DS-620 Data Visualization

Chapter 7 Summary.

Valerii Klymchuk

August 19, 2015

0. EXERCISE 0

7 Tensor Visualization

Tensor data encode some spatial property that varies as a function of position and direction, such as curvatureof a three-dimensional surface at a given point and direction. Every point in a tensor dataset carries a 3× 3matrix. Material properties such as stress and strain in 3D volumes, are described by stress tensors. Diffusionof water in tissues can be described by a 3× 3 diffusion tensor matrix. In human brain diffusion is strongerin the direction of the neural fibers and weaker across fibers. By measuring the diffusion, we can get insightinto complex structure of neural fibers in the human brain. The measurement of the diffusion of water inliving tissues is done by a set of techniques known as diffusion tensor magnetic resonance imaging(DT-MRI). The process that constructs visualizations of the anatomical structures of interest starting fromthe measured diffusion data is known as diffusion tensor imaging (DTI).

Intrinsic structure of the tensor data can be exploited by computations called principal componentanalysis.

7.1 Principal Component Analysis

We have shown that we can compute the normal curvature at some point x0 in some direction s in the tangentplane as the second derivative ∂2f/∂s2 of f using the two-by-two Hessian matrix of partial derivatives of f.Minimal and maximal values of the curvature at a given point are invariant to the choice of the (direction)local coordinate system since they depend only on the surface shape at a given point.

The direction in the tangent plane for which the normal curvature has extremal values are the solutionsof the following equation:

Hs = λs.

For 2× 2 matrices we can solve the equation analytically, obtaining two solutions λ1 and s1 and λ2 ands2, respectively:

The surface has minimal curvature in the direction s1 and maximal curvature in the direction s2. Alongall directions in the tangent plane orthogonal to the surface normal, the curvature takes values between theminimal and maximal ones.

The solutions si are called the principal directions, or eigenvectors of the tensor H, and values λiare called eigenvalues. For n×n symmetric matrix, the principal directions are perpendicular to each otherand form directions in which the quantity reaches extremal values.

In the case of a 3D surface given by an implicit function f(x, y, z) = 0 in global coordinates, we have a3× 3 Hessian matrix of partial derivatives, which has 3 eigenvalues and three eigenvectors that we computeby solving equation. A good method is the Jacobi iteration method, which solves Equation numerically forarbitrary-size n× n real symmetric matrices.

If we order the eigenvalues in decreasing order λ1 > λ2 > λ3, the corresponding eigenvectors e1, e2 ande3, also called the major, medium, and minor eigenvalues , that have following meaning. In case of

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curvature tensor, e1, e2 are tangent to the given surface and give the directions of maximal and minimalnormal curvature on the surface, and e3 is equal to the surface normal.

7.2 Visualizing Components

The simples way to visualize a tensor dataset is to treat it as a set of scalar datasets. Given a 3× 3 tensormatrix we can consider each of its nine components hij as a separate scalar field.

Each component of the tensor matrix is visualized using grayscale colormap that maps scalar value toluminance. Note, that due to the symmetry of the tensor matrix, there are only 6 different images inthe visualization (h12 = h21, h13 = h31, h23 = h32). In general, the tensor matrix components encode thesecond-order partial derivatives of our tensor-encoded quantity with respect to the global coordinate system.

7.3 Visualizing Scalar PCA Information

A better alternative to visualizing the tensor matrix components is to focus on data derived from thesecomponents that has a more intuitive physical significance.

Diffusivity. The mean of the measured diffusion over all directions at that point is measures as theaverage of the diagonal entries: 1

3 (h11 + h22 + h33).Anisotropy. Recall, that eigenvalues give the values of the extremal variations in directions of eigen-

vectors (extremal variations). In case of diffusion data, the eigenvalues can be used to describe the degreeof anisotropy of the tissue at a point (different diffusivities in different directions around the point).

A set of metrics proposed by Westin, estimates the certainties cl, cp, and cs that a tensor has a linear,planar, or spherical shape, respectively. If the tensor’s eigenvalues are λ1 ≥ λ2 ≥ λ3, the respective certaintiesare

cl =λ1 − λ2

λ1 + λ2 + λ3

cp =2(λ2 − λ3)

λ1 + λ2 + λ3

cs =3λ3

λ1 + λ2 + λ3

.

A simple way to use the anisotropy metrics proposed previously is to directly visualize the linear certaintycl scalar signal.

Another frequently used measure for the anisotropy is the fractional anisotropy, which is defined as

FA =

√3

2

√∑3i=1(λi − µ)2

λ21 + λ22 + λ23,

where µ = 13 (λ1 + λ2 + λ3) is the mean diffusivity.

