cse788 mathemacal$and$algorithmic$ foundaons$for...
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CSE 788 Mathema-cal and Algorithmic
Founda-ons for Data Visualiza-on Winter 2011 Han-‐Wei Shen
Class Objec-ves
• Give you an overview of data visualiza-on research – Focus more on scien-fic data
• Overview the basic mathema-cs and algorithms
• Introduce you the state-‐of-‐the-‐art research in visualiza-on
• Provide you possible future research direc-ons
Reference Books
• There is no required textbook • Reference books: – The visualiza-on handbook by Charles D. Hansen and Christopher R. Johnson (Elsevier)
– Mathema-cal methods in the physical sciences by Mary L. Boas (Wiley)
– Research papers published in various top visualiza-on conferences and journals (IEEE Vis, IEEE Transac-ons on Visualiza-on and Computer Graphics, EuroVis, Pacific Vis, etc)
Reference Books
• Principles of mathema-cal analysis, by Walter Rudin
• Free PDF: Introduc-on to real analysis by William Trench
• Elementary differen-al geometry by Bare] O’neill
Grading
• Homeworks and paper summaries: 25% • Presenta-on: 15 % • Labs: 20% • Final project: 40%
All done individually
Another Class
• CSE 888, Wednesday 11:30 am. to 12:30 pm. • Extension of this class • We will read addi-onal papers related to the topics discussed in this class
• Student presenta-ons
What is Visualiza-on?
• A branch of computer graphics research • A process of conver-ng numerical data into visual images
• Focused more on data analysis than pre]y pictures
Visual Analysis Pipeline
Data acquisi-on/genera-on
Data prepara-on
Feature extrac-on and visual mapping
Rendering Analysis
Images and Anima-ons
Data Acquisi-on and Genera-on
• Spa-al Data – Medical imaging: CT, MRI, DT imagery
– Numerical simula-ons: PDEs – Other devices: Radar, LiDAR, etc
Examples of PDEs
• PDE: Par-al Differen-al Equa-ons • General forms: solve the func-on
in the form of:
• Result: Mul-variate scalar and vector func-ons
u(x1, x2, ..., xn)
F (x1, ...xn, u,∂
∂x1u, ...
∂
∂xnu,
∂2
∂x1∂x2u, ...) = 0
Examples of PDEs • Heat equa-on:
• Wave equa-on:
• Laplace equa-on:
∂U
∂t= α
∂2U
∂x2
∂2U
∂t2= c2 ∂2U
∂x2
∂2φ
∂x2+
∂2φ
∂y2= 0
Scien-fic Compu-ng and Numerical Methods
Data Acquisi-on and Genera-on
• Non-‐spa-al Data – Network Data (Graphs and trees) – Text Data (documents) – Matrix data (spread sheets)
Visual Analysis Pipeline
Data acquisi-on/genera-on
Data prepara-on
Feature extrac-on and visual mapping
Rendering Analysis
Images and Anima-ons
Data Prepara-on
• Reconstruc-on • Smoothing/De-‐noising • Re-‐sampling • Transforma-on (cosine, wavelet, Fourier transform) • Projec-on to lower dimensions • Compression/down-‐sampling • Par--oning/Bricking/Distribu-on • Mul--‐resolu-on hierarchy • Data layout and stripping • Histogram/entropy calcula-on • ... and more
Visual Analysis Pipeline
Data acquisi-on/genera-on
Data prepara-on
Feature extrac-on and visual mapping
Rendering Analysis
Images and Anima-ons
Feature Extrac-on • Generic features: isosurfaces, streamlines, pathlines, cri-cal points (local extreme points)
• Specific features: vortex cores, material boundary, flow separa-on lines
Visual Analysis Pipeline
Data acquisi-on/genera-on
Data prepara-on
Feature extrac-on and visual mapping
Rendering Analysis
Images and Anima-ons
Visual Mapping and Rendering • The process of conver-ng data to visual forms
• Volume rendering – Op-cal models – Transfer func-ons
• Polygon rendering (e.g. for isosurfaces) – Ray tracing – Raster graphics using GPUs
• Advanced illumina-on and stylized rendering
f(s,∇s) = (r, g, b, α)
Illumina-on and NPR
Visual Analysis Pipeline
Data acquisi-on/genera-on
Data prepara-on
Feature extrac-on and visual mapping
Rendering Analysis
Images and Anima-ons
Analysis
• The recent focus in visualiza-on research • The purpose of visualiza0on is insight, not pre5y pictures (paraphrased from Richard Hamming’s “the purpose of compu0ng is insight, not numbers”
Analysis
• Topological – Surface topology: Reeb graph, contour trees – Vector field topology: loca-ons and types of cri-cal points (sink, source, saddles)
• Geometrical – Space curves: curvature, torsion – Surface: first and second fundamental forms, various types of curvatures
• Sta-s-cal – Histogram, various types of distribu-on models – Informa-on theory: entropy measures
Visual Analysis Pipeline
Data acquisi-on/genera-on
Data prepara-on
Feature extrac-on and visual mapping
Rendering Analysis
Images and Anima-ons
Images and Anima-ons
• The output of the visual data analysis pipeline • Analysis of image and anima-on output can be used to adjust the visualiza-on parameters
• Many metrics available to evaluate the quality of visualiza-on
• Visual computa-on can be accelerated by considering the final image proper-es (visibility, projec-on, etc) – image space method
Image Based Visualiza-on Algorithm
• Example: image based streamline seeding
Topics this quarter
• Differen-al calculus basics (par-al differen-al) – Real func-on of one or mul-ple variables
• Level sets (isosurfaces) of scalar func-ons • Topological analysis of level sets • Differen-al geometry basics – Curves and surfaces
• Direct volume rendering – Op-cal model, basic ray tracing algorithms, transfer func-ons
• Quick overview of level set and PDEs
Scalar data visualiza-on and analysis
Topics this quarter (cont’d)
• Vector calculus basics • Par-cle tracing and numerical ODE (ordinary differen-al equa-on)
• Vector field topology: cri-cal point classifica-on
• Flow visualiza-on techniques • Quick overview of tensor data visualiza-on
Vector data visualiza-on and analysis
Topics this quarter (cont’d)
• Random variables and distribu-ons • Mul-variate distribu-ons
• Es-ma-on
• Informa-on theory and entropy measures
• Applica-ons in visualiza-on
Sta-s-cs based data visualiza-on and analysis