Isosurface Extractions (II) 2D Isocontour 3D Isosurface

Isosurface cell search Isosurfaec cells: cells that contain isosurface. min < isovalue < max Marching cubes algorithm performs a linear search to locate the isosurface cells – not very efficient for large-scale data sets.

Isosurface Cells For a given isovalue, only a smaller portion of cells are isosurface cell. For a volume with n x n x n cells, the average number of the isosurface cells is nxn (ratio of surface v.s. volume) n n n

Efficient isosurface cell search Problem statement: Given a scalar field with N cells, c1, c2, …, cn, with min-max ranges (a1,b1), (a2,b2), …, (an, bn) Find {Ck | ak < C < bk; C=isovalue}

Efficient search methods Spatial subdivision (domain search) Value subdivision (range search) Contour propagation

Domain search Subdivide the space into several subdomains, check the min/max values for each subdomain If the min/max values (extreme values) do not contain the isovalue, we skip the entire region Min/max Complexity = O(Klog(n/k))

Range Search (1) Subdivide the cells based on their min/max ranges Global minimum Global maximum Hierarchically subdivide the cells based on their min/max ranges Isovalue

Range Search (2) Within each subinterval, there are more than one cells To further improve the search speed, we sort them. Sort by what ? Min and Max values G1 Max Min M5 M2 M6 M4 M1 M3 M7 M8 M11 M10 M9 m5 m1 m6 m3 m8 m7 m2 m9 m11 m4 m10 G2 Isosurface cells = G1 G2

Range Search (3) A clean range subdivision is difficult … Difficult to get an optimal speed ?

Range Search (4) Span Space : Instead of treating each cell as a range, we can treat it as a 2D point at (min, max) This space consists of min and max axes is called span space Any problem here?

Span Space What are the isosurface cells? max How to search them? min

Span Space Search (1) With the point representation, subdividing the space is much easier now. Search method 1: K-D tree subdivision (NOISE algorithm) K-d tree: A multi-dimensional version of binary tree Partition the data by alternating between each each of the dimensions at each level of the tree

NOISE Algorithm (K-d tree) Median point Min Construction Max left right ? max up … down … … * One node per cell min

NOISE Algorithm (Query) Median point Min If ( isovalue < root.min ) check the ?? Subtree If (isovalue > root.min) Check the ?? Subtree Don’t forget to check the root ‘s interval as well. Max left right ? up … down … … Complexity = O( N + k)

Span Space Search (2) Search Method (2): ISSUE O(log(N/L)) O(1) Complexity = ? ? O(log(N/L))

… … Back to Range Search Sort all the data points Interval Tree: (x1,x2,x3,x4,…. , xn) Let d = x (mid point) n/2 Id I left I right We use d to divide the cells into three sets Id, I left, and I right Id : cells that have min < d < max I left: cells that have max < d I right: cells that have min > d … …

… … Interval Tree Id I right I left Now, given an isovalue C If C < d If C > d 3) If C = d Id I left I right … … Complexity = O(log(n)+k) Optimal!! Id : cells that have min < d < max I left: cells that have max < d I right: cells that have min > d

Range Search Methods In general, range search methods all are superfast – two order of magnitude faster than the marching cubes algorithm in terms of cell search But they all suffer a common problem … Excessive extra memory requirement!!!

Contour Propagation Basic Idea: Given an initial cell that contains isosurface, the remainder of the isosurface can be found by propagation FIFO Queue A B C C C D …. Initial cell: A Enqueue: B, C Dequeue: B Enqueue: D … A B D C E Breadth-First Search

Challenges Need to know the initial cells! For any given isovalue C, finding the initial cells to start the propagation is almost as hard as finding the isosurface cells. You could do a global search, but …

Solutions Extrema Graph (Itoh vis’95) 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 isovalue 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

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 isocontour is intersect by at leat one of the arcs.

Extrema Graph (3) E1 E2 Extreme Graph: { E, A: E: extrema points A: Arcs conneccts E } a1 a2 a3 E3 E4 a5 An ‘arc’ consists of cells that connect extrema points (we only store min/max of the arc though) a4 E7 a7 E5 a6 E6 E8

Extrema Graph (4) Algorithm: Given an isovalue Search the arcs of the extrema graph (to find the arcs that have min/max contains the isovalue Walk through the cells along each of the arcs to find the seed cells Start to propagate from the seed cells …. There is something more needs to be done…

We are not done yet … What ?! We just mentioned that all the closed isocontours will intersect with the arcs connecting the extrema points How about non-closed isocontours? (or called open isocontours)

Extrema Graph (5) Boundary Cells!! These open isocontours will Contours missed These open isocontours will intersect with ?? cells

Extrema Graph (6) Algorithm (continued) Given an isovalue Search the arcs of the extrema graph (to find the arcs that have min/max contains the isovalue Walk through the cells along each of the arcs to find the seed cells Start to propagate from the seed cells Search the cells along the boundary and find seed cells from there Propagate open isocontours