A related measure is the relative anisotropy, defined as

RA =

√3

2

√∑3i=1(λi − µ)2

λ1 + λ2 + λ3.

Methods in this section reduce the visualization of a tensor field to that of one or more scalar quantities.These can be examined using any of the scalar visualization methods such as color plots, slice planes, andisosurfaces.

7.4 Visualizing Vector PCA Information

Let’s say, we are interested only in the direction of maximal variation of our tensor-encoded quantity. Forthis we can visualize the major eigenvector field using any of the vector visualization methods in Chapter 6.Vectors can be uniformly seeded at all points where the accuracy of the diffusion measurements is above a

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certain confidence level. The hue of the vector coloring can indicate their direction, by using the followingcolormap:

R = |e1 · x|G = |e1 · y|B = |e1 · z|

.

The luminance can indicate the measurement confidence level. A relatively popular technique in thisclass is to simply color map the major eigenvector direction.

Visualizing a single eigenvector or eigenvalue at a time may not be enough. In many cases the ratios ofeigenvalues, rather than their absolute values, are of interest.

7.5 Tensor Glyphs

We sample the dataset domain with a number of representative sample points. For each sample point, weconstruct a tensor glyph that encodes the eigenvalues and eigenvectors of the tensor at that point. Fora 2 × 2 tensor dataset we construct a 2D ellipse whose half axes are oriented in the directions of the twoeigenvectors and scaled by the absolute values of the eigenvalues. For a 3 × 3 tensor we construct a 3Dellipsoid in a similar manner.

Besides ellipsoids, several other shapes can be used: like parallelepipeds (cuboids), or cylinders insteadof ellipsoids. Smooth glyph shapes like those provided by the ellipsoids provide a less-distracting picture,than shapes with sharp edges, such as the cuboids and cylinders.

Superquadric shapes are parameterized as functions of the planar and linear certainty metrics cl and cp,respectively.

Another tensor glyph used is an axes system, formed by three vector glyphs that separately encodethe three eigenvectors scaled by their corresponding eigenvalues. This method is easier to interpret for 2Ddatasets, however in 3D they create too much confusion due to spatial overlap.

Eigenvalues can have a large range, so directly scaling the tensor ellipsoids by their values can easily leadto overlapping and (or) very thin or very flat glyphs. We can solve this problem as we did for vector glyphsby imposing a minimal and maximal glyph size, either by clamping or by using a nonlinear value-to-sizemapping function.

7.6 Fiber Tracking

In case of a DT-MRI tensor dataset, regions of high anisotropy in general, and of high values of the cllinear certainty metric in particular, correspond to neural fibers aligned with the major eigenvector e1. Ifwe want to visualize the location and direction of such fibers, it is natural to think of tracking the directionof this eigenvector over regions of high anisotropy by using the streamline technique. First, a seed region isidentified. This is a region where the fibers should intersect, so it can be detected by thresholding one of theanisotropy metrics presented in section 7.3. Second, streamlines are densely seeded in this region and traced(integrated) both forward and backward in the major eigenvector field e1 until a desired stop criterion isreached (minimal value of anisotropy reached, or a maximal distance from other tracked fibers).

After the fibers are tracked, they can be visualized using the stream tubes technique. The constructedtubes can be colored to show the value of a relevant scalar field, the major eigenvalue, anisotropy metric, orsome other quantity scanned along with the tensor data.

Focus and context. Fiber tracks are most useful when shown in context of the anatomy of the brainstructure being explored.

Fiber clustering. Given two fibers a = a(t) and b = b(t) with t ∈ [0, 1] we first define distance:

d(a, b) =1

2N

N∑i=1

(||a(i/N), b||+ ||b(i/N), a||) ,

as symmetric mean distance of N sample points on a fiber to the (closest points on) other fiber. Thedirectional similarity of two fibers is defined as the inverse of the distance. Using the distance, the trackedfibers are next clustered in order of increasing distance, i.e. from the most to the least similar, until the desired

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number of clusters is reached. For this simple bottom-up hierarchical agglomerative technique introduced inSection 6.7.3 for vector fields can be used.

Tracking challenges. First, tensor data acquired via the current DT-MRI scanning technology containsin practice considerable noise and has a sampling frequency that misses several fine-scale details. Moreover,tensors are not directly produced by the scanning device, but obtained via several processing steps, of whichprincipal component analysis is the last one. All these steps introduce extra inaccuracies in the data, whichhave to be accounted for. The PCA estimation of eigenvectors can fail if the tensor matrices are not close tobeing symmetric. Even if PCA works, fiber tracking needs a strong distinction between the largest eigenvalueand the other two ones, in order to robustly determine the fiber direction.

