A Maximum Enhancing Higher-Order Tensor Glyph

Home , Glyph

Eurographics/ IEEE-VGTC Symposium on Visualization 2010 Volume 29 (2010), Number 3 G. Melançon, T. Munzner, and D. Weiskopf (Guest Editors)

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 represented 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 implementation 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. Here, the eigenvectors ei form an orthonormal basis of the underlying vector space, and both The polar plot is related to the HOME glyph through definitions of positive definiteness are equivalent. p(v) = (h(v) · v)v. Based on two-dimensional experiments in [SWS09], it has been conjectured that at stationary points Since tensors of odd order exhibit antipodal antisymme- of the homogeneous form (i.e., points at which the derivative try, F(−v)= −F(v), the notion of positive definiteness is f ′ vanishes), the following properties hold: limited to even orders. Similar to symmetric second-order tensors, symmetric higher-order tensors have a decomposi- 1. The polar plot and the HOME glyph coincide. tion into a sum of rank-1 terms [CGLM08]: 2. For both glyphs, the distance to the origin is extremal. r 3. The HOME glyph has sharper maxima, while minima are T = ∑ λi ei ⊗···⊗ ei (7) more pronounced in the polar plot. i=1 l factors Intuitively, this is explained by the observation that at non- | {z } stationary points, h(v) deflects the input vector v towards a In this case, the vectors ei can still be normalized to unit maximum in the homogeneous form. This is also the basis length, but they no longer need to be pairwise orthogonal. of the power method for finding the largest eigenvector of Consequently, a rank-1 decomposition with all λi > 0 is still a matrix, and its generalization to higher-order tensors with a sufficient, but no longer a necessary condition for a posi- a definite homogeneous form [KR02]. When implementing tive homogeneous form. Intuitively, negative terms can now the glyph by deforming a uniformly tesselated sphere, as be balanced by non-orthogonal positive ones and still create described in Section 5, it automatically concentrates mesh peaks which are so sharp that they cannot be expressed with vertices in the high-curvature regions around sharp peaks. positive terms alone. Figure 3 illustrates this in 2D, both for a matrix and a fourth- Therefore, in the higher-order case, a positive rank-1 de- order tensor. composition is a stronger notion of positive definiteness than To formalize these results, and to transfer them to three a positive homogeneous form. For the HOME glyph, we will dimensions, we express v in spherical coordinates to obtain impose this stronger condition. As illustrated by the self- a parametrization of the polar plot p and the HOME glyph h intersection of the 2D HOME glyph in Figure 2 (b), F(v) > 0 in terms of θ and φ. Let hθ and hφ denote partial derivatives is not sufficient to guarantee that the glyph is well-behaved. of this surface parametrization, fθ and fφ partial derivatives The reasons for this will become clear in the formal treat- of the angular part of the homogeneous form. The principal ment of the glyph properties in Appendix A. h h curvatures of h are given as κ1 and κ2. By the convention in [Top06], maxima have negative principal curvatures. 3.3. Definition and Properties of the Glyph With this, the three properties of the HOME glyph can be The standard ellipsoid t(v) for symmetric second-order ten- formalized as follows: If fθ(θ,φ)= fφ(θ,φ)= 0, sors T can be expressed as the transformation of the unit 1. p(θ,φ)= h(θ,φ). sphere under the linear mapping defined by T: 2. ∂kp(θ,φ)k/∂θ = ∂kp(θ,φ)k/∂φ = ∂kh(θ,φ)k/∂θ = t(v)= T · v with kvk = 1 (8) ∂kh(θ,φ)k/∂φ = 0. p 3. if f (θ,φ) 6= 0, fθθ(θ,φ) 6= 0, and fφφ(θ,φ) 6= 0, then κ1 > h p h The Higher-Order Maximum Enhancing (HOME) glyph κ1 and κ2 > κ2.

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

non-antipodal peaks might also receive the same color. We use XYZ-RGB despite this fact, since it is widely accepted by radiologists and domain scientists, and it can be shown that any coloring that avoids this problem will exhibit unde- Figure 4: Coloring based on surface partitioning (b) con- sirable discontinuities [PP99]. veys the number and direction of maxima more clearly The exact boundaries of the individual regions are less than simple rainbow coloring (a). An additional smoothing important, and sharp edges would draw undue attention to step (c) blurs the boundaries between regions. them (Figure 4 (b)). Therefore, we perform a small amount of smoothing to blur these transitions (Figure 4 (c)).

