Required Readings Further Reading Data Reduction HyperSlice: Visualization of scalar functions of many variables. how to reduce amount of stuff to draw? Chapter 8: Attribute Reduction Methods Jarke J. van Wijk and Robert van Liere. Proc. IEEE Visualization crosscuts view composition considerations Lecture 10: Attribute Reduction 1993, p 119-125. Glimmer: Multilevel MDS on the GPU. Stephen Ingram, Tamara Methods Interactive Hierarchical Dimension Ordering, Spacing and Filtering item reduction Munzner and Marc Olano. IEEE TVCG, 15(2):249-261, Mar/Apr for Exploration Of High Dimensional Datasets. Jing Yang, Wei 2009. last time Information Visualization Peng, Matthew O. Ward and Elke A. Rundensteiner. Proc. InfoVis rows of table CPSC 533C, Fall 2011 2003. attribute reduction A Data-Driven Reflectance Model. Wojciech Matusik, Hanspeter Tamara Munzner Pfister, Matt Brand and Leonard McMillan. Proc. SIGGRAPH this time 2003 columns of table UBC Computer Science methods for both Wed, 12 October 2011 filtering, aggregation, ordering 1 / 44 2 / 44 3 / 44 4 / 44 Attribute Reduction Methods Slicing: High-Dimensional Functions Slicing: HyperSlice Slicing/Cutting: Spatial Data camera metaphors HyperSlice: matrix of orthogonal 2D slices 4D function � 3 i =0 w i / (1 + | x − p i | 2 ) easy to understand: spatial data, 3D to 2D, axis aligned slicing, cutting, projection each panel is display and control: drag to change slice diagonals = standard graph filtering, ordering, aggregation simple 3D example for attributes as opposed to items x 5 dimensionality reduction uncovering hidden structure x 4 estimating true dimensionality x 3 generating synthetic dimensions linear mappings x 2 nonlinear mappings displaying low-dimensional spaces x 1 scatterplots, SPLOMS, landscapes x 3 x 4 x 5 [Fig 0. Rieder et al. Interactive Visualization of Multimodal Volume Data for x 1 x 2 Neurosurgical Tumor Treatment. Computer Graphics Forum (Proc. EuroVis 2008) [Fig 1, 2. van Wijk and van Liere. HyperSlice: Visualization of scalar functions of 27(3):1055–1062, 2008. many variables. Proc. IEEE Visualization 1993] [Fig 4. van Wijk and van Liere. HyperSlice: Visualization of scalar functions of many variables. Proc. IEEE Visualization 1993] 5 / 44 6 / 44 7 / 44 8 / 44 Slicing: HyperSlice Projections Attribute Filtering Attribute Ordering orthographic: remove all information about filtered dims filtering, but for attributes rather than items ordering, but for attributes rather than items satellite orbit eccentricity: x pos, y pos, x vel, grav const hypercube: 3D to 2D, 4D to 3D (video) unfiltered vs filtered SPLOM Hierarchical Clustering Explorer perspective: some info about filtered dims remains [Fig 4. Yang et al. Interactive Hierarchical Dimension Ordering, Spacing and Filtering [Fig 4. van Liere and van Wijk. Visualization of Multi-Dimensional Scalar Functions [http://en.wikipedia.org/wiki/File:Lat%C3%A9co%C3%A8re 28.svg, for Exploration Of High Dimensional Datasets. Proc. InfoVis 2003] [Fig 1. Seo and Shneiderman. A Rank-by-Feature Framework for Unsupervised http://en.wikipedia.org/wiki/File:Railroad-Tracks-Perspective.jpg] Using HyperSlice. CWI Quarterly, 7(2), June 1994, 147-158. ] Multidimensional Data Exploration Using Low Dimensional Projections. Proc. IEEE 9 / 44 10 / 44 11 / 44 InfoVis 2004, p 65-72.] 12 / 44 Dimensionality vs Attribute Reduction Uncovering Hidden Structure Estimating True Dimensionality Showing Dimensionality Estimates vocab use in field not consistent measurements indirect not direct scree plots as simple way: error against # dims error for low-dim projection vs high-dim original dimension/attribute real-world sensor limitations original dataset: 294 dims no single correct answer; many metrics proposed attribute reduction: reduce set with filtering measurements made in sprawling space estimate: almost all variance preserved with < 20 dims cumulative variance that is not accounted for includes orthographic projection documents, images strain: match variations in distance (vs actual distance values) dimensionality reduction: create smaller set of new dims DR only suitable if (almost) all information could be stress: difference between interpoint distances in high set size is smaller than original, new dims completely conveyed with fewer dimensions and low dimensions synthetic how do you know? need to estimate true dimensionality clarification: includes dimensional aggregation to check if different than original! � P ij ( d ij − δ ij ) 2 includes some projections (but not all) stress ( D , ∆) = P ij δ 2 ij vocab: projection/mapping D : matrix of lowD distances ∆: matrix of hiD distances δ ij [Fig 2. Ingram et al. DimStiller: Workows for dimensional analysis and reduction. Proc. VAST 2010, p 3-10] 13 / 44 14 / 44 15 / 44 16 / 44
