iso-contouring and level-sets

Iso-Contouring and Level-Sets

Roger Crawfis

Contributors: Roger Crawfis, Han-Wei Shen, Raghu Machiraju, Torsten Moeller, Huang Zhiyong, Fan Ding, and Charles Dyer

04/22/23 R. Crawfis, Ohio State Univ. 2

Iso-contour/surface Extractions

2D Iso-contour 3D Iso-surface


Contouring - The Problem

Extracting an iso- surface from an implicit function, that is,

Extracting a surface from volume data (discrete implicit function), f (x ,y ,zT


More Formally

A scalar visualization technique that creates curves (in 2D) or surfaces (in 3D) representing a constant scalar value across a scalar field.

Contour lines are called isovalue lines or isolines.

Contour surfaces are called isovalue surfaces or isosurfaces


Contouring a 2D structured grid with contour line value = 5

0 1 1 3 2

1 3 6 6 3

3 7 9 7 3

2 7 8 6 2

1 2 3 4 3

1. Using interpolation to generate points along edges with the constant value

2. Connect these points into contours using a few different approaches. One of the approaches:

. Detects edge intersection . Tracks this contour as it moves across cell boundary . Repeat for all contours


2D Contouring – Not so easy

AnnotationsSmooth Curves

Topographic Map of Jerusalem (Contour interval 10 meters)

North is at the top of the map. The Mount of Olives is on the far right, Mount Zion on the left. Mount Moriah rises as a long ridge at the south end of the City of David and continues on past the present Temple Mount, and reaches its highest point outside the Northern walls of the Old City, at the top of the map.

http://www.skullandcrossbones.org/articles/solomontemple/solomontemple2.htm


Quantitative and Qualitative In this figure, we can

actually read off any value to about three significant digits.

Hard and tedious problem to determine where to place labels.

Usually done as vector graphics, rather than raster graphics for precision.

http://omnimap.com/catalog/cats/fish/fish-contour.htm


Divide and Conquer

Partition space into rectangular sub-regions.


Dividing Cubes/Squares Generates Points and

renders them Rendering points is

faster than polygons Principle - Divide a

square until contour passes thru the cell

1 2 3 4 3

2 7 8 6 2

3 7 9 7 3

1 3 6 6 3

0 1 1 3 2

Iso-value=5


Dividing Cubes

1. Create a cell2. Determine if the cell is a surface cell3. Subdivide surface cells to image resolution4. Determine intensity or color of refined cell5. Output a point for each remaining surface cell


Step 3: Subdivide Subdivide each cell such that the resolution is

higher than the image resolution. For example, when a 1282 gridded field is rendered to a 5122 image, cells containing the contour will be divided to 4x4 smaller squares.

classify subdivide output points


Surface Tracking Given function

S(t,x,y,z) that returns 1 if (x,y,z) is on the surface t and 0 otherwise

Needed: a seed cell p that is on the surface.

1 2 3 4 3

2 7 8 6 2

3 7 9 7 3

1 3 6 6 3

0 1 1 3 2

Iso-value=5

Seed cell


Surface TrackingLet Q be a queue of cells.

Push p onto QFlag p as “visited”While Q is not empty do

Pop q from Q and output it.For each cell v in the neighborhood of q doIf S(t, v) = 1 then

If v was not visited thenPush v onto QFlag it as “visited”

end {while}


Tracking and Smooth Curves

Recall that the gradient to a scalar field is normal to the isolines.

Therefore, the curve will be towards the direction perpendicular to the gradient.

Can also use the gradient to calculate tangents for each point on the curve to generate smooth curves.


Restricted Topological Cases

Filled circle vertices indicate scalar value is above the contour valueUnfilled ones which scalar values are below the contour values

•A closed curve can not be contained in the cell.•The curve can only enter and exit the cell once per edge.


Not Allowed


Linear Topology The restricted set is topologically equivalent to linear

segments. Topologically equivalent if we restrict the edge

intersections to lie at the mid-points of each edge.


