Data we are discussing in the class. Scalar/vector/tensor. Heterogeneous. Scientific data. 3D+time (n<4) Information data
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1 Data we are disussing in the lass Soure: VIS, Universit of Stuttgart Sientifi data 3D+time (n<4 Salar/vetor/tensor Information data nd(n>3 Heterogeneous
2 Salar Data Analsis
3 Salar Fields The approimation of ertain salar funtion in phsial spae f(,,z. Disrete representation. Visualization primitives: olors, transparen, iso-ontours (2D, isosurfaes (3D, 3D tetures.
4 3D Surfae 3D surfaes an be onsidered some sub-sets orresponding to ertain iso-values of some salar funtions (e.g. a distane field. Therefore, analzing the harateristis of the surfaes an help understand the salar funtion the represent. Soure:
5 Shape Analsis What features do people are about? Computing orientation and range of the shape Computing etrema(protrusion Finding handles and holes Finding ridges and valles or other feature lines Computing skeletal representation
6 Sub-Topis Compute bounding bo Compute Euler Charateristi Estimate surfae urvature Line desription for onveing surfae shape Morse funtion and surfae topolog--reeb graph Salar field topolog--morse-smale omple
7 Bounding Bo
8 Shape Desriptors We need enters and varianes of all points on the surfae.
9 Shape Desriptors We need enters and varianes of all points on the surfae. There are different definitions of enters
10 Center of Mass ( i, i ( j, j
11 Center of Mass Center of Mass ( i, i ( j, j N N i i = ( = 1, i 1 N N i
12 Geometri Center Geometri enter ( i, i ( j, j ( min + ma min + i i, i 2 2 ma i
13 Centers Geometri enter min ( Center of Mass N N i i = i + ma i min i + ma, 2 2 = 1 i 1 (, N N i Whih one is better and wh? i
14 Variane What are the dominant diretions? ( i, i ( j, j
15 Variane What are the dominant diretions? ( i, i ( j, j How are the defined? Priniple omponents analsis
16 Variane What are the dominant diretions?, ( i i, ( j j = i i i i i i i i i i i i V ( ( ( ( ( ( ( ( Covariane matri
17 Variane What are the dominant diretions? ( i, i ( j, j Major, minor eigenvetors give the diretions
18 Variane, ( i i, ( j j [ ] k k k k k k k k k k = 2 2 ( ( ( ( ( (, ( k k p =
19 Variane, ( i i, ( j j [ ] k k k k k k k k k k = 2 2 ( ( ( ( ( (, ( k k p = It is a voting sheme
20 Bounding Boes ( i, i ( j, j Ais-aligned bounding bo Oriented bounding bo
21 Oriented Bounding Boes ( i, i ( j, j
22 Oriented Bounding Boes ( i, i ( j, j
23 Oriented Bounding Boes ( i, i ( j, j
24 Oriented Bounding Boes ( i, i ( j, j
25 Oriented Bounding Boes ( i, i ( j, j How do ou etend this to 3D?
26 Compute Euler Charateristis
27 Surfae Representation Consider a disrete polgonal representation V number of verties E number of edges F number of faes Euler harateristis L = V-E+F Euler harateristiis a topologial invariant, a number that desribes a topologial spae's shape or struture regardless of the wa it is bent. (Wiki
28 Elementar Collapse on Edges
29 Elementar Collapse on Edges
30 Elementar Collapse on Edges
31 Elementar Collapse on Edges
32 Elementar Collapse on Edges
33 Elementar Collapse on Edges
34 Elementar Collapse on Edges
35 Elementar Collapse on Edges
36 Elementar Collapse on Edges
37 Elementar Collapse on Edges
38 Elementar Collapse on Edges V=E for a losed and simple planar urve.
39 Elementar Collapse on Edges V=E for a losed and simple planar urve. What about 3D surfaes? Need to onsider the merging of the faes!
40 Elementar Collapse for Faes V=6 E=7 F=2 L=? V=6 E=6 F=1 L=?
41 Euler Charateristi of a Cube V=1 E=0 F=1 L=? V=8 E=12 F=6 L=?
42 Dual of a Heahedron
43 Dual of a Heahedron Does this dual hange the Euler harateristi?
44 What Do Different Euler Charateristis Tell us? L=?
45 What Do Different Euler Charateristis Tell us? L=0
46 What Do Different Euler Charateristis Tell us? What is its Euler harateristi?
47 What Do Different Euler Charateristis Tell us? What is its Euler harateristi?
48 What Do Different Euler Charateristis Tell us? L = -2! What about 3D surfaes?
49 What Do Different Euler Charateristis Tell us? For 3D surfaes, L=V-E+F=?
50 Aknowledge Part of the materials are provided b Prof. Eugene Zhang at Oregon State Universit
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