7.7 Illustrative Fiber Rendering

While some approaches are easy to implement, they give “raw” view on the fiber data, which has severalproblems:

• Region structure: Fibers are one-dimensional objects. However, to better understand the structure ofthe DTI tensor field, we would like to see the linear anisotropy regions with fibers and planar anisotropyregions rendered as surfaces.

• Simplifications: Densely-seeded datasets can become highly cluttered, so it is hard to discern the globalstructure implied by the fibers. A simplified visualization of fibers can be useful in understandingrelative depth of fibers.

• Context: Showing combined visualizations of fibers and tissue density can provide more insight intothe spatial distribution and connectivity patterns implied by fibers.

A set of simple techniques can address the above goals.Fiber generation. We densely seed the volume and trace fibers using Diffusion Toolkit. Each resulting

fiber is represented as polyline consisting of an ordered set of 3D vertex coordinates.Alpha blending. One simple step to reduce occlusion and see inside the fiber volume is to use additive

alpha blending (Section 2.5). However, fibers need to be sorted back-to-front as seen from the viewingangle. One simple way to do this efficiently is to transform all fiber vertices in eye coordinates, i.e., in acoordinate frame, where the x- and y-axes match the screen x- and y-axes, and $z-%axis is parallel to theview vector, and next to sort them based on their z value. Sorting has to be executed every time we changethe viewing direction.

Anisotropy simplification. Alpha blending reduces occlusion, but it akts in a global manner. Wehowever, are specifically interested in regions of high anisotropy. To emphasize such regions we next modulatethe colors of the drawn fiber points by the value of the combined linear and planar anisotropy

ca = cl + cp = 1− cs =λ1 + λ2 − 2λ3λ1 + λ2 + λ3

, where cl, cp,and $c s $are the linear, planar and spherical anisotropy metrics. IF we render fiber pointshaving ca > 0.2 color-coded by direction, and all other fiber points in gray. The image shows well the fibersubset which passes through regions of linear and (or) planar anisotropy, i.e., separates interesting from lessinteresting fibers. Using anisotropy to cull fiber fragments after tracing is a less aggressive, and offers morechances for meaningful fiber fragments to exist in the final visualization, without having to be very precisein the selection of the anisotropy threshold used.

Illustrative rendering.Here we construct stream tube-like structures around the rendered fibers. However instead of using 3D

space stream tube algorithm, we densely sample all fiber polylines, and render each resulting vertex withan OpenGL sprite primitive that uses a small 2D texture. The texture encodes the shading profile of asphere, i.e., is bright at the middle and dark at the border. Compared to streamtubes, the advantage of thistechnique is that it is much simpler to implement, and also much faster, since there is no need to constructcomplex 3D tubes, we only render one small 2D texture per fiber vertex.

A second option for illustrative (simplified) rendering of fiber tracks entails using the depth-dependenthalos method presented for vector field streamlines in Section 6.8. Depth dependent halos effectively merge

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dense fiber regions into compact black areas, but separate fibers having different depth by a thin white haloborder. Together with interactive viewpoint manipulation, this helps users in perceiving the relative depthsof different fiber sets.

Fiber bundling. We still cannot easily visualy distinguish regions of linear and planar anisotropy fromeach other. WE cannot visually classify dense fiber regions as being (a) either thick tubular fiber bundles or(b) planar anisotropy regions covered by fibers.

In order to simplify the structure of the fiber set, we apply a clustering algorithm, as follows: Givena set of fibers, we first estimate a 3D fiber density field ρ : R3 → R+, by convolving the positions ofall fiber vertices, or sample points, with a 3D monotonically decaying kernel, such as Gaussian or convexparabolic function. Next, we advect each sample point upstream in the normalized gradient ∇ρ/||∇ρ|| of thedensity field, and recompute the density ρ of the new fiber sample points. Iterating this process 10..20 timeseffectively shifts the fibers towards their local density maxima. In other words kernel density estimationcreates compact fiber bundles that describe groups of fibers which are locally close to each other.

The bundled fibers occupy much less space, thus allow a better perception of the structure of the brainconnectivity pattern they imply. However effective in reducing spatial occlusion and thereby simplifyingthe resulting visualization, fiber bundling suffers from two problems. First, planar anisotropy regions, arereduced to a few one-dimensional bundles. This conveys wrong impression. Second, bundling effectivelychanges the positions of fibers. As such, fiber boundless should be interpreted with great care, since theyhave limited geometrical meaning.