Establishing these properties ensures that the HOME 4.2. Modulation by Peak Sharpness glyph reproduces extrema of the homogeneous form, and that maxima are indeed sharper than in the polar plot. A for- In case of a second-order tensor T, XYZ-RGB colors are mal proof is provided in Appendix A. generally modulated by a measure of linearity in order to avoid indicating a principal direction when it is not clearly ∗ 4. Coloring by Peaks defined. One common measure is Westin’s cl [WMK 99], defined from sorted eigenvalues λ1 ≥ λ2 ≥ λ3: Since the full tensor information is expressed in the glyph λ1 − λ2 geometry alone, it is up to the individual application to use cl = (10) color or texture to emphasize specific properties of the tensor λ1 field, or to encode additional information. In general, the magnitude of the second derivatives is an In diffusion imaging, the degree of anisotropy and the ori- indicator of how well-localized a maximum of a function is. entation of the tensor are especially relevant, so they are fre- For an order-l tensor T , the Hessian H(v) of the homoge- quently used to determine the color of the glyph. It is cus- neous form F(v) at v is given as tomary to employ an XYZ-RGB color coding of the princi- l−2 pal direction, in which the spatial x, y, and z axes are mapped H(v)= l(l − 1)T · v (11) to the red, green, and blue color channel, respectively. In this section, we will transfer this way of supporting the percep- Since we are interested in the restriction of F(v) to the tion of tensor orientation with color to higher-order tensors. unit sphere, we express v in spherical coordinates such that e1 = ∂v(θ,φ)/∂θ and e2 = ∂v(θ,φ)/∂φ are orthonormal tan- gents at the peak. We then look for the Hessian Hθφ of the 4.1. Partitioning of the Glyph Surface spherical function f (θ,φ). It can be computed by stacking Higher-order tensors may have more than one peak, so it is e1 and e2 into the columns of a 3 × 2 matrix P. Then, the no longer possible to indicate a single direction by a uni- Hessian Hθφ at a stationary point of f (θ,φ) is given as form color. A common trick is to instead color each surface θφ T point individually, based on the vector to the center of the H = P HP − l f (θ,φ)I (12) ∗ glyph [Tuc04, TCGC04, PPvA 09]. If the glyph has narrow where I denotes a 2 × 2 identity matrix. A derivation of this and clearly separated lobes, this simple scheme often colors Equation is given in Appendix B. each of them approximately uniformly. However, it fails if θφ the tensor has a rounder shape. For example, the color green Let µi be the eigenvalues of H . At a maximum, µ2 ≤ is dominant in Figure 4 (a) only because the glyph is viewed µ1 ≤ 0. How well a peak is localized depends on the less from the y direction—there is no maximum in that direction. sharp direction, given by µ1. As part of Appendix A, it is shown that if the tensor has a positive decomposition, Therefore, we propose a coloring based on an explicit par- µ1 ≥ −l f (θ,φ) (Eq. (19)). Therefore, µ1 can be converted titioning of the surface, in which surface points are grouped into a peak sharpness measure PS ∈ [0,1] by normalizing it together if the same maximum is reached from them when as follows: following a gradient ascent. Then, each maximum deter- −µ mines the color for its whole region of influence. In the ter- PS = 1 (13) l f (θ,φ) minology of Morse-Smale complexes [EHNP03], our parti- tions resemble descending manifolds. However, we refrain Figure 5 presents two examples in which the peak sharp- from depicting saddles, since they are not considered as rel- ness measure is used to modulate the saturation of XYZ- evant features in the context of HARDI. RGB colors. In (a), the maxima on an almost-planar fourth- XYZ-RGB coloring accounts for the antipodal symme- order tensor are desaturated strongly. At the same time, the try F(v) = F(−v) of even-order tensors by assigning the sharp peaks in (b) retain their saturated colors. Despite the same color to antipodal peaks. Unfortunately, additional, fact that saturation varies in the XYZ-RGB colors already,

