Tensor Visualisation
Visualisation : Lecture 14Tensor VisualisationVisualisation – Lecture 14Taku KomuraInstitute for Perception, Action & BehaviourSchool of InformaticsTaku KomuraTensors 1
Visualisation : Lecture 14Reminder : Attribute Data Types Scalar– Vector– Taku Komuracolour mapping, contouringlines, glyphs, stream {lines ribbons surfaces}Tensor–complex problem : active area of research–today : simple techniques for tensor visualisationTensors 2
Visualisation : Lecture 14What is a tensor ? A tensor is a data of rank k defined in n-dimensional space (ℝn)––generalisation of vectors and matrices in ℝn—Rank 0 is a scalar—Rank 1 is a vector—Rank 2 is a matrix—Rank 3 is a regular 3D arrayk : rank defines the topological dimension of the attribute—–n : defines the geometrical dimension of the attribute—Taku KomuraTopological Dimension: number of independent continuous variables specifyinga position within the topology of the datai.e. k indices each in range 0 (n-1)Tensors 3
Visualisation : Lecture 14Tensors in ℝ 3Here we limit of discussion to tensors in ℝ3–In ℝ3 a tensor of rank k requires 3k numbers—A tensor of rank 0 is a scalarA tensor of rank 1 is a vector— A tensor of rank 2 is a 3x3 matrix— A tensor of rank 3 is a 3x3x3 cube—[] [V1V V2V3 Taku KomuraT 11 T 21 T 31T T 12 T 22 T 32T 13 T 23 T 33(30 1)(31 3)(9 numbers)(27 numbers)]We will only treat rank 2 tensors – i.e. matricesTensors 4
Visualisation : Lecture 14Where do tensors come from? Stress/strain tensors– DT-MRI– molecular diffusion measurementsThese are represented by 3x3 matrices–Taku Komuraanalysis in engineeringOr three orthogonal eigenvectors and threecorresponding eigenvaluesTensors 5
Visualisation : Lecture 14Stress Tensor Say we are to apply force from various directions to asmall boxThe stress at the surface can be represented by thestress tensor:––σxx,σyy σzz indicates a ‘normal’ stressin x,y,z direction, respectivelyThe rest indicates a shear stress–Taku KomuraTensors 6
Visualisation : Lecture 14Stress Tensor A ‘normal’ stress is a stress perpendicular (i.e. normal) to aspecified surfaceA shear stress acts tangentially to the surface orientation–– Stress tensor : characterised by principle axes of tensorWe can compose a 3x3 matrix called Stress Tensor representingthe stress added to the boxThis is for computing how much the shape gets distorted by the stressTaku KomuraTensors 7
Visualisation : Lecture 14Computing Eigenvectors fromthe Stress Tensor———3x3 matrix results in Eigenvalues (scale) of normal stress along eigenvectors(direction)form 3D co-ordinate system (locally) with mutually perp. Axesordering by eigenvector referred to as major, medium and minoreigenvectorsTaku KomuraTensors 8
Visualisation : Lecture 14Diffusion Tensor-MRI : diffusiontensor Water molecules have anisotropic diffusion in the body dueto the cell shape and membrane propertiesTaku Komura–Neural fibers : long cylindrical cells filled with fluid–Water diffusion rate is fastest along the axis–Slowest in the two transverse directions–brain functional imaging by detecting the anisotropy–Again, we can represent the diffusion tensor by theeigenvectors and eigenvaluesTensors 9
Visualisation : Lecture 14Tensors : Visualisation Methods For visualization of tensors, we have to visualize–three vectors orthogonal to each other–At every sample point in the 3D spaceVector methods–hedgehogs–Streamline, hyper-streamlineGlyphs–Taku Komura3D ellipses particularly appropriate(3 modes of variation)Tensors 10
Visualisation : Lecture 14Hedgehogs Using hedgehogs to draw the three eigenvectors– Taku KomuraThe length is the stress valueGood for simple cases as above–Applying forces to the box–Green represents positive, red negativeTensors 11
Visualisation : Lecture 14Hedgehogs Taku KomuraNot good if–The grid is coarse–The stress is non-uniform, non-linear across the objectTensors 12
Visualisation : Lecture 14Tensor Visualisation byColormap Visualise just the major eigenvectorsas a vector field–alternatively medium or minoreigenvectore.g. Major eigenvectordirection visualised with(u,v,w) (r,g,b)colourmap.Source: R. SierraTaku KomuraTensors 13
Visualisation : Lecture 14Streamlines for tensor visualisation Each eigenvector defines a vector field Using the eigenvector to create the streamline–Taku KomuraWe can use the Major vector, the medium and theminor vector to generate different streamlinesTensors 14
Visualisation : Lecture 14Streamlines for MRI For DT-MRI, major vectorindicates nerve pathways or stressdirections.Visualization of the brain nerves bythe streamlines based on the majoreigenvectors of the water diffusionhttp://www.cmiv.liu.se/Taku KomuraTensors 15
Visualisation : Lecture 14Hyper-streamlines Construct a streamline fromvector field of majoreigenvectorEllipse from of 3orthogonal eigenvectors Form ellipse together with mediumand minor eigenvector– [Delmarcelle et al. '93]Swept along StreamlineForce applied hereboth are orthogonal to streamline directionSweep ellipse along streamline–Hyper-Streamline(type of stream polygon)Visualizing the information of thethree eigenvectors altogether !Taku KomuraTensors 16
