lecture 4 : Isosurface Extraction

Isosurface Definition

Isosurface (i.e. Level Set ) :

Constant density surface from a 3D array of data C(w) = { x | F(x) - w = 0 }( w : isovalue , F(x) : real-valued function , usually 3D volume data )

< ocean temperature function > < two isosurfaces (blue,yellow) >

isosurfacing

Isosurface Triangulation

• Idea:– create a triangular mesh that will approximate the iso-surface– calculate the normals to the surface at each vertex of the triangle

• Algorithm:– locate the surface in a cube of eight pixels – calculate vertices/normals and connectivity– march to the next cube

Marching Cubes

• [Lorensen and Cline, ACM SIGGRAPH ’87]• Goal

– Input : 2D/3D/4D imaging data (scalar)– Interactive parameter : isovalue selection– Output : Isosurface triangulation

isosurfacing

Isosurface Extraction

2. Isocontouring [Lorensen and Cline87,…]

• Definition of isosurface C(w) of a scalar field F(x)

C(w)={x|F(x)-w=0} , ( w is isovalue and x is domain R3 )

( Isocontour in 2D function: isovalue=0.5 )

• Marching Cubes for Isosurface Extraction

1. Dividing the volume into a set of cubes

2. For each cubes, triangulate it based on the 2^8(reduced to 15) cases

0.7 0.6 0.75 0.4

0.40.80.40.6

0.4 0.3 0.35 0.25

1.0 0.8 0.4 0.3

0.7 0.6 0.75 0.4

0.40.80.40.6

0.4 0.3 0.35 0.25

1.0 0.8 0.4 0.3

0.7 0.6 0.75 0.4

0.40.80.40.6

0.4 0.3 0.35 0.25

1.0 0.8 0.4 0.3

Surface Intersection in a Cube

• assign ZERO to vertex outside the surface• assign ONE to vertex inside the surface• Note:

– Surface intersects those cube edges where one vertex is outside and the other inside the surface

Surface Intersection in a Cube

• There are 2^2=256 ways the surface may intersect the cube• Triangulate each case

Patterns

• Note:– using the symmetries reduces those 256 cases to 15 patterns

Marching Cubes table : 15 Cases

• Using symmetries reduces 256 cases into 15 cases

Surface intersection in a cube

• Create an index for each case:

• Interpolate surface intersection along each edge

Calculating normals

• Calculate normal for each cube vertex:

• Interpolate the normals at

the vertices of the triangles:

Summary

• Read four slices into memory• Create a cube from four neighbors on one slice and four

neighbors on the next slice• Calculate an index for the cube• Look up the list of edges from a pre-created table• Find the surface intersection via linear interpolation• Calculate a unit normal at each cube vertex and

interpolate a normal to each triangle vertex• Output the triangle vertices and vertex normals

Ambiguity Problem

Trilinear Function

• Trilinear Function

• Saddle point– Face saddle– Body saddle

Trilinear Isosurface Topology

Triangulation

Acceleration Techniques

• Octree• Interval Tree• Seed Set and Contour Propagation

• How to handle large isosurfaces?– Simplification– Compression– Parallel Extraction & Rendering

• How to choose isovalue?– Contour Spectrum

Interval Tree for Isocontouring

• Interval Tree– An ordered data structure that holds intervals– Allows us to efficiently find all intervals that overlap with any

given point (value) or interval

– Time Complexity of query processing : O (m + log n) • Output-sensitive• n : total # of intervals• m : # of intervals that overlap (output)

– Time Complexity of tree construction : O (nlog n)

• How can we apply interval tree to efficient isocontouring?

Seed Set for Isocontouring

• Main Idea– Visit the only cells that intersect with isocontour– Interval tree for entire data can be too large– Use the idea of contour propagation

• Seed Set– A set of cells intersecting every connected component of every

isocontour

• Seed Set Generation : refer to Bajaj96 paper

Seed Set Isocontouring Algorithm

• Algorithm– Preprocessing

• Generate a seed set S from volume data• Construct interval tree of seed set S

– Online processing• Given a query isovalue w,• Search for all seed cells that intersect with isocontour with

isovalue w by traversing interval tree• Perform contour propagation from the seed cells that were

found from interval tree.

Contour Propagation

• Given an initial cell which contains the surface of interest• The remainder of the surface can be efficiently traced

performing a breadth-first search in the graph of cell adjacencies

< Contour Propagation >

Seed Set Generation (k seeds from n cells)

• 238 seed cells

• 0.01 seconds

Domain Sweep

177 seed cells0.05 seconds

Responsibility Propagation

59 seed cells1.02 seconds

O(n) O(n) O(n log n)O(k) O(k) O(n)

Time

Space

? ? 2 kmink =

Test

Range Sweep

Seed Set Generation

Seed Set Computation using Contour Tree

Contour Tree generates minimal seed set generation

Contour Tree

h(x,y)

x

y

• Definition : a tree with (V,E)– Vertex ‘V’

• Critical Points(CP) (points where contour topology changes , gradient vanishes)

– Edge ‘E’: • connecting CP where an infinite contour class is created and CP where the

infinite contour class is destroyed.• contour class : maximal set of continuous contours which don’t contain

critical points

Contour Tree

2D Example

• Height map of Vancouver

Contour Tree

Join Tree

Split Tree

Merge to Contour Tree

• Merge Join Tree and Split Tree to construct Contour Tree [Carr et al. 2010]

+ =

Properties• Display of Level Sets Topology (Structural Information)

– Merge , Split , Create , Disappear , Genus Change (Betti number change)

• Minimal Seed Set Generation

• Contour Segmentation– A point on any edge of CT corresponds to one contour component

Contour Tree Drawing and UI

Hybrid Parallel Contour Extraction

• Different from isocontour extraction• Divide contour extraction process into

– Propagation• Iterative algorithm -> hard to optimize using GPU• multi-threaded algorithm executed in multi-core CPU

– Triangulation• CUDA implementation executed in many-core GPU

33< propagation > < performance of our hybrid parallel algorithm >

Hybrid Parallel Contour Extraction

Results

Interactive Interface with Quantitative Information

• Geometric Property as saliency level– Gradient(color) + Area (thickness)

36

Segmentation of Regions of Interest

• Mass Segmentation from Mammograms• Minimum Nesting Depth (MND)

– Measured for each node of contour tree– MND = min (depth from current node to terminal node of every

subtree)– High MND contour represents the boundaries of distinctive

regions with abrupt intensity changes retaining the same topology

– Successfully applied to mass detection from 400 mammograms in USF database.

37

Salient Isosurface Extraction

• How to select isovalue?

• Contour Spectrum– [ Bajaj et al. VIS97 ]– shows quantitative properties (area, volume, gradient) for all

isovalues– allows semi-automatic isovalue selection

Isovalue Selection

• The contour spectrum allows the development of an adaptive ability to separate interesting isovalues from the others.

Contour Spectrum (CT scan of an engine)

• The contour spectrum allows the development of an adaptive ability to separate interesting isovalues from the others.

Salient Contour Extraction Using Contour Tee

41

Motivation

• Infinitely many isocontours defined in an image• An isocontour may have many contours• Contour– Connected component of an isocontour– Often represents an independent structure

42

Ex) mammogram (X-ray exam of female breast)

Motivation

• Salient Contour Extraction – Useful for segmentation, analysis and visualization of

regions of interest– Can be applied to CAD(Computer Aided Diagnosis) for

detecting suspicious regions

43mass (tumor) dense tissue dense tissuebreast boundary pectoral muscle

3D Examples

44

<Head MRI> <isocontour> <ventricle contour>

<mass segmentation from breast MRI>

Past Contour Tree Approach

• Contour Tree– Represents topological changes of contours

according to isovalue change.– Property

• structure (topology) of level sets• contour extraction• seed set generation for fast extraction

45

Our Approach

• Interactive Contour Tree Interface– Performance Improvement of Extraction Process– Utilizing Quantitative Information

• Development of Saliency Metric– MND(Minimum Nesting Depth)– Apply to medical images

46

Hybrid Parallel Contour Extraction

• Different from isocontour extraction• Divide contour extraction process into

– Propagation• Iterative algorithm -> hard to optimize using GPU• multi-threaded algorithm executed in multi-core CPU

– Triangulation• CUDA implementation executed in many-core GPU

47< propagation > < performance of our hybrid parallel algorithm >

Interactive Interface with Quantitative Information

• Geometric Property as saliency level– Gradient(color) + Area (thickness)

48

Saliency Metric

• Minimum Nesting Depth (MND)– Measured for each node of contour tree– MND = min (depth from current node to terminal node of every

subtree)– High MND contour represents the boundaries of distinctive

regions with abrupt intensity changes retaining the same topology

– Successfully applied to mass detection from 400 mammograms in USF database.

49