Using Symmetry to Reduce to 5

Contour ambiguity: either solidor dash lines look OK


2D Iso-contour (0)

Remember bi-linear interpolation

p2 p3

p0 p1

P =?p4 p5

To know the value of P, we can first compute p4 andP5 and then linearly interpolateP


2D Iso-contour (1)

Consider a simple case: one cell data set

The problem of extracting an iso-contour is an inverse of value interpolation. That is:

p2 p3

p0 p1

Given f(p0)=v0, f(p1)=v1, f(p2)=v2, f(p3)=v3

Find the point(s) P within the cell that have values F(p) = C


2D Iso-contour (2)

p2 p3

p0 p1

We can solve the problem based on linear interpolation

(1) Identify edges that contain points P that have value f(P) = C

(2) Calculate the positions of P

(3) Connect the points with lines


2D Iso-contouring – Step 1

(1) Identify edges that contain points P that have value f(P) = C

v1 v2

If v1 < C < v2 then the edge contains such a point



(2) Calculate the position of P

Use linear interpolation:

P = P1 + (C-v1)/(v2-v1) * (P2 – P1)v1 v2

Pp1 p2

C



p2 p3

p0 p1

Connect the points with line(s)

Based on the principle of linear variation, all the points on the line have values equal C


Inside or Outside?

Just a naming convention

1. If a value is smaller than the iso-value, we call it “Outside”2. If a value is greater than the iso-value, we call it “Inside”

p2 p3

p0 p1- +

inside cell

p2 p3

p0 p1-

outside cell


Contouring in 3D

Treat volume as a set of 2D slices Apply 2D Contouring algorithm on each

slice. Or given as a set of hand-drawn

contoursStitch the slices together.


Contour Stitching•Problem:Problem:

Given: 2 two-dimensional Given: 2 two-dimensional closedclosed curves curves Curve #1 has Curve #1 has mm points points Curve #2 has Curve #2 has nn points points

Which point(s) does vertex Which point(s) does vertex ii on curve one correspond to on curve one correspond to on curve two?on curve two? ?

?

i


A Solution

Fuchs, et. al. Optimization problem 1 stitch consists of:

2 spans between curves1 contour segment

Triangles of {Pi,Qj,Pi+1} or {Qj+1,Pi,Qj}Consistent normal directions

Pi Pi+1

Qj


Fuchs, et. al.

Left span PiQj => go up

Right span (either) Pi+1Qj => go down PiQj+1 => go down


Fuchs, et. al.

Constraints Each contour segment is used once and

only once. If a span appears as a left span, then it

must also appear as a right span. If a span appears as a right span, then it

must also appear as a left span.


Fuchs, et. al.

This produces an acceptable surface (from a topological point of view) No holes

We would like an optimal one in some sense.


Fuchs et. al.

Graph problemVertices Vij = span between Pi and

QjEdges are constructed from a left

span to a right span. Only two valid right

spans for a left span.PPii

QQjj QQj+1j+1

PPi+1i+1


Fuchs, et. al.

Organize these edges as a grid or matrix.

PP

QQ

i

j

PiQjPi+1QjPiQj+1


Fuchs, et. al.

Acceptable graphs Exactly one vertical arc between Pi and Pi+1 Exactly one horiz. arc between Qj and Qj+1 Either

indegree(Vij) = outdegree(Vij) = 0both > 0

• (if a right, also has to be a left)


Fuchs, et. al.

Claim: An acceptable graph, S, is weakly

connected.Lemma 2 Only 0 or 1 vertex of S has an indegree = 2.

E.g., Two cones touching in the center. All other vertices have indegree=1 (recall indegree = outdegree)


Fuchs, et. al.

Thereom 1: S is an acceptable surface if and only if: S has one and only one horizontal arc

between adjacent columns. S has one and only one vertical arc

between adjacent rows. S is Eulerian (closed walk with every arc

only once).


Fuchs, et. al.

Number of arcs is thus m+n.Many possible solutions still!!!Associate costs with each edge Area of resulting triangle Aspect ratio of resulting triangle


Fuchs, et. al.

Note that Vi0 is in S for some i.Note that V0j is in S for some j. With the weights (costs), we can

compute the minimum path from a starting node Vi0. Since we do not know which Vi0, we compute

the paths for all of them and take the minimum.Some cost saving are achievable.


Marching Cubes

Predominant method used today.Efficient and simpleIt was independently reported by

Wyvill and McPeeters in 1986, Lorenson and Cline in 1987


Marching Cubes

Treat each cube individual No 2D contour curves

Allow intersections only on the edges or at vertices.

Pre-calculate all of the necessary information to construct a surface.


Marching Cubes

Consider a single cube All vertices above the

contour threshold All vertices below Mixed above and

below+ +

+

+ +

+-

+


Marching Cubes

Binary label each node => (above/below)

Examine all possible cases of above or below for each vertex.

