrary index permutations. The remainder of this paper is organized as follows: Af- The inner product T · v between an order-l tensor T and ter reviewing related work in Section 2, we introduce the a vector v produces a tensor T of order (l − 1): Higher-Order Maximum Enhancing glyph and discuss its e T = T v (2) formal properties in Section 3. In Sections 4 and 5, we treat i1i2...il−1 i1i2...il il e coloring by peaks and address implementation issues, re- In Eq. (2), the Einstein summation convention implies spectively. Finally, applications to HARDI data are shown in summation over all values of the repeated index il. Perform- Section 6, before Section 7 concludes the paper and points ing the inner product l times (T ·l v) produces a scalar. This out possible directions of future research. mapping is called the homogeneous form F(v) of the tensor: F(v)= T ·l v = T v v ···v (3) 2. Related Work i1i2...il i1 i2 il Several previous authors have tried to visually emphasize There is a one-to-one mapping between symmetric ten- maxima in the polar plot: Tuch [Tuc04] visualizes a trans- sors and their homogeneous forms [CM96], so the full ten- formed tensor to emphasize peaks, while trying to avoid ex- sor information is conveyed by plotting F(v). Traditionally, aggerating noise-related features. Hlawitschka and Scheuer- this is done using the polar plot mann [HS05] render the glyphs semi-transparently and add p(v)= F(v)v (4) arrows that point towards the maxima. Similarly, Descoteaux et al. [DDKA09] use lines to indicate maxima. In our work, When rewriting v in spherical coordinates, the contribu- the shape of the glyph itself is sharpened around the peaks, tion of the radius r can be separated from the angular part making them visually more prominent. Perception of the f (θ,φ) of the homogeneous form: number and direction of maxima is further improved by a ˆ l new coloring scheme. F(v(r,θ,φ)) = f (r,θ,φ)= r f (θ,φ) (5) Therefore, it is sufficient to plot p(v) for vectors on the Other works have concentrated on interactive rendering ∗ of the polar plot: Shattuck et al. [SCB∗08] precompute a unit sphere (kvk = 1). Some authors (including [PPvA 09, set of multi-resolution slice images suitable for exploration DDKA09]) prefer to express f (θ,φ) in a spherical harmon- over the internet. Peeters et al. [PPvA∗09] present a GPU- ics basis of order l, with odd-order coefficients set to zero. based ray casting approach. We found that, alternatively, a That basis defines the same space of functions on the sphere relatively simple geometry-based approach allows for inter- as symmetric tensors, and the relation between the respective active data exploration. coefficients is linear [ÖM03]. In this paper, we rely on tensor notation, since it lends itself more easily to a generalization A textbook by Strang [Str98] suggests to visualize a sym- of second-order tensor ellipsoids. metric second-order tensor T by the quadratic surface T {v|v Tv = 1} (1) 3.2. Generalizing Positive Definiteness Corresponding algebraic surfaces have been defined from The second-order tensor ellipsoid is only applied to positive higher-order tensors by Qi [Qi06]. However, since maxima definite tensors. A symmetric second-order tensor T is posi- in the homogeneous form lead to minima in these surfaces tive definite if its homogeneous form F(v) > 0 for all v 6= 0. c 2010 The Author(s) Journal compilation c 2010 The Eurographics Association and Blackwell Publishing Ltd. T. Schultz and G. Kindlmann / A Maximum Enhancing Higher-Order Tensor Glyph Figure 2: Sharp peaks can cause self-intersections in the Figure 3: The polar plot p(v) preserves the angles between HOME glyph (b) even if the homogeneous form is positive, vectors v that are distributed uniformly over the unit circle as revealed by the polar plot (a). This problem cannot occur (a). In the tensor ellipse t(v) and the HOME glyph h(v), the if the tensor has a positive rank-1 decomposition. vectors are deflected towards maxima. Alternatively, positive definiteness is defined by the fact that h(v) generalizes Eq. (8) to symmetric tensors of even order all eigenvalues λi that occur in the spectral decomposition l. It transforms the unit circle under the mapping given by n (l − 1) applications of the inner tensor-vector product: T = λiei ⊗ ei (6) ∑ l−1 i=1 h(v)= T · v with kvk = 1 (9) are larger than zero.