Visualisation : Lecture 14Tensor Glyphs Ellipses–rotated into coordinatesystem defined by eigenvectors of tensor–axes are scaled by the eigenvalues–very suitable as 3 modes of variationEllipseEigenvector axesClasses of tensor:–(a,b) - large major eigenvalue—–(c,d) - large major and medium eigenvalue—–ellipse approximates a lineellipse approximates a plane(e,f) - all similar - ellipse approximates a sphereTaku KomuraTensors 17
Visualisation : Lecture 14Diffusion Tensor VisualisationThe brain consists of different types oftissues with different diffusionsAnisotropic tensors indicate nerve pathwayin brain: Blue shape – tensor approximates aline. (nerves) Yellow shape – tensor approximates aplane.Yellow transparent shape – ellipseapproximates a sphereColours needed due to ambiguity in 3Dshape Baby's brain imageTaku Komura(source: R.Sierra)Tensors 18
Visualisation : Lecture 14Example : tensor glyphs Glyphs with similar positional and modes of shape variation toellipsoid used for MRI diffusion tensor visualisation–disambiguates orientation–coloured by tensor class—interpolated between classes[Westin et al. '02]Taku KomuraTensors 19
Visualisation : Lecture 14Stress EllipsesForce applied here Taku KomuraForce applied to dense 3D solid– resulting stress at 3D positionin structureEllipses visualise the stresstensorTensor Eigenvalues:–Large major eigenvalue indicatesprinciple direction of stress–‘Temperature’ colourmapindicates size of majoreigenvalue (magnitude of stress)Tensors 20
Visualisation : Lecture 14Interpolation of Tensors Taku KomuraHow do we interpolate over tensors ?Can simply interpolate over eigen-componentsindividually:But if it represents specific information (e.g.nerve pathway) then shape preserving methodsare preferred:Tensors 21
Visualisation : Lecture 14Summary Tensor Visualisation–challenging–for common rank 2 tensors in ℝ3–common sources stress / strain / MRI dataa number of methods exist via eigenanalysisdecomposition of tensors—Taku Komura—3D glyphs – specifically ellipsoids—vector and scalar field methods—hyper-streamlinesTensors 22
Taku Komura Tensors 3 Visualisation : Lecture 14 What is a tensor ? A tensor is a data of rank k defined in n-dimensional space (ℝn) – generalisation of vectors and matrices in ℝn — Rank 0 is a scalar — Rank 1 is a vector — Rank 2 is a matrix — Rank 3 is a regular 3D array – k: rank defines the topological dimension of the attribute — Topological Dimension: number of .
02 - tensor calculus 1 02 - tensor calculus - tensor algebra tensor calculus 2 tensor the word tensor was introduced in 1846 by william rowan hamilton . it was used in its current meaning by woldemar voigt in 1899. tensor calculus was deve-loped around 1890 by
LA, a tensor contraction, and a possible tensor contractions design using Einstein summation notation and a Domain Speci c Embedded Language (or DSEL ) . Right: Illustration of tensor contractions needed and viable approaches to solve them in machine learning. Figure 1 Left illustrates the notion of tensor and tensor contraction in DLA, as well .
Scientific and information visualisation Research on using visualisation to facilitate information and knowledge processing has advanced greatly in the past two decades. Card et al. (1999) classify visualisation techniques into two categories – scientific visualisation and information visualisation – based on the nature of the data
– VTK is a set of methods (toolkit) that implement a variety of visualisation operations – Implements a visualisation pipeline – Platform independent (we use linux, DICE) – Object-orientated visualisation – Program in C or Java or use an interpreted language such as Tcl/Tk or Python – VTK also implements basic tools for visualisation:
on tensor decomposition, which aims to dissolve tensors into separable representations, while Song et al.(2019) reviewed tensor completion, which aims to impute the unobserved entries of a partially observed tensor. Tensor decomposition and tensor completion are both fundamen-tal problems i
tensor. Similarly, the vector is a rst order tensor, and a scalar is a zeroth order tensor. The order of the tensor equals the number of free indices (see SectionA.3) that the tensor has, and it can also be interpreted as the dimensionality of the array needed to represent the
studies relating to whether visualisation may help L2 learners’ writing. Using visualisation as a learning strategy, this paper reports on how visualisation training might affect a group of Chinese intermediate EFL learners’ narrative writing. Quantitative data from the pre-test and post-test did not
(CCSS) for Writing, beginning in early elementary, will be able to meet grade-level writing goals, experience success throughout school as proficient writers, demonstrate proficiency in writing to earn an Oregon diploma, and be college and career-ready—without the need for writing remediation. The CCSS describe ―What‖ writing skills students need at each grade level and K-12 Writing .