8 vertices implies 256 possible cases.

+ +

+

+ +

+-

+


How many cases?

Now we have 8 vertices

So it is: 2 = 2568

How many unique topological cases?


Case Reduction

Value Symmetry

+

+

_ _

_

__

_+

+

_

_

+

+

+

+


Case Reduction

Rotation Symmetry

+

+

_ _

_

__

__

_

+

+

_ _

__


Case Reduction

Mirror Symmetry

+

+

_ +

_

+_

__

+

+

+

+ _

__

By inspection, we can reduce 256 15


Marching Cube - Summary

Create a cubeClassify each voxelBuild an indexLookup edge listInterpolate triangle verticesCalculate and interpolate normals


Step 1: Create a Cube

Consider a cube defined by eight data values: four from slice K, and four from slice K+ 1


Step 2: Classify Each Voxel

Binary classification of each vertex of the cube as to whether it lies outside the surface (voxel value < isosurface

value) or inside the surface (voxel value <=

isosurface value).


Step 3: Build an Index

Use the binary labeling of each voxel to create an 8-bit index: 8 vertices - 256 cases Requires a canonical

ordering of the cube.


Step 4: Look-up the Topology

Given the index for each cell, a table lookup is performed to identify the edges that intersect the iso-surface.

Also requires a canonical ordering of the edges: e1, e2, …, e12 0

12

3

14

e1, e3, e5…

Index intersection edges

e1

e2

e3

e4e5

e6

e7

e8

e9 e10

e11 e12


Step 4: Look-up the Topology

all 256 cases could be derived from 15 base cases. Difficult to determine rotational symmetry and map it correctly to the table.


Step 4: Example Index = 10001101 triangle 1 = e4,e7,e12 triangle 2 = e1, e7, e4 triangle 3 = e1, e6, e7 triangle 4 = e1, e10,

e6

e1e10

e6

e7e12

e4

v1 v2

v7

e2


Step 5: Interpolate Triangle Vertices

For each edge, find the vertex location along the edge using linear interpolation of the voxel values.


Step 6: Compute Normals

Calculate the normal at each cube vertex

Use linear interpolation to interpolate the polygon vertex normal


Why is it called marching cubes?

Linear search through cells •Row by row, layer by layer•Reuse the interpolated points for adjacent cells

No neighborhood informationNo neighborhood informationis required!!!is required!!!


Problems

Ambiguities: Are the curves below right or wrong?


Problems

Inconsistencies: Does the result produce the correct topology?

An iso-contour surface should always be a manifold (with boundary). We know whether these cases are right

or wrong.wrong


2D Contouring Ambiguity

Break contour Join contour

or

Either choice isacceptable becausecontour lines are continuousand closed

Independent of otherchoices


Marching Cubes

Topological inconsistencies in the 15 cases Turns out positive and negative are

not symmetric.

+ +

+

+ +

+- -

+ -

-

+ +

---


Topological Inconsistencies

The problem is that the edge intersections match, but the face intersections do not.

In 2D we only had edge intersections, so this was not a problem.

In 4D and higher the problem gets even worse.


A Bad Example

Resolution:Favor positive pathways thru the voxel (or negative, but be consistent).

Case 3 Case 3

Case 3Case 6

(compliment symmetry)


Ambiguous Cases Ambiguous cases:

3, 4, 6, 7, 10, 12, 13 Adjacent vertices:

different states Diagonal vertices:

same state Case 4 is not usually

considered ambiguous, but we could favor a tunnel through the long diagonal.

or

or


Ambiguous Cases

The ambiguous cases lead to a continuous manifold, but due to improper sampling of the volume, a (somewhat arbitrary) decision can be made on whether to close two curves or connect them.

Asymptotic Decider - by Nielson and Hamann (IEEE Vis’91) examines neighboring cells to determine this choice.


Asymptotic Decider

Based on bilinear interpolation over faces

B01

B00 B10

B11

(s,t)B(s,t) = (1-s, s) B00 B01

B10 B111-t t

The contour curves of B:

{(s,t) | B(s,t) = } are hyperbolas


Asymptotic Decider (2)

(0,0)

(1,1)

Where the hyperbolasgo through the cell depends on the valuesat the corners, I.e., B00, B01, B10, B11



(0,0)

(1,1)

Asymptote

(ST

If B(ST



(1,1)

Asymptote

(ST

(0,0)

If B(ST



(1,1) (ST

(0,0)

S B00 - B01 B00 + B11 – B01 – B10

T B00 – B10 B00 + B11 – B01 – B10

B(ST B00 B11 + B10 B01 B00 + B11 – B01 – B10



Based on the result of asymptotic decider, we expand the marching cube case 3, 6, 12, 10, 7, 13(These are the cases with at least one ambiguous faces).