To address the first problem, we can modify the fiber bundling algorithm and instead of using an isotropicspherical kernel to estimate the fiber density, we can use an ellipsoidal kernel, whose axes are originated alongthe directions of the eigenvectors of the DTI tensor field, and scaled by the reciprocals of the eigenvalues ofthe same field. In linear anisotropy regions, fibers will strongly bundle towards the local density center, butbarely shift in their tangent directions. In planar anisotropy regions, fibers will strongly bundle towards theimplicit fiber-plane, but barely shift across this plane. Additionally, we use the values of cl and cp to renderthe above two fiber types differently. For fiber points located in linear anisotropy regions (cl large) we renderpoint sprites using spherical textures, for planar anisotropy regions we render 2D quads perpendicular to thedirection of the eigenvector corresponding to the smallest eigenvalue, i.e., tangent to the two underlying fiberplane. So that in linear anisotropy regions we see tube-like structures, in planar regions - planar structures.

7.8 Hyperstreamlines

First, we perform principal component analysis to decompose the tensor field into three eigenvector fieldsei and three corresponding scalar eigenvalue fields λ1 ≥ λ2 ≥ λ3. Next, we construct streamtubes in themajor eigenvector field e1. At each point along such a streamtube, we now wish to visualize the medium andminor eigenvectors e2 and e3. For this instead of using a circular cross section of constant size and shape,we now use an elliptic cross section, whose axes are oriented along the directions of the medium and minoreigenvectors e2 and e3 and scaled by λ2 and λ3, respectively.

The local thickness of the hyperstreamlines gives the absolute values of the tensor eigenvalues, whereasthe ellipse shape indicates their relative values as well as the orientation of the eigenvector frame along astreamline.

Besides ellipses, other shapes can be used for the cross section. In general, hyperstreamlines providebetter visualizations than tensor glyphs. However, appropriate seed points and hyperstream length must bechosen to appropriately cover the domain, which can be a delicate process. Moreover, scaling of the crosssections must be done with care, in order to avoid overly thick hyperstreamlines that cause occlusion or evenself-injection. For this we can use scaling techniques in Section 6.2.

7.9 Conclusion

Tensor data can be visualized by reducing it to one scalar or vector field, which is then depicted by specificscalar or vector visualization techniques. The scalar or vector fields can be the direct outputs of the PCAanalysis (eigenvalues and eigenvectors) or derived quantities, such as various anisotropy metrics. Alterna-tively, tensors can be visualized by displaying several of the PCA results combined in the same view, suchas done by the tensor glyphs or hyperstreamlines.

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What have you learned in this chapter?Chapter provides an overview of a number of methods for visualizing tensor data. It explains principal

component analysis as a technique used to process a tensor matrix and extract from it information thatcan directly be used in its visualization. It forms a fundamental part of many tensor data processing andvisualization algorithms. Section 7.4 shows how the results of the principal component analysis can bevisualized using the simple color-mapping techniques. Next parts of the chapter explain how same data canbe visualized using tensor glyphs, and streamline-like visualization techniques.

In contrast to Slicer, which is a more general framework for analyzing and visualizing 3D slice-based datavolumes, the Diffusion Toolkit focuses on DT-MRI datasets, and thus offers more extensive and easier to useoptions for fiber tracking.

What surprised you the most?

• New rendering techniques, such as volume rendering with data-driven opacity transfer functionsare also being developed to better convey complex structures emerging from the tracking process.

• Fiber tracking in DT-MRI datasets is an active area of research.

• Fiber bundling is a promising direction for the generation of simplified structural visualizations of fibertracts for DTI fields.

What applications not mentioned in the book you could imagine for the techniques ex-plained in this chapter?

A hologram 3D glyph can be used in rendering (i.e., semitransparent hyperstreamlines) to mask discon-tinuities caused by regular tensor glyphs.

Anisotropic bundled visualization of the fiber dataset. We can render the bundled fibers with a translu-cent sprite texture rendered with alpha blending, while using a kernel of small radius to estimate thefiiber density ρ. Instead of using an anisotropic sphrical kernel to estimate the fiber density, we use an ellip-soidal kernel, whose axes are oriented along the directions of the eigenvectors of the DTI tensor field, andscaled by the reciprocals of the eigenvalues of the same field. In linear anisotropy regions, fibers will stronglybundle towards the local density center, but barely shift in their tangent directions. In planar anisotropyregions, fibers will strongly bundle towards the implicit fiber-plane, but barely shift across this plane. We usethe values of cl and cp to render the above two fiber types differently. For fibers in linear anisotropy regions(cl large), we render point sprites using sphere textures. For fiber points located in planar anisotropy regions(cp large), we render translucent 2D quads oriented perpendicular to the direction of the eigenvectorcorresponding to the smallest eigenvalue.