Linear Dimensionality Reduction: PCA Nonlinear Dimensionality Reduction DR in Visualization: Tasks Example: DR for CG Reflectance Model principal components analysis many techniques proposed find/verify new/synthetic dimensions goal: simulate how light bounces off materials to make describe location of each point as linear combination of MDS, charting, Isomap, LLE, TSNE,... are the new dimensions believable? realistic pictures weights for each axis optimization problem ex: data-driven reflectance model computer graphics: BRDF (reflectance) finding axes: first with most variance, second with next pro: can handle curved rather than linear structure find/verify clusters idea: measure what light does with real materials most, ... con: lose all ties to original dimensions is there clear cluster structure in the new low-dim space? does it match a conjectured clustering (color-coded)? new dimensions cannot be easily related to originals ex: glimmer [Fig 2. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [http://en.wikipedia.org/wiki/File:GaussianScatterPCA.png] 17 / 44 18 / 44 19 / 44 20 / 44 Capturing Material Reflectance Goal: Image Synthesis Need For Low-Dimensional Model Dimensionality Reduction: Linear step 1: create new renderings with CG objects that look how to do step 2 simulation of new materials? measurement: interaction of light with real materials first try: PCA, linear DR technique (spheres) like captured materials 104 materials * 4M pixels = 400 million dimensions result: error falls off sharply CG teapot looks just like real hematite result: 104 high-res images of material model much too hi-dim to be useful good results for step 1 around 45 dims each image 4M pixels step 2 problem: physically impossible intermediate goal: much more concise model that humans can points when simulating new materials understand/use to generate computer graphics images specular highlights cannot have holes! allow users to tweak meaningful knobs: how shiny, how greasy, how metallic, what color... step 2: simulate completely new materials rusty, greasy, ... dimensionality reduction to the rescue [Fig 7, 9. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [Fig 5. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [Fig 6, 1. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] 21 / 44 22 / 44 23 / 44 24 / 44 Dimensionality Reduction: Nonlinear Dimensionality Reduction: Nonlinear Finding Semantics for Synthetic Dimensions Understanding Synthetic Dimensions second try: charting, nonlinear DR second try: charting, nonlinear DR look for meaning in scatterplots crosscheck meaning better if data embedding is curved not flat scree plot suggests 10-15 dims each synthetic dimension named by people, not by arrows show simulated images (teapots) made from note that dim estimate depends on technique used! algorithm model points represent real-world images (spheres) check if those match dimension semantics people inspect images corresponding to points to decide if axis could have a meaningful name [Fig 12,16. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [Fig 10. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [Fig 11. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [Fig 12. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] 25 / 44 26 / 44 27 / 44 28 / 44 Understanding Synthetic Dimensions Understanding Synthetic Dimensions Nonlinear Dimensionality Reduction Spring-Based MDS: Naive repeat for all points Specular-Metallic Diffuseness-Glossiness MDS: multidimensional scaling compute spring force to all other points confusingly, large family of things all called MDS difference between high dim, low dim distance some linear, some nonlinear! move to better location using computed forces classical: minimize strain compute distances between all points early formulation equivalent to PCA (linear) O ( n 2 ) iteration, O ( n 3 ) algorithm spectral methods: approximate eigenvectors distance scaling: minimize stress nonlinear optimization force simulation (mass-spring) [Fig 14,16. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] [Fig 13,16. Matusik et al. A Data-Driven Reflectance Model. SIGGRAPH 2003] 29 / 44 30 / 44 31 / 44 32 / 44
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