Marching Cubes Demo

Animating the contour value

Special functions for contouring

Varying speeds and numbers of triangles

D:\OfficePC\MSDEV\projects\OpenGL\Debug\OpenGL.exe


Marching Cubes

Data Structures/Tablesstatic int const HexaEdges[12][2] = { {0,1}, {1,2}, {2,3}, {3,0}, {4,5}, {5,6}, {6,7}, {7,4}, {0,4}, {1,5}, {3,7}, {2,6}};

typedef struct { EDGE_LIST HexaEdges[16];} HEXA_TRIANGLE_CASES;

/* Edges to intersect. Three at a time form a triangle. */

static const HEXA_TRIANGLE_CASES HexaTriCases[] = {{-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}, /* 0 */{ 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}, /* 1 */{ 0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}, /* 2 */{ 1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}, /* 3 */

...


Marching Cubes - How simple /* Determine the marching cubes index */ for ( i=0, index = 0; i < 8; i++) if (val1[nodes[i]] >= thresh) /* If the nodal value is above the */ index |= CASE_MASK[i]; /* threshold, set the appropriate bit. */

triCase = HexaTriCases[index]; /* triCase indexes into the MC table. */ edge = triCase->HexaEdges; /* edge points to the list of intersected edges */

for ( ; edge[0] > -1; edge += 3 ) /* stop if we hit the -1 flag */ { for (i=0; i<3; i++) /* Calculate and store the three edge intersections */ { vert = HexaEdges[edge[i]]; n0 = nodes[vert[0]]; n1 = nodes[vert[1]]; t = (thresh - val1[n0]) / (val1[n1] - val1[n0]); tri_ptr[i] = add_intersection( n0, n1, t ); /* Save an index to the pt. */ } add_triangle( tri_ptr[0], tri_ptr[1], tri_ptr[2], zoneID); /* Store the triangle */ }}


Iso-Contouring – Advanced Issues

1. Efficiently determining which cells to examine.

2. Using iso-contouring as a slicing mechanism

3. Iso-contouring in higher dimensions4. Texturing and coloring of iso-contours5. Polygonal simplification of contours.6. Choosing a good iso-value


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


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

nn


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.


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}


Efficient search methods

1. Spatial subdivision (domain search)

2. Value subdivision (range search) 3. Contour propagation


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) )


Span Space

MinimumThreshold

Threshold

High Gradient

Low Gradient


Span Space Given: iso-value v Candidate cells:

Minimum cell value less than the iso-value.

Maximum cell value greater than the iso-value.

Different algorithms query this span-space with different efficiencies.

min

max

v

v

Possiblecandidates


Span Space Create two lists:

Store (in order) the minima of each cell into the first list

Store (in order) the maxima of each cell into the second list.

Can easily get the set of cells with minimum less than the iso-value.

min

max

v


Span Space

Can easily get the set of cells with maximum greater than the iso-value.

Need the intersection of the two sets.

Not very efficient.min

max

v

v


Span Space - Active List

Given: iso-value v andmaximum cell range, r

gracefully update to new value nv

min

max

v

v

v-r

v+r

nvnv-r

nv

nv+r


Span Space – NOISE Algorithm

Minimum

• K-d TreesK-d Trees

Split Min. axisSplit Max. axis


Span Space

Search Method (2): ISSUE, discretized span space

O(log(N/L))

O(log(N/L))

O(1)

?

Complexity = ?


Range Search: Interval Tree

Interval Tree:

I

I left I right

Sort all the data points(x1,x2,x3,x4,…. , xn)Let = x mid point)n/2

We use to divide the cells into threesets II left, and I right

Icells that have min < max

I left: cells that have max < I right: cells that have min >

… …


Basic Idea:

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

Contour Propagation

A

BD

CE

Initial cell: A

Enqueue: B, C

Dequeue: B

Enqueue: D

…

FIFO QueueA

B C

C

C D

….