1. EXERCISE 1

In data visualization, tensor attributes are some of the most challenging data types, due to their highdimensionality and abstract nature. In Chapter 7 (and also in Section 3.6), we introduced tensor fields bygiving a simple example: the curvature tensor for a 3D surface. Give another example of a tensor fielddefined on 2D or 3D domains. For your example

• Explain why the quantity you are defining is a tensor

• Explain how the quantity you are defining varies both as a function of position but also of direction

• Explain what are the intuitive meanings of the minimal, respective maximal, values of your quantityin the directions of the respective eigenvectors of your tensor field.

• Stress on a material, such as a construction beam in a bridge can be an example of tensor field. Stressis a tensor because it describes things happening in two directions simultaneously.

Another example is the Cauchy stress tensor T, which takes a direction v as input and produces thestress T (v) on the surface normal to this vector for output:

σ = [T e1 , T e2 , T e3 ] =

σ11 σ12 σ13σ21 σ22 σ23σ32 σ32 σ33

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whose columns are the stresses (forces per unit area) acting on the e1, e2, and e3 faces of the cube.Other examples of tensors include the diffusion of water in tissues, strain tensor, the conductivity tensor,

and the inertia tensor. Moment of inertia is a tensor too because it involves two directions: the axis ofrotation, and the position of the center of mass.

• quantity describing water diffusivity varies as a function of position (coordinates of the point) anddirection of measurement.

• minimal and maximal values of this quantities are achieved in directions of corresponding eigenvalues:e1, e2 which are tangent to the given surface and give the directions of maximal and minimal TENSORVALUE on the surface, and e3 is equal to the surface normal.

2. EXERCISE 2

Consider a 2-by-2 symmetric matrix A with real-valued entries, such as the Hessian matrix of partialderivatives of some function of two variables. Now, consider the two eigenvectors x and y of the matrix A,and their two corresponding eigenvalues λ and µ. We next assume that these eigenvalues are different. Provethat the two eigenvectors x and y are orthogonal.

Hints: There are several ways to prove this. One way is to use that the matrix is symmetric, henceA = AT . Next, use the algebraic identity < Ax, y >=< x,AT y >, where < a, b > denotes the dot productof two vectors a and b. To prove that x is orthogonal to y, prove that < x, y >= 0. Based on definition ofeigenvectors and eigenvalues: Ax = λx, Ay = µy.

Let us multiply both sides of the first equation by y and both sides of the second equation by x, so weget: Axy = λxy, Ayx = µyx. Substituting one from another yilds: Axy − Ayx = λxy − µyx, which means(λ− µ)xy = 0.

In the equation above λ 6= µ, which means that xy = 0, there fore vectors x and y are orthogonal.

3. EXERCISE 3

One basic solution for visualizing eigenvectors of a 3-by-3 tensor, such as the one generated from adiffusion-tensor MRI scan, is to color-code its (major) eigenvector using a directional colormap. Figure 7.6(also shown below) shows such a colormap, where we modulate the basic color components R, G, and B toindicate the orientation of the eigenvector with respect to the x, y, and z axes respectively. For the sametask of directional color-coding of a tensor field, imagine a different colormap, which, in your opinion, maybe more intuitive than the red-green-blue colormap proposed here.

Vector color coding is easier to understand in HSV system.

The hue value H we calculate as H = arctg( |e1·z||e1·x| , to encode the direction of e1 eigen vector in x × z

view plane.Saturation S = 1 |e1 · y| of the vector coloring can encode its orientation along y axis orthogonal to

the view plane. We can use additive alpha blending in which fibers need to be sorted back-to-front asseen from the viewing angle. One simple way to do this efficiently is to transform all fiber vertices in eyecoordinates, i.e., in a coordinate frame, where the x− and z−axes match the screen x and y axes, and yaxis is parallel to the view vector, and next to sort them based on their y value. Sorting has to be executedevery time we change the viewing direction. High values of S will correspond to low values of |e1 · y|, whichwill result in vectors oriented along y axis depicted in shades of white.

The luminance V indicates the measurement confidence level. So that bright vectors indicate highconfidence measurement levels, whereas dark vectors indicate low confidence level.

