Morphological Analysis of 3D Scalar Fields based on Morse Theory and Discrete Distortion Mohammed...

Morphological Analysis of 3D Scalar Fields based on Morse

Theory and Discrete Distortion

Mohammed Mostefa Mesmoudi

Leila De Floriani

Paola Magillo

Dept. of Computer Science, University of Genova, Italy

Outline

1. Motivations

2. Background notions

3. Discrete distortion

4. Experimental results

5. Future work

Outline

• Motivations

• Background notions

• Discrete distortion

• Experimental