Breadth-First Search


Challenges

Need to know the initial cells!

For any given iso-value C, findingthe initial cells to start the propagation is almost as hard as finding the iso-surface cells.

You could do a global search, but …


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 smallerthan G.

We will only talk about extrema graph due to time constraint


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.


Extrema Graph (2)


Extrema Graph (3)

E1 E2

E3 E4

E7

E5E6 E8

a2a3

a4a5

a6a7

a1

Extreme Graph:

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

An ‘arc’ consists of cells that connectextrema points (we only store min/max of the arc though)


Extrema Graph (4)

Algorithm:

Given an iso-value1) Search the arcs of the extrema graph (to find the arcs that have

min/max values that contain the iso-value2) Walk through the cells along each of these arcs to find the

seed cells3) Start the iso-contour propagation from the seed cells 4) ….

There is something more that needs to be done…


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)


Extrema Graph (5)

Contours missed

These open iso-contours will intersect with ?? cells

Boundary Cells!!


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 - 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


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-valuesWhich 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.


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, .Present these functions to the user as an

aid in picking contour values.

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


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


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

• The horizontal axis spans the scalar values

• Plot of a set of signatures (length, area, gradient ...) as functions of the scalar value .

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


Graphical User Interface for time varying data

(,t ) --> c

The color, c, is mapped to the magnitude of a signature function of time, t, and isovalue

t

c

low

high

magnitude

• The horizontal axis spans the scalar value dimension .

• The vertical axis spans the time dimension t .


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


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

(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


Real Time Quantitative Queries(agricultural yield data)

size and position of the region with unsatisfactory production

size and size and position of the position of the region where region where wrong data wrong data acquisition acquisition occurredoccurred

Spectrum space is a useful space to visualize by itself.


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.


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.

an isocontour with three connected components two of which are about to merge}

•number of components per number of components per isocontourisocontour•which isocontours merge together which isocontours merge together or split while modifying the or split while modifying the isovalue.isovalue.


Other Topics

Interval Volumes4D Iso-contouringSmooth surfacesPolygon ReductionTriangle Strip generationTime-varying iso-contoursTexture map parameterization

Penny Rheingans

Level Sets

Roger Crawfis

Slides collected from: Fan Ding, Charles Dyer, Donald Tanguay, Theodore Kim and Roger Crawfis


Motivation:


Easy Case – Use Marching Cubes


Input Data Noisy


Non-uniform Exposure


Intensity Varies


Multiple problems


Applying Marching Cubes (thresholding)


Applying A Threshold


Four (contour) Levels


What To Do?

User Intervention!!! We see it!! It’s right there!!! Well, the edges get pretty fuzzy .

Two step process:1. Draw an initial curve (or surface)

within the desired region.2. Expand that interface outward towards

the edge of our desired region.


Interfaces

An interface (or front) is a boundary between two regions: “inside” and “outside.”

In 2-D, an interface is a simple closed curve:


Propagating InterfacesHow does an interface evolve over time?At a specific moment, the speed function F (L, G, I) describesthe motion of the interface in the normal direction.


Propagating Interfaces

• Local – depend on local geometric information (e.g., curvature and normal direction)• Global – depend on the shape and position of the front (e.g., integrals along the front, heat diffusion)• Independent – do not depend on the shape of the front (e.g., an underlying fluid velocity that passively transports the front)

Speed F(L,G,I) is a function of 3 types of properties:


Motion Under Curvature

Example: Motion by curvature. Each piece moves perpendicularto the curve with speed proportional to the local curvature.

largepositivemotion

smallnegativemotion



Curvature κ is the inverse of theradius r of the osculating circle.



http://math.berkeley.edu/~sethian/Applets/java_curve_flow.html


Functional Representation

Eulerian framework: define fixed coordinate system on the world.For every world point x, there is (at most) one value y = ft(x).Falling snow example:

)(0 xfy t )(1 xfy t


Functional RepresentationHowever, many simple shapes are multivalued;they are not functions regardless of the orientation of thecoordinate system.


Parametric RepresentationSpatially parameterize the curve x by s so that at time t the curve is xt(s), where 0 s S and the curve is closed: xt(0) = xt(S).

Points on initial curve.

Gradient (wrt time) is thespeed in normal direction.

Normal is perpendicularto curve, as is curvature.