4. EXERCISE 4

Tensor glyphs are a generalization of vector glyphs which attempt to convey three vectors (the eigenvectorsof the tensor-field to be explored) at a given point over its domain. In Section 7.5 (Figure 7.8, also shownbelow), four kinds of tensor glyphs are proposed: ellipsoids, cuboids, cylinders, and superquadrics. Proposea different kind of tensor glyph. Sketch the glyph. For your proposed glyph, explain:

• How the glyph’s properties (shape, shading, color) convey the directions and magnitudes of the threeeigenvectors

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• How it is possible, by looking at the shape, to understand which is the direction of the major eigenvector,medium eigenvector, and minor eigenvector

• Which are, in your opinion, the advantages and (or) disadvantages of your proposal as compared tothe ellipsoid, cuboid, cylinder, and superquadric glyphs.

I can imagine a glyph, constructed of a union of dots, dispersed in the eigenvector basis (eigenvectorsscaled by corresponding eigenvalues) and turned around its center of mass, according to linear, planar andspherical probabilities.

Figure 1: Elliptic toroid point cloud glyph.

• This glyph’s elliptic shape easily conveys the directions and magnitudes of the three eigenvectors. Wecan use shading, and color to depict extra characteristics that straighten the insight, such as confidencelevel or orientation.

Figure 2: Elliptic point cloud torus, turned around it’s center of mass.

• By looking at the shape, we understand that half axes of our elliptic torus cloud are scaled with theeigenvalues, and it rotated by the matrix, which has eigenvectors, as columns. Direction of the major,medium and minor eigenvectors are depicted by longest medium and, and minor half axes of the elliptictoroid in 3D space. We translate the projection of the resulting glyph onto the viewing plane.

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The advantages are:

• Smooth elliptic shape provides a less distracting picture, and creates less discontinuities, than shapeswith sharp edges, such as the cuboids and cylinders.

• 2D projection of a point cloud will better convey a non-ambiguous 3D orientation for eigenvaluescorresponding to equal eigenvalues, when viewed from certain angles, compared to a regular ellipsoidglyph.

• overlapping clouds, perhaps, will result in more dense (more saturated, brighter and more visible)areas, which only straighten our visual insight (they are predicted by data not from one, but frommultiple sample points), instead of creating occlusion and clutter.

5. EXERCISE 5

One way to visualize a symmetric 3D tensor field is to reduce it, by principal component analysis (PCA), toa set of three eigenvectors (v1, v2, v3), whose corresponding lengths are given by three eigenvalues (λ1, λ2, λ3).Such eigenvectors can be visualized, among other methods, by using vector glyphs. In this context, answerthe two questions below:

• If we use vector glyphs, and since we have three eigenvectors, and all of them encode relevant informa-tion for the tensor field, why do we usually choose to visualize just the major-eigenvector field, ratherthan drawing a single image containing vector glyphs?

• Oriented glyphs such as arrows are typically preferred against unoriented ones (e.g. lines) when vi-sualizing vector fields. Why do not we use such oriented glyphs, but prefer unoriented glyphs, whenvisualizing eigenvector fields?

Answers:

• We truly can use another tensor glyph in practice, called axis system, formed by three vector glyphsthat separately encode the three eigenvectors scaled by their corresponding eigenvalues. However, for3D datasets they create too much confusion due to the 3D spatial overlap, whereas the rounded convexellipsoid shapes tend to be more distinguishable even with small amount of overlap. Also, we often areinterested only in the direction of change, which is always determined by the largest eigenvalue.

• Eigenvectors have an unoriented nature. A tensor is independent of any chosen frame of reference. Ingeneral, any scalar function f(λ1, λ2, λ3) that only depends on the eigenvalues again is an invariant.As a consequence, also every scalar function of invariants is an invariant itself. Eigenvectors have nomagnitude and no orientation (are bidirectional).

We use the term direction space for the feature space that consists of directions.The full direction information is represented as a triple of points. Because eigenvectors are normalized,

no additional scaling is needed and all points lie on the surface of the unit sphere. In general, we are onlyinterested in a single direction or in two selected directions. For a single direction, the direction space is a2D feature space with a spherical basis. Due to the unoriented nature of the eigenvectors, the space furtherreduces to a hemisphere.

Symmetric tensors are separated into shape and orientation. Here, shape refers to the eigenvalues andorientation to the eigenvectors. Symmetric tensors can be represented as diagonal matrices. The basis forsuch a representation is given by the eigenvectors corresponding to the diagonal matrix. For symmetric ten-sors, the eigenvalues are all real, and the eigenvectors constitute an orthonormal basis. The diagonalizationgenerally is computed numerically via singular value decomposition (SVD) or principal component analysis(PCA).