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

Tensor order Polar Plot HOME glyph 4 68 ms (202 ms) 70 ms (224 ms) 6 76 ms (231 ms) 86 ms (274 ms) Table 1: Time needed to generate 5577 glyphs, on a quad- Figure 5: Color modulation decreases the saturation at core and on a single-core CPU (in brackets). broad peaks that lack a clear direction (a). Sharp peaks (b) are still marked by highly saturated colors. v, our tesselation also contains the antipodal point −v. Thus, we only need to compute the transformed positions for half we modulate saturation rather than brightness because dark of the vertices and may scale them by −1 to obtain the other colors would make it difficult to appreciate glyph shapes. half. Moreover, all glyphs are independent from each other, which makes it straightforward to generate them in parallel At the maximum of a second-order tensor, f (θ,φ)= λ 1 on modern multi-core CPUs. and µ1 = 2(λ2 − λ1). Thus, peak sharpness coincides with cl from Eq. (10) for order l = 2. However, cl characterizes Table 1 summarizes the time it takes to generate 5577 a second-order tensor as a whole, while PS is specific to a glyphs with 642 vertices each from an axial slice of HARDI particular peak of a higher-order tensor. data on a 2.67 GHz Intel Xeon Quad-Core, for different tensor orders and glyph types. The timings include discrete Note that we use the name peak sharpness rather than peak normal estimation [Tau95]. Even when restricting our im- anisotropy, since the latter term has been used to denote the plementation to a single CPU core, it creates approximately degree to which the sharpness of a peak varies as a function ∗ 27600 glyphs per second. Compared to the 1 15 glyphs per of the direction in its tangent plane [KKA07, SCH 07]. . second reported for the same refinement level in [PPvA∗09], this amounts to a speedup of factor 24000. 5. Interactive Implementation

Our implementation allows for interactive data exploration, 5.2. Coloring even though it is based on standard triangle meshes, along with a few simple optimizations. This may seem surprising To implement the segmentation-based coloring from Sec- in the light of previous work, which used GPU programming tion 4, each local maximum is assigned an XYZ-RGB color, for interactive ellipsoid splatting [Gum03] and ray casting of with the saturation modulated according to peak sharpness ellipsoids [SWBG06] and polar plots [PPvA∗09]. (Eq. (13)). Then, we move from each uncolored vertex to the neighbor that is furthest from the origin and repeat this However, unlike in molecule visualization [SWBG06], recursively until a colored vertex is met. The resulting color where the information mainly lies in the spatial relation is copied to the original vertex and all vertices on the path. of the displayed atoms, the utility of tensor glyphs hinges This procedure is iterated until no uncolored vertices are left. on the discernability of their individual shapes, which requires a larger size in image space. Consequently, the num- In a second pass, the color of each vertex is blended with ber of geometric objects is generally lower in tensor visual- the colors on its one-ring. For efficiency, we perform simple ization than in molecule visualization. Moreover, we found averaging in RGB space. Coloring the above 5577 glyphs in the geometry-based implementations used for comparison in this way takes 60 ms (210 ms on a single CPU core). part of the literature [PPvA∗09] to be highly inefficient. 5.3. Efficient Rendering 5.1. Creating the Geometry While exploring a tensor dataset, the user typically either Both the polar plot and the HOME glyph can be expressed as looks at a large-scale overview for orientation, or zooms into deformed spheres, which we approximate by refined icosa- a part of the data to examine individual tensors. In the first hedra. In each refinement step, each edge is bisected at case, each glyph is projected to a small area on the screen, its center, thus subdividing each triangle into four smaller so coarser tesselations can be used without a visible degrada- ones. New vertices are projected back to the surface of the tion of image quality. In the second case, only a small num- sphere. To faithfully represent tensors of orders four and ber of glyphs is rendered at the same time. Two common six, we use three subdivision steps, generating 642 vertices acceleration methods exploit this situation: The use of dif- on the sphere. This corresponds to “tesselation order 7” in ferent levels of detail, and culling of objects that fall outside [PPvA∗09]. the current view frustum. Once a suitable triangulation of the sphere is given, glyphs We use bounding spheres to implement both techniques. are generated by evaluating Eq.s (4)or(9), respectively. To Their radius is given by the global maximum of the homoge- speed up this process, we exploit the fact that for each vertex neous form, as found during glyph generation. First, glyphs