Parametric RepresentationFor motion under curvature, speed F depends only onlocal curvature κ – the equation of motion is thus:

where curvature is

and the normal is

2/322 )()(

ss

sssssst yx

yxxys

2/1222/122 )()(

)(ss

Tss

ss

Tss

t yxxy

yxyx

xxsn

),())(()( snsFsxt ttt


Particle MethodsIn order to compute, discretize the parameterization into moving particles which reconstruct the front.Known under a variety of names: marker particle techniques, string methods, nodal methods.

= # mesh particles

st

sS

= time step

= parameterization step

),( ni

ni yx = location of point iΔs at time nΔt

Δs


Problem Statement

Generally, given: An initial front Equations that govern its evolution

How do we simulate the front’s evolution?

Called an ‘initial value problem’ Given the initial position Solve for a position at a future time


More formally:

Given some initial front Г:


More formally: (2)

And a function F that specifies the velocity of the front in the normal direction:


More formally: (3)

Solve for Г at some future timeLevel set methods are used to track

an interfaceWater/air interface, for example


What’s Wrong with the Obvious Solution?

Why is a level set method necessary?

There seems to be a more intuitive way to solve this problem


Marker/String Methods

Why not just connect some control points (in 3D, triangulate):


Marker/String Methods (2)

And run the simulation on the points?


Ocean Waves

Think of an air/water interface with two waves racing towards each other:


Ocean Waves (2)

What happens to the control points when the waves collide?


Shocks

Event known as a ‘shock’Below formation called a

‘swallowtail’


Shocks (2)

How to fix up the control points?Fixing swallowtails known as ‘de-

looping’ Very difficult Some methods exist in 2D No robust 3D methods so far


Changing Topology

Example: two fires merge into a single fire.


Changing Topology

Buoys!

In particle methods:• Difficult (and expensive) to detect and change the particle chains• Much more difficult as dimensionality increases


Difficulties With Particle MethodsInstabilityLocal singularitiesManagement of particles: remove,

redistribute, connect


Level Set FormulationRecast problem with one additional dimension – the distance from the interface.

And then use Marching Cubes to extract the surface.

),(0 yxz t


Level Set Methods

Contour evolution method due to J. Sethian and S. Osher, 1988 www.math.berkeley.edu/~sethian/level_set.html Difficulties with snake-type methods Hard to keep track of contour if it self-

intersects during its evolution Hard to deal with changes in topology


The level set approach: Define problem in 1 higher dimension Define level set function z = (x,y,t=0)

where the (x,y) plane contains the contour, andz = signed Euclidean distance transform value (negative means inside closed contour, positive means outside contour)


How to Move the Contour?

Move the level set function, (x,y,t), so that it rises, falls, expands, etc. Contour = cross section at z = 0


Level Set Surface

The zero level set (in blue) at one point in time is a slice of the level set surface (in red)


Level Set Surface

Later in time the level set surface (red) has moved and the new zero level set (blue) defines the new contour


Level Set Surface


Level Set FormulationThe interface always lies at the zeroth level set of the function ,i.e., the interface is defined by the implicit equation t (x, y) = 0.


How to Move the Level Set Surface?

1. Define a velocity field, F, that specifies how contour points move in time Based on application-specific physics such as time,

position, normal, curvature, image gradient magnitude2. Build an initial value for the level set function,

(x,y,t=0), based on the initial contour position3. Adjust over time; current contour defined by

(x(t), y(t), t) = 0

0yΦ

xΦF

tΦ

0ΦFtΦ

2122


Speed Function εK1(k)FFF(K) 10

εK1x,ykF(K) I

x,yIG-

I

σI

σ ek

x,yIG11k


Example: Shape Simplification

F = 1 – 0.1 where is the curvature at each contour point


Example: Segmentation

Digital Subtraction Angiogram F based on image gradient and contour curvature


Example (cont.)

Initial contour specified manually


Result – segmentation using Fast marching

No level set tuning


Results – vein segmentation

No level set tuning With level set tuning


Results – vein segmentation continued

Original Our result Sethian’s result (Fast marching + (Level set only) Level set tuning)


Result – segmentation using Fast marching

No level set tuning


Results – brain segmentation continued

No level set tuning With level set tuning


Results – brain image segmentation

# of iterations = 9000 # of iterations = 12000 Fast marching only, no level set tuning

iso-contouring and level-sets

Documents

ohio state univ

iso surface

surface cells

contour surfaces

contour lines

surface cell3

surface trackinglet

contour line value