1 Domain search Span Space High Gradient Subdivide the space - - PDF document

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Iso-Contouring – Advanced Issues

• 1. Efficiently determining which cells to

examine.

• 2. Using iso-contouring as a slicing

mechanism

• 3. Iso-contouring in higher dimensions
• 4. Texturing and coloring of iso-contours
• 5. Polygonal simplification of contours.
• 6. Choosing a good iso-value

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Iso-surface cells: cells that contain iso- surface. min < iso-value < max Marching cubes algorithm performs a linear search to locate the iso-surface cells – not very efficient for large-scale data sets.

Iso-surface cell search

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Iso-surface Cells

For a given iso-value, only a smaller portion

• f the cells contain part of the iso-surface.

For a volume with n x n x n cells, the average number of the iso-surface cells is O(n x n) (ratio of surface v.s. volume)

n n n

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Efficient Searching

With < 10% of the voxels contributing to the surface, it is a waste to look at every voxel. A voxel can be specified in terms of its interval, its minimum and maximum values.

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Efficient iso-surface cell search Problem statement:

Given a scalar field with N cells:

⌧c1, c2, …, cn

With min-max range:

⌧(a1,b1), (a2,b2),…, (an, bn)

Find {Ck | ak < C < bk; C=iso-value}

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Efficient search methods

• 1. Spatial subdivision (domain search)
• 2. Value subdivision (range search)
• 3. Contour propagation

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Domain search

• Subdivide the space into several sub-domains, check the

min/max values for each sub-domain

• If the min/max values (extreme values) do not contain the

iso-value, we skip the entire region

Min/max

Complexity = O( k log(n/k) )

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Span Space

Mi Minimum nimum

Threshold Threshold High Gradient Low Gradient 4/21/2003

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Span Space - Representing

Mi Minimum nimum

• K

K-

• d Trees

d Trees

Split Min. axis Split Max. axis

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Basic Idea: Given an initial cell that contains iso-surface, the remainder of the iso-surface can be found by propagation

Contour Propagation

A B D C E Initial cell: A Enqueue: B, C Dequeue: B Enqueue: D … FIFO Queue A B C C C D …. Breadth-First Search

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Challenges

Need to know the initial cells!

For any given iso-value C, finding the initial cells to start the propagation is almost as hard as finding the iso-surface cells. You could do a global search, but …

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Solutions

(1) Extrema Graph (Itoh vis’95) (2) Seed Sets (Bajaj volvis’96)

Problem Statement: Given a scalar field with a cell set G, find a subset S G, such that for any given iso-value C, the set S contains initial cells to start the propagation. We need search through S, but S is usually (hopefully) much smaller than G. We will only talk about extrema graph due to time constraint

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Extrema Graph (1)

Basic Idea: If we find all the local minimum and maximum points (Extrema), and connect them together by straight lines (Arcs), then any closed Iso-contour is intersected by at least one of the arcs.

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Extrema Graph (2)

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Extrema Graph (3)

E1 E2 E3 E4 E7 E5 E6 E8 a2 a3 a4 a5 a6 a7 a1 Extreme Graph: { E, A: E: extrema points A: Arcs conneccts E } An ‘arc’ consists of cells that connect extrema points (we only store min/max of the arc though)

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Extrema Graph (4)

Algorithm: Given an iso-value 1) Search the arcs of the extrema graph (to find the arcs that have min/max values that contain the iso-value 2) Walk through the cells along each of these arcs to find the seed cells 3) Start the iso-contour propagation from the seed cells 4) …. There is something more that needs to be done…

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We are not done yet …

What ?! We just mentioned that all the closed iso-contours will intersect with the arcs connecting the extrema points How about non-closed iso-contours? (or called open iso- contours)

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Extrema Graph (5)

Contours missed

These open iso-contours will intersect with ?? cells

Boundary Cells!!

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Extrema Graph (6)

Algorithm (continued) Given an iso-value:

1) Search the arcs of the extrema graph (to find the arcs that have min/max values that contain the iso-value. 2) Walk through the cells along each of the arcs to find the seed cells. 3) Start the iso-contour propagation from the seed cells.

4) Search the cells along the boundary and find additional seed cells. 5) Propagate from these new seed cells for the open iso-contours.

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Extrema Graph

Efficiency - Number of cells visited:

extrema graph - N0.33 boundary - N 0.66 Iso-surface - N 0.66

based on tetrahedra - will create more surface triangles ... should extract the same number of cells/ triangulation as Marching Cubes

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Extrema Graph

Storage Costs:

Extrema graph – very small amount of memory for most datasets. For unstructured grids, I need to be able to access my neighbors. This requires a very expensive data structure, more than the span-space data structures.

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Selecting iso-values

Which iso-values should I examine to best comprehend my dataset? Some data has very specific values:

The aluminum structure in my simulation will start to fail at a pressure

• f 5672psi.

The molecular state changes from a liquid to a gas at 100° C.

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Contour Spectrum

Basic Idea: Calculate and present to the user several properties of an iso-contour. Do this for all iso-contours. This leads to several functions in terms of the iso-value, α. Present these functions to the user as an aid in picking contour values.

These slides are from Baja, Pascucci and Schikore (IEEE Visualization 1997)

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Outline

User Interface Signature Computation Real Time Quantitative Queries Rule-based Contouring Topological Information Future Directions

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Graphical User Interface Graphical User Interface for Static Data for Static Data

• The horizontal axis

spans the scalar values α

• Plot of a set of

signatures (length, area, gradient ...) as functions of the scalar value α.

• Vertical axis spans normalized ranges of each signature.
• White vertical bars mark current selected isovalues.

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Graphical User Interface for time varying data

(α,t ) --> c The color, c, is mapped to the magnitude

• f a

signature function of time, t, and isovalue α α t c low high magnitude

• The horizontal axis spans the scalar value

dimension α.

• The vertical axis spans the time dimension t .

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User Interface - MRI of a human torso -

• In real time the exact value of each signature is displayed.
• The isocontour that bounds the region of interest is obtained by

selecting the maximum of the gradient signature.

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Signature Computation

Consider a terrain of which you want to compute the length of each isocontour and the area contained inside each isocontour

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Signature Computation

• The length of each

contour is a C0 spline function.

• The area inside/outside

each isocontour is a C1 spline function.

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Signature Computation

In general, the size (surface area) of each iso-contour of a scalar field of dimension d is a spline function of d-2 continuity. The size (volume) of the region inside/ outside is given by a spline function of d-1 continuity

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Real Time Quantitative Queries

(agricultural yield data)

• size and position
• f the region

with unsatisfactory production

• size and position

size and position

• f the region
• f the region

where wrong where wrong data acquisition data acquisition

• ccurred
• ccurred

Spectrum space is a useful space to visualize by itself.

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Rule-based Contouring

(CT scan of an engine)

• The contour

spectrum allows the development of an adaptive ability to separate interesting isovalues from the

• thers.

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Rule-based Contouring (foot of the Visible Human)

• The contour

spectrum allows the development of an adaptive ability to separate interesting isovalues from the

• thers.

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Topological information.

an isocontour with three connected components two

• f which are about to merge}
• number of components per

number of components per isocontour isocontour

• which

which isocontours isocontours merge together or merge together or split while modifying the split while modifying the isovalue isovalue. .

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Other Topics

Interval Volumes 4D Iso-contouring Smooth surfaces Polygon Reduction Triangle Strip generation Time-varying iso-contours Texture map parameterization

Penny Rheingans