6. EXERCISE 6

Consider a smooth 2D scalar field f(x, y), and its gradient ∇f , which is a 2D vector field. Consider nowthat we are densely seeding the domain of f and trace streamlines in ∇f , upstream and downstream. Where

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do such streamlines meet? Can you give an analytic definition of these meeting points in terms of values ofthe scalar field f?

Hints: Consider the direction in which the gradient of a scalar field points.They can meet in critical points, where quantity f reaches its extremal values: sinks downstream, and

sources upstream, where ||∇f || = 0 and values of f are close to maxima or minima.

7. EXERCISE 7

Consider that we have a (dense) point cloud P = {pi} of N 3D points, which are the samples of a 3Dsmooth and non-intersecting surface. Many methods exist for the reconstruction of a meshed surface fromsuch an unorganized point cloud. However, several such methods require to know the orientation of thesurface normal ni at each sample point pi. Describe in detail a method to compute this normal orientationbased on principal component analysis applied to P.

There are two possibilities:

• Obtain the underlying surface from the acquired point cloud by using surface meshing techniques, andthen computing the surface normals from the mesh by averaging;

• Infer the surface normals from the point cloud dataset directly.

The problem of determining the normal to a point on the surface is approximated by the problem ofestimating the normal of a plane tangent to the surface, which in turn becomes a least-square plane fittingestimation problem:

• Collect some nearest neighbors of pi, for instance 12;

• Fit a plane to pi and its 12 neighbors;

• Use the normal of this plane as the estimated normal for pi.

Surface normal at a point can be estimated from the surrounding point neighborhood support of the point(also called k -neighborhood). The solution for estimating the surface normal can be reduced to PrincipalComponent Analysis of a covariance matrix created from the nearest neighbors of the query point. Morespecifically, for each point pi, we assemble the covariance matrix C as follows:

C =1

3

k∑i=1

(pi − p) · (pi − p)T , C · ~sj = λj · ~sj , j ∈ 1, 2, 3,

where k = 12 is the number of point neighbors considered in the neighborhood of pi, and p represents the3D centroid of the nearest neighbors, λj is the j-th eigenvalue of the covariance matrix, and ~sj is the j-theigenvector. Principal vector s3 is perpendicular to the tangent plane, and specifies our estimated normaln = 1

λ3s3 if scaled by corresponding eigenvalue λ3.

In general, because there is no mathematical way to solve for the sign of the normal, its orientationcomputed via Principal Component Analysis (PCA) as shown above is ambiguous, and not consistentlyoriented over an entire point cloud dataset. There is a question of the right scale factor: given a sampledpoint cloud dataset , what are the correct k values that should be used in determining the set of nearestneighbors of a point?

8. EXERCISE 8

Given a 2D shape, represented as a binary image, or alternatively as a (densely sampled) 2D polyline,an important tool in graphics and visualization is finding the so called oriented bounding box (OBB) of thisshape. In 2D, the OBB is a (possibly not axis-aligned) rectangle, which encloses the shape as tightly aspossible. Present a way of computing an OBB, given an unordered set of 2D points S = {pi} which denselysample the boundary of such a 2D shape, based on principal component analysis (PCA).

Given a blob of points S = {pi}, PCA allows to compute a covariance matrix for the point set. Theeigenvectors of this matrix specify orthogonal OBB’s half-axis e1 and e2. The average of the points is theOBB’s center: χ =

∑pi/N.

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The OBB itself can be defined as rectangle who’s center coincides with χ and four vertices are computedas follows:

a =χ+ ~e1 + ~e2,

b =χ− ~e1 + ~e2,

c =χ− ~e1 − ~e2,

d =χ+ ~e1 − ~e2,

9. EXERCISE 9

(Hyper)streamline tracing, or tractography, is one of the best known methods for visualizing a 3D tensorfield such as the ones produced by 3D diffusion tensor magnetic resonance imaging (DT-MRI). Both theseeding strategy and the streamline tracing stop criterion have to be carefully set in function of thecharacteristics of the DT-MRI field to obtain useful visualizations. Describe one typical strategy for seedingand one for stopping the tracing, and explain how they are related to the DT-MRI field values.

First, a seed region is identified. This is a region where fibers should intersect, so it can be detectede.g., by thresholding one of the anisotropy metrics presented in Section 7.3. Second, streamlines are denselyseeded in this region and traced (integrated) both forward and backward in the major eigenvector field e1

until a desired stop criterion is reached.The stop criterion is, in practice, a combination of various conditions, each of which describes one

desired feature of the resulting visualization. These can contain, but are not limited to, a minimal valueof the anisotropy metric considered beyond which the fiber structure becomes less apparent), themaximal fiber length, exiting or entering a predefined region of interest specified by the user(which can present previously segmented anatomical structure), and a maximal distance from othertracked fibers (beyond which the current fiber “strays” from a potential bundle structure that is the targetof the visualization).

