Iso-surface cell search. Iso-surface Cells. Efficient Searching. Efficient search methods. Efficient iso-surface cell search. Problem statement:

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1 Iso-Contouring Advanced Issues Iso-surface cell search 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 4/21/2003 R. Crawfis, Ohio State Univ. 73 Iso-surface cells: cells that contain isosurface. 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. 4/21/2003 R. Crawfis, Ohio State Univ. 74 Iso-surface Cells For a given iso-value, only a smaller portion of the cells contain part of the iso-surface. For a volume with n x n x n cells, the n average number of the iso-surface cells is O(n x n) (ratio of surface v.s. volume) 4/21/2003 R. Crawfis, Ohio State Univ. 75 n n 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. 4/21/2003 R. Crawfis, Ohio State Univ. 76 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) Efficient search methods 1. Spatial subdivision (domain search) 2. Value subdivision (range search) 3. Contour propagation Find {Ck ak < C < bk; C=iso-value} 4/21/2003 R. Crawfis, Ohio State Univ. 77 4/21/2003 R. Crawfis, Ohio State Univ. 78

Domain search Span Space 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

2 Domain search Span Space 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 High Gradient Low Gradient Min/max Threshold Complexity = O( k log(n/k) ) 4/21/2003 R. Crawfis, Ohio State Univ. 79 Threshold Minimum 4/21/2003 R. Crawfis, Ohio State Univ. 80 Span Space - Representing K-d d Trees Contour Propagation Basic Idea: Given an initial cell that contains iso-surface, the remainder of the iso-surface can be found by propagation Initial cell: A FIFO Queue A Minimum Split Min. axis Split Max. axis 4/21/2003 R. Crawfis, Ohio State Univ. 81 E B C A D Enqueue: B, C Dequeue: B Enqueue: D. 4/21/2003 R. Crawfis, Ohio State Univ. 82 B C C C D Breadth-First Search Challenges Solutions 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 (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. 4/21/2003 R. Crawfis, Ohio State Univ. 83 We will only talk about extrema graph due to time constraint 4/21/2003 R. Crawfis, Ohio State Univ. 84

3 Extrema Graph (1) Extrema Graph (2) 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. 4/21/2003 R. Crawfis, Ohio State Univ. 85 4/21/2003 R. Crawfis, Ohio State Univ. 86 Extrema Graph (3) Extrema Graph (4) E1 E3 a4 E5 E2 a1 a3 E6 a6 a2 E4 a5 E7 a7 E8 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) 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 4/21/2003 R. Crawfis, Ohio State Univ. 87 4/21/2003 R. Crawfis, Ohio State Univ. 88 We are not done yet Extrema Graph (5) 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 isocontours) Contours missed These open iso-contours will intersect with?? cells Boundary Cells!! 4/21/2003 R. Crawfis, Ohio State Univ. 89 4/21/2003 R. Crawfis, Ohio State Univ. 90

4 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. Extrema Graph Efficiency - Number of cells visited: extrema graph - N 0.33 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 4/21/2003 R. Crawfis, Ohio State Univ. 91 4/21/2003 R. Crawfis, Ohio State Univ. 92 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. 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 of 5672psi. The molecular state changes from a liquid to a gas at 100 C. 4/21/2003 R. Crawfis, Ohio State Univ. 93 4/21/2003 R. Crawfis, Ohio State Univ. 94 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) Outline User Interface Signature Computation Real Time Quantitative Queries Rule-based Contouring Topological Information Future Directions 4/21/2003 R. Crawfis, Ohio State Univ. 95 4/21/2003 R. Crawfis, Ohio State Univ. 96

Graphical User Interface for Static Data Vertical axis spans normalized ranges of each signature. White vertical bars mark current selected isovalues.

97 Graphical User Interface for time varying data high (α,t ) --> c c The color, c, is mapped to the t magnitude of a signature α function of time, t, and isovalue α The horizontal axis spans the

98 magnitude User Interface - MRI of a human torso - In real time the exact value of each signature is displayed.

Signature Computation Consider a terrain of which you want to compute the length of each isocontour and the area contained inside each isocontour 4/21/2003 R. Crawfis, Ohio State Univ.

5 Graphical User Interface for Static Data Vertical axis spans normalized ranges of each signature. White vertical bars mark current selected isovalues. The horizontal axis spans the scalar values α Plot of a set of signatures (length, area, gradient...) as functions of the scalar value α. 4/21/2003 R. Crawfis, Ohio State Univ. 97 Graphical User Interface for time varying data high (α,t ) --> c c The color, c, is mapped to the t magnitude of a signature α function of time, t, and isovalue α The horizontal axis spans the scalar value low dimension α. The vertical axis spans the time dimension t. 4/21/2003 R. Crawfis, Ohio State Univ. 98 magnitude 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. Signature Computation Consider a terrain of which you want to compute the length of each isocontour and the area contained inside each isocontour 4/21/2003 R. Crawfis, Ohio State Univ. 99 4/21/2003 R. Crawfis, Ohio State Univ. 100 Signature Computation Signature Computation The length of each contour is a C 0 spline function. The area inside/outside each isocontour is a C 1 spline function. 4/21/2003 R. Crawfis, Ohio State Univ. 101 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 4/21/2003 R. Crawfis, Ohio State Univ. 102

Real Time Quantitative Queries (agricultural yield data) size and position of the region with

the development of an adaptive ability to separate interesting isovalues from the others.

space to visualize by itself. 4/21/2003 R. Crawfis, Ohio State Univ. 103 4/21/2003 R.

104 Rule-based Contouring (foot of the Visible Human) The contour spectrum allows Topological

number of components per isocontour which isocontours merge together or split while modifying the

an isocontour with three connected components two of which are about to merge} 4/21/2003 R.

6 Real Time Quantitative Queries (agricultural yield data) size and position of the region with unsatisfactory production Rule-based Contouring (CT scan of an engine) The contour spectrum allows the development of an adaptive ability to separate interesting isovalues from the others. size and position of the region where wrong data acquisition occurred Spectrum space is a useful space to visualize by itself. 4/21/2003 R. Crawfis, Ohio State Univ /21/2003 R. Crawfis, Ohio State Univ. 104 Rule-based Contouring (foot of the Visible Human) The contour spectrum allows the development of an adaptive ability to separate interesting isovalues from the others. Topological information. number of components per isocontour which isocontours merge together or split while modifying the isovalue. an isocontour with three connected components two of which are about to merge} 4/21/2003 R. Crawfis, Ohio State Univ /21/2003 R. Crawfis, Ohio State Univ. 106 Other Topics Interval Volumes 4D Iso-contouring Smooth surfaces Polygon Reduction Triangle Strip generation Time-varying iso-contours Texture map parameterization Penny Rheingans 4/21/2003 R. Crawfis, Ohio State Univ. 107

Scalar Algorithms: Contouring

Scalar Algorithms: Contouring Computer Animation and Visualisation Lecture tkomura@inf.ed.ac.uk Institute for Perception, Action & Behaviour School of Informatics Contouring Scaler Data Last Lecture...