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

# in Level of Detail # Glyphs Low High Framerate 5577 5577 0 88 fps (34 fps) 5577 0 477 >100 fps (72 fps) 12846 12846 0 41 fps (15 fps) 12846 299 2786 42 fps (14 fps) Figure 6: Even when the polar plot hardly shows a differ- ence between the original ODF and a positive rank-k ap- Table 2: Framerates at different stages of navigation in a proximation (a), this step is indispensible for an intersection- tensor dataset, with and without (in brackets) VBOs. free visualization of HOME glyphs (b).

outside the viewport are culled by computing signed dis- Unlike diffusion tensors, spherical deconvolution ODFs tances of their center to the six planes that bound the view do not model a physical process. Rather, they describe a frustum: If the point is outside and its distance is larger than fiber distribution which is inferred from the acquired diffu- the radius of the bounding sphere, then the glyph is discarded sion weighted images (DWIs) based on a particular model of without further processing. the signal that arises from a single fiber bundle. A high and a low level of detail are given by three and two refinements of the icosahedron, respectively. This way, 6.1. Enforcing a Positive Decomposition only one set of vertices has to be kept for each glyph, and the coarser version simply uses a subset of them. The level of de- Conceptually, spherical deconvolution ODFs can be ex- tail is chosen based on the projected radius of the bounding pected to meet the positive decomposition assumption from sphere. A simple and efficient formula for its approximative Section 3.2. When using a deconvolution kernel that maps computation under perspective projections is derived in Sec- the single fiber response to a rank-1 tensor (as in [SS08]), tion 3.2 of [LRC∗03]. ODFs that do not possess a positive decomposition correspond to fiber densities that contain negative fiber contribu- During interactive exploration, the same glyphs are typ- tions, which do not make sense anatomically. ically rendered many times, from varying positions. In this case, OpenGL Vertex Buffer Objects (VBOs) can be used to In practice, ODFs are not even non-negative, which is keep glyph geometry in video memory. Setting up VBOs for commonly attributed to noise. Therefore, we approximate 5577 glyphs takes 55 ms, but speeds up the rendering itself them with a positive sum of rank-1 terms before visualiza- by a factor of ≈ 2.5. tion. This step is analogous to the clamping of negative lobes in the polar plot, which is common in the spherical deconvo- Rendering performance on a 1800 1000 viewport using × lution literature [TCGC04]. an NVIDIA Quadro FX 1800 is summarized in Table 2. The results correspond to a single slice of HARDI data, and to To perform the positive rank-k approximation, we make three orthogonal slices, both at an overview and a zoomed- two modifications to the algorithm proposed in [SS08]: First, in level. Even though creating HOME glyphs takes slightly we replace the heuristic for estimating the number of cross- longer than creating polar plots, and order 6 requires more ing fibers with a rule that accepts a rank-k approximation if, computations than order 4, the rendering of all glyphs is compared to the previous rank-(k − 1) approximation, it de- equally efficient. Due to differences in graphics hardware creases the norm of the residual by more than ε times the and because the viewport size is not specified, an exact com- original tensor norm. Second, we ensure that only positive parison to [PPvA∗09] is not possible. However, our results rank-1 terms are added. are approximately three orders of magnitude faster than re- In most cases, this approximation only leaves a small ported for their geometry-based implementation and com- residual, and the clamped polar plot of the result looks al- pare favorably to their ray-tracing based results. most identical to the original ODF (cf. Figure 6 (a)). How- ever, using the approximation avoids confusing and mislead- 6. Application to HARDI ing self-intersections in the HOME glyph, like they become apparent in 2D in Figure 2(b) and in 3D by the presence of In High Angular Resolution Diffusion Imaging (HARDI), unlit backfaces on the left hand side of Figure 6(b). there exist different types of spherical functions that can be represented as higher-order tensors. The sharpest of Spherical deconvolution ODFs were estimated from them are created by spherical deconvolution [TCGC04] 60 DWIs at b = 1000s/mm2. Figure 7 shows a detail of a or, equivalently, the “sharpening deconvolution transform” coronal slice, at the transition between corpus callosum and [DDKA09]. In our examples, we focus on orientation den- pyramidal tract. The top row uses polar plots with clamped sity functions (ODFs) that arise from this technique. There- negative lobes, the bottom row shows HOME glyphs after fore, we demonstrate an additional visual sharpening, after a positive approximation, with the coloring from Section 4. sharpening on the data itself has already been performed. Two high-resolution images in the supplementary material