10. EXERCISE 10

Hyperstreamlines visualize a tensor field by constructing streamlines in the vector field given by themajor eigenvector of the tensor field. The medium and minor eigenvectors are encoded, at each pointalong a hyperstreamline, by using an ellipse whose half-axes are oriented along the medium and minoreigenvectors, and respectively scaled to reflect the sizes of the medium and minor eigenvalues. Propose adifferent hyperstreamline construction, whose cross-section would not be an ellipse, but a different shape.

Hints: Think about other tensor glyph shapes. Discuss the advantages and/or disadvantages of yourproposal as compared to hyperstreamlines that use an elliptic cross-section.

For example, we can use a cross, whose arms are scaled and rotated to represent the medium and minoreigenvectors. Superquadric tensor glyphs are a more sophisticated approach that resolves some ambiguity.

11. EXERCISE 11

Fiber clustering is a method that, given a set of 3D curves computed e.g., by tracing streamlines alongthe major eigenvector of a tensor field, partitions (or clusters) this fiber-set into subsets of fibers thatare very similar in terms of spatial location and curvature. Fiber clustering is useful into highlightingsets of similar fibers and thereby potentially simplifying the resulting visualization. However, using justgeometric attributes to compare fibers ignores other information, such as encoded by the medium and minoreigenvectors and the corresponding eigenvalues. Propose an alternative similarity function for fibers that,apart from the geometric information, would also consider similarity of the medium and minor eigenvectorsand eigenvalues. Describe your similarity function in (mathematical) detail, and discuss why it would producea different (and potentially more insightful) clustering of tensor fibers.

12. EXERCISE 12

Image-based flow visualization (or IBFV) is a method that depicts a vector field by means of an animatedluminance texture, which gives the impression to ‘flow’ along the vector field (see Section 6.6.1). Imagine an

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extension of IBFV that would be used to visualize 2D tensor fields. The idea is to use the major eigenvectorfield to construct the IBFV animation, and, additionally, encode the minor eigenvector an (or) eigenvalue inother attributes of the resulting visualization, such as color, luminance, or shading. How would you modifyIBFV to encode such additional attributes?

Hints: Take care that modifying luminance may adversely affect the result of IBFV, e.g., destroy theapparent flow patterns that convey the direction of the major eigenvector field.

We can use same Noise texture advected in the direction of main eigenvector field. After obtainingresulting texture N ′, we can color it based on orientation of the minor eigenvector.

13. EXERCISE 13

Consider a point cloud that densely samples a part of the surface of a sphere of radius R, defined inpolar coordinates θ, φ by the ranges [θmin, θmax] and [φmin, φmax]. The ‘patch’ created by this sampling isshown in the figure below. Given the three points a, b, c indicated in the same figure, describe what are thethree eigenvectors of the principal component analysis (PCA) applied to the points’ covariance matrix forsmall neighborhoods of each of these three points. The neighborhood sizes are indicated by the circles in thefigure. For this, indicate which are the directions of these eigenvectors, and (if possible from the providedinformation), which are their relative magnitudes.

Figure 3: Point cloud sampling a sphere patch with three points of interest.

For the sphere λ1 = λ2 > 0. In this case we can only determine the minor eigenvector s3, and vectors s1and s2 can be any two orthogonal vectors in the tangent plane, which are also orthogonal to s3.

Principal vector s3 is perpendicular to the tangent plane, and its magnitude equals λ3,and it coinsideswith the normal to abc plane at location C, when scaled by it’s corresponding eigenvalue: n = 1

λ3s3.

Let’s define a centroid as follows:

C =1

3

3∑i=1

(pi − p) · (pi − p)T , C · ~sj = λj · ~sj , j ∈ {1, 2, 3},

where k = 3 is the number of point-neighbors considered in the neighborhood of pi, so that p representsthe 3D centroid of the nearest neighbors, λj is the j-th eigenvalue of the covariance matrix, and ~sj the j-theigenvector.

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PCA can yild following magnitudes for first two principal directions: λ1 = λ2 = r, so that r = |C − a| =|C − b| = |C − c| is a radius of circumscribed circle of the triangle abc., and we can choose s1 = vecC − a,(b,or c accordingly). Vector s2 = s⊥1 belongs to plane abc and is orthogonal to s1 having same magnitude|s2| = |s1| = r.

R = (sinθ cosϕ, sinθ sinϕ, cosθ)

Figure 4: Polar to Cartesian coordinate transformation in 3D.

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