volume graphics (lecture 5 : contour tree / contour spectrum) lecture notes acknowledgement : j....
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Volume Graphics(lecture 5 : Contour Tree / Contour Spectrum)
lecture notes acknowledgement : J. Snoeyink, C. Bajaj
Bong-Soo Sohn
School of Computer Science and Engineering
Chung-Ang University
Contour Tree
h(x,y)
x
y
• Definition : a tree with (V,E)– Vertex ‘V’
• Critical Points(CP) (points where contour topology changes , gradient vanishes)
– Edge ‘E’: • connecting CP where an infinite contour class is created and CP where the
infinite contour class is destroyed.• contour class : maximal set of continuous contours which don’t contain
critical points
Properties• Display of Level Sets Topology (Structural Information)
– Merge , Split , Create , Disappear , Genus Change (Betti number change)
• Minimal Seed Set Generation
• Contour Segmentation– A point on any edge of CT corresponds to one contour component
Contour tree
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Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Minimal Seed Set Contour Tree
f
Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Minimal Seed SetContour Tree(local minima)
f
Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Minimal Seed SetContour Tree
(local maxima)
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Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Minimal Seed Set Contour Tree
fEach seed cell corresponds to a monotonic
path on the contour treeEach seed cell corresponds to a monotonic
path on the contour tree
Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Minimal Seed Set Contour Tree
f
For a minimal seed set each seed cell corresponds to a path that is not covered by
any over seed cell
For a minimal seed set each seed cell corresponds to a path that is not covered by
any over seed cell
Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Minimal Seed Set Contour Tree
f
Current isovalue
Each connected component of any isocontour corresponds exactly to one
point of the contour tree
Each connected component of any isocontour corresponds exactly to one
point of the contour tree
Optimal Single-Resolution Isocontouring
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
Range sweep
Optimal Single-Resolution Isocontouring
Evolution of level sets
• Watch contours of L(v) as v increases
Evolution of level sets
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Contour tree
Evolution of level sets
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Contour tree
Evolution of level sets
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Contour tree
Evolution of level sets
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Contour tree
Evolution of level sets
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Contour tree
Evolution of level sets
Contours appear, merge, split, & vanish
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Contour tree
Morse theory & Contour trees
• Karron et al 95, digital Morse theory• Bajaj et al 97–99, seed sets• Van Kreveld et al 97, 2-D contour tree• Tarasov & Vyalyi 98, 3-D contour tree
How to compute contour trees without sweeping all level sets
• Build join and split trees
• Merge to form contour tree
• Works in all dimensions...
The join tree
• Represents set {p | f(p) >= x}
• Can be computed w/out the surface of the level set.
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Join tree construction
• Sort data by value
• Maintain union/find for comp. of {p | f(p) >= x}
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Union/Find
• Data structure for integers 1..n supporting:
– Initialize() each integer starts in its own group• for i = 1..n, g(i) = i;
– Union(i,j) union groups of i and j• g(Find(i)) = Find(j);
– Find(i) return name of group containing i• group = i; • while group != g(group), group = g(group); // find group• while i != group,
{nx = g(i), g(i) = group, i = nx} // compress path
• Does n union/finds in O(n a(n)) steps [T72]
Join tree construction
• Sort data by value
• Maintain union/find for comp. of {p | f(p) >= x}
• O(n+ t·α(t) ) after sort
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The split tree
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• Represents the set {p | f(p) <= x}
• Reverse join tree computation
Merge join and split trees
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Merge join and split trees
• Algorithm
1. Add nodes from other tree
• Nodes have correct up degree in join & down degree in the split tree.
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2. Identify leaves
• A leaf in one tree, with up/down degree < 1 in other tree
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3. Add to contour tree
• A leaf in one tree, with up/down degree < 1 in other tree
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3. Add to contour tree
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3. Add to contour tree
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3. Add to contour tree
• Note that 4 is not a leaf
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3. Add to contour tree
• The merge time is proportional to the tree size; much smaller than the data
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4. Ends with Contour tree
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Advantages
• Doesn’t need the surface structure to build the contour tree.
• The merge becomes simple, and works on a subset of the points
• The tree structure can advise where to choose the threshold value
• Can cut diff. branches at diff. values.
Flexible contouring
• Threshold diff. areas at diff. values
Flexible contouring
• Threshold diff. areas at diff. Values
• Guided by the Contour Tree
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Contour Spectrum
• User Interface
• Signature Computation
• Real Time Quantitative Queries
• Topological Information
• Semi-Automatic Isovalue Selection
Input 3D Fuction
• Consider a terrain of which you want to compute the length of each isocontour and the area contained inside each isocontour.
Contour Spectrum :GUI for static data
• Plot of a set of signatures (length, area, gradient ...) as functions of the scalar value .
Contour Spectrum :GUI for time-varying data
(,t ) --> c
The color c is mapped to the magnitude of a signature function of time t and isovalue low
high
mag
nitu
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Isovalue Selection
Copyright: Chandrajit Bajaj, CCV, University of Texas at Austin
• The contour spectrum allows the development of an adaptive ability to separate interesting isovalues from the others.
Signature Computation
– The length of each contour is a c0 spline function.
The area inside/outside each
isocontour is a C1 spline function.
Signature Computation
• In general the size of each isocontour of a scalar field of dimension d is a spline function of d-2 continuity.
• The size of the region inside/outside is given by a spline function of d-1 continuity