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 7: Fourth-order tensors from a HARDI dataset. The relatively ﬂat appearance of the HOME glyphs (bottom) with respect to the polar plot (top) conveys the fact that the displayed ﬁber distributions stay mostly within the image plane.

of this paper provide a whole-slice comparison of our novel Figure 8: Sixth-order tensors showing a fiber with small, glyph-based visualization against the state of the art. larger, and anisotropic spread (from left to right). The In order to determine which glyph conveys the properties HOME glyphs (bottom) make it much easier to distinguish of fiber distributions more effectively, we will first consider these cases than the polar plot (top). a number of synthetic examples, for which the underlying fiber configurations are known, before we return to real data. shown both from the side and by looking orthogonally onto 6.2. Inspecting Fiber Spread the peak, to maximize the impression of peak anisotropy. In addition to inferring dominant fiber directions, spherical In the polar plot, the width of the lobe grows with the mag- deconvolution enables conclusions about fiber spread, i.e., nitude of its peak. Since less fiber spread leads to a larger the variance around the main direction. Fiber spread is often density at the dominant direction, the lobe becomes wider as stronger in some principal direction than orthogonal to it, fiber spread decreases, which is counter-intuitive. This prob- and it has been proposed that exploiting such information lem does not occur in the HOME glyph, where higher vari- can reduce the leakage of probabilistic tractography methods ance around the main direction leads to a wider peak. More- into anatomically implausible regions [SCH∗07]. over, the polar plot makes it very difficult to appreciate the spread anisotropy, whereas the aspect ratio of the lobe of the Anisotropic fiber spread has been modeled using the HOME glyph is close to the ratio of λ2/λ3. Bingham distribution [KKA07, SCH∗07]: 1 T p(v,B)= ev Bv (14) 6.3. Inspecting Fiber Crossings C(B) Since it is known that ODF maxima present a biased estimate where B is a symmetric 3 × 3 matrix with eigenvalues λ1 ≥ of fiber directions in case of crossings [TCGC04, SS08], it λ2 ≥ λ3 and corresponding eigenvectors e1, e2, e3, and C(B) may appear problematic to use a maximum enhancing glyph is a constant that makes p(v,B) integrate to unity over the for ODFs. To examine this potential problem, we created unit sphere. We set λ1 = 0, since p(v,B) is invariant under DWIs of synthetic fiber crossings at varying angles and vol- addition of any real constant to the eigenvalues of B. Then, ume fractions, and again estimated spherical deconvolution e1 is the dominant fiber direction, e2 is the principal direction models of order six. of fiber spread, and the magnitudes of λ and λ define the 2 3 Figure 9 compares the resulting ODFs to the fiber di- amount and anisotropy of fiber spread. rections used in the simulation (gray lines). ODF maxima To produce a synthetic model of a spreading fiber, we have are highlighted by colored cylinders. It becomes clear that drawn 1000 samples on the sphere uniformly at random and the HOME glyph only develops very sharp, pin-like lobes generated a single fiber model for each of them, weighted when maxima are well-separated (left, 78◦ crossing). In such according to the Bingham distribution. From this mixture, cases, little bias occurs. As the angle becomes smaller, bias we then created 60 DWIs at b = 1000s/mm2 and estimated increases, but maxima are emphasized less. a linear spherical deconvolution model of order six. The lobe of the HOME glyph on the right (60◦ crossing) From left to right, Figure 8 shows fiber spread with λ2 = exhibits a steep slope, tangential to the true fiber directions. λ3 = −15 (small and isotropic), λ2 = λ3 = −7.5 (larger This is also true for other crossing angles and tensor orders, and isotropic), and λ2 = −5, λ3 = −15 (anisotropic), re- and is understood by the fact that given a rank-1 decompo- spectively. Polar plots (top) and HOME glyphs (bottom) are sition in the form of Eq. (7), the HOME glyph of an order-l

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 10: In real data (order six), sharp peaks of the HOME glyph (a) correspond well to rank-1 terms. Wide lobes can be interpreted as ﬁber spread (b) or as unresolved crossings (c).

7. Conclusion and Future Work The one-to-one relation between ellipsoids and positive def- Figure 9: Results on synthetic crossings show that sharp inite second-order tensors provides a well-established way peaks of the HOME glyph (left) are well-aligned with of reasoning about these abstract mathematical objects in true fiber directions (gray lines). When maxima are biased a graphic and intuitive way. In this work, we have studied (right), glyph shape emphasizes them less. the Higher-Order Maximum Enhancing (HOME) glyph, a recent generalization of ellipsoids to higher-order tensors [SWS09]. Our theoretical contributions are a formal proof of its properties and a clarification of the exact conditions under which it can be applied. tensor can be re-written as Even though the mathematical proofs may appear intim- r idating, implementing the new glyph is almost as simple as l−1 ∗ h(v)= ∑ λi(v · ei) ei (15) for the polar plot. Contrary to previous findings [PPvA 09], i=1 we have demonstrated that relatively simple optimizations allow triangle-based implementations to remain interactive up to several ten thousands of glyphs. In case of a two-fiber crossing, r = 2, e1 and e2 are the fiber directions (unit-length, but not necessarily orthogonal), Finally, we have made contributions to a particular appli- and λ1 and λ2 reflect their respective volume fractions. In cation, the visualization of data from High Angular Resolu- l−1 Eq. (15), the contribution of e1 is weighted by cos(φ) , tion Diffusion Imaging (HARDI). We have transferred the where φ is the angle between v and e1. Obviously, any vector traditional way of coloring diffusion ellipsoids to higher- v ⊥ e1 is projected unto e2. However, not only is the weight order tensor models. In a side-by-side comparison to the of e1 zero, but also the first l − 2 derivatives of this weight state of the art, we have demonstrated that the HOME glyph with respect to φ. Thus, h(v) deviates from the direction of visualizes situations of fiber spread and crossings more ef- e2 only very slowly as φ increases. This leads to the good fectively than the polar plot that has been used previously. agreement of the glyph shape with the true fiber direction. When discussing our results with a domain scientist, the On real data, we compared glyph shape to the rank-k ap- expert found our approach of assigning colors based on peak proximation from [SS08], which provides unbiased fiber es- directions beneficial and emphasized the importance of in- timates. In this case, we use cylinders to indicate rank-1 teractive glyph-based visualizations for analyzing orienta- terms rather than ODF maxima, and scale their length with tion density functions. relative volume fractions. In these experiments, sharp peaks We expect that the HOME glyph will prove helpful both of the HOME glyph corresponded well to terms of the ap- for the exploration of data that can be modelled with higher- proximation (Figure 10 (a)). order tensors and to clarify more general mathematical prop- A priori, it is unclear if wide ODF lobes (Figures 10 (b) erties of higher-order tensors. As part of our future work, we and (c)) are due to high fiber spread or to a crossing that has are interested in finding scalar invariants that parametrize the not been resolved by the measurement or the model. Varying shape of our glyph in an intuitive way. the parameters of the approximation allows one to explore both options. We expect that such a combined visualization Acknowledgements of ODFs and inferred fiber directions over a region of inter- est will help to find parameters for model selection that pro- We would like to thank Alfred Anwander (Max Planck In- duce coherent and anatomically plausible fiber directions. stitute for Human Cognitive Brain Science, Leipzig, Ger-

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 many) for providing the HARDI dataset and giving feed- The homogeneous form Fi of an individual rank-1 term ⊗l l back on our results. Our implementation makes use of Teem Ti = λivi can be written as Fi(v) = fi(ξ) = λi cos (ξ), (teem.sf.net). This work was supported by a fellowship where ξ denotes the angle between v and vi. Since within the Postdoc Program of the German Academic Ex- ′′ l l−2 2 change Service (DAAD). fi (ξ)= λi h−l cos (ξ)+ l(l − 1)cos (ξ)sin (ξ)i, (20) it follows that Appendix A: Proof of the HOME Glyph Properties ′′ fi (ξ)/l ≥ − f (ξ) if λi > 0. (21) This appendix uses fundamental concepts from differential geometry [Top06] to prove the three properties of the HOME According to Section 3.2, we assume that the given tensor glyph which were stated in Section 3.3. For this, we express can be written as a sum of rank-1 terms with positive λi, p(v) from Eq. (4) and h(v) from Eq. (9) in the parametric so Eq.s (19) hold by linearity of the derivative. Therefore, T form that results from writing v in spherical coordinates: nh = (1,0,0) . sin(θ)cos(φ) At the stationary point, the coefficients of the first (E, F, p(θ,φ)= f (θ,φ)sin(θ)sin(φ) (16) G) and second fundamental forms (L,M,N) are: cos θ 2 2  ( )  Ep = f Fp = 0 Gp = f The convention in this paper is to use θ ∈ [0,π] as the Lp = fθθ − f Mp = 0 Np = fφφ − f inclination from the positive z axis, and φ ∈ [0,2π) as the 2 2 azimuth angle from the positive x axis. Eh = ( f + f /l) Fh = 0 Gh = f + f /l θθ φφ l − 2 l − 2 Standard calculus confirms that L = f − f M = 0 N = f − f h l θθ h h l φφ − 1 T ·l 1 v = ∇F(v) (17) l Since the eigensystems of all fundamental forms coincide, where F is the homogeneous form of the tensor T (Eq. (3)). it is simple to compare corresponding principal curvatures. We write F in spherical coordinates (cf. fˆ in Eq. (5)) and In case of direction θ, the condition for property three reads compute its gradient in Cartesian coordinates via the chain − f − f l 2 f − f rule. This involves partial derivatives ∂{r,θ,φ}/∂{x,y,z}, θθ l θθ 2 > 2 . (22) which can be taken from handbooks like [BSMM04] or from f ( f + fθθ/l) computer algebra software. We observe from Eq. (5) that l−1 Straightforward simplification transforms this into ∂ fˆ(r,θ,φ)/∂r = lr f (θ,φ) and fix r = 1 to obtain 2 2 ((2l − 1) f + fθθ) fθθ/l > 0. Using Eq. (21), it is easy to h(θ,φ)= p(θ,φ) verify that for f > 0 and fθθ 6= 0, the condition in Eq. (22) indeed holds. Direction φ is treated in complete analogy. sin(φ) f (θ,φ)cos(θ)cos(φ) − f (θ,φ) This concludes the proof of property three.  θ φ sin(θ)  1 cos(φ) + fθ(θ,φ)cos(θ)sin(φ)+ fφ(θ,φ) (18) l  sin(θ)   − fθ(θ,φ)sin(θ)  Appendix B: Hessian in Spherical Coordinates We obtain second derivatives of the spherical function Without loss of generality, we choose the coordinate sys- f (θ,φ) from the easy-to-compute Hessian H of the homoge- tem such that the stationary point under investigation is at neous function F(v) (Eq. (11)) by expressing v in spherical (θ = π/2,φ = 0). For brevity, omitting function parameters coordinates, and applying the chain rule to compute θ and φ will imply function evaluation at that point, and ∂ f (θ,φ) ∂F(v(r,θ,φ)) ∂v we will omit the terms fθ = fφ = 0 completely. Moreover, = = ∇F(v) · (23) we rotate the coordinate system such that the mixed second ∂θ ∂θ ∂θ derivatives of f vanish ( fθφ = fφθ = 0). Taking the second derivative requires both the chain rule Under these conditions, p = h = ( f ,0,0)T, so the first and the product rule: property follows immediately. The equations for the first ∂2F(v(r,θ,φ)) ∂vT ∂v ∂2v T T = H + ∇F(v) · (24) derivatives of p and h are pθ = (0,0,− f ) , pφ = (0, f ,0) , ∂θ∂φ ∂θ ∂φ ∂θ∂φ T T hθ = (0,0,− f − fθθ/l) , and hφ = (0, f + fφφ/l,0) . From this, it is clear that p·pθ = p·pφ = h·hθ = h·hφ = 0, which The other second derivatives are given in complete anal- proves property two. ogy. Since we chose coordinates such that v, e1 = ∂v/∂θ and e = ∂v/∂φ form an orthonormal basis, ∂2v/(∂θ)2 = In order to compute principal curvatures, we need to de- 2 ∂2v/(∂φ)2 = −v and ∂2v/(∂θ∂φ)= 0. From property one termine the normal vector, n = h ×h /kh ×h k. For this, h θ φ θ φ of the HOME glyph and Equation (17), it is clear that at we will first show that stationary points of f (θ,φ), ∇F(v) · v = lF(v). Therefore, θφ − f − fθθ/l ≤ 0 and f + fφφ/l ≥ 0. (19) collecting all second derivatives in H results in Eq. (12).

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

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c 2010 The Author(s) Journal compilation c 2010 The Eurographics Association and Blackwell Publishing Ltd.