Automatic Isosurface Propagation Using an Extrema Graph and Sorted Boundary Cell Lists

Introduction Visiting non-intersected cells is very

time-consuming in huge volumes.

Elimination of non-intersected cells outside the process is more effective.

The number of visited cells is less than O(n)

Propagation Algorithm (1)

Propagation Algorithm (2)

A

BD

CE

Initial cell: A

Enqueue: B, C

Dequeue: B

Enqueue: D

…

FIFO Queue

A

B C

C

C D

….

Breadth-First Search

Basic Idea:

Given an starting cell that contains isosurface, the remainder of the isosurface can be found by propagation

Propagation Algorithm (3)

Challenges

Need to know the starting cells!

For any given isovalue C, findingthe starting cells to start the propagation.

You could do a global search, but …

Solution ： Extrema Graph & Boundary Cell Lists

Extremum Point Extremum points ard defined as grid-

points whose scalar values are higher or lower than the values of all adjacent grid-points.

Generating an Extrema Graph

Searching for starting cells:

Find all the local minimum and maximum points, and connect them together by straight lines (Arcs).

The closed isosurface is intersected by at least one of the arcs.

Extrema Graph

Extreme Graph:

{ E, A: E: extrema points A: Arcs conneccts E }

Problem ：Holes!!

Hole

Generating Boundary Cell Lists

Hole

The open isosurface is intersected by visiting boundary cells in order.

Generating IsosurfacesSearching for starting cells

Outline of the AlgorithmVoid main(){

/* Preprocess */ExtremumPointExtraction();GenerateGraph();GenerateBoundList();

/* Isosurfacing process*/while(1){

Specify an isovalue C;GenerateSurface(C);

}}

nO

)()( 3

2

3

1

nOnO

About Arc

Image of Isosurface

Volume Thinning for Automatic Isosurface Propagation

Topology of an extrema graph and an isosurface

Boundary cells are not necessary if there is a cycle around a through-hole

Image thinning method

(a)

(b)

(c)

(d)

(e)

(f)

(g)

p p p p p1 2 3

4

567

8

Volume thinning method

We initially assumes that a seed set of a volume contains all cells in the volume.

The extremum point will never be eliminated from the seed set during the process.

Finally, the seed set form a one-cell-wide skeleton.

Bubble-like layerExtremum point

Elimination of Bubble-Like Layers of Cells

Visited cells in the isosurfacing process

Preprocess

Outline of the Algorithm

nOVoid main(){

/* Preprocess */ExtremumPointExtraction();VolumeThinning();

/* Isosurfacing process */while(1){ Specify an isovalue; Extract isosurface cells from the extrema

skeleton; IsosurfacePropagation();}

})( 3

1

nO

Image of Volume Thinning Process

Image of Volume Thinning Process

Image of Isosurface(1)

Image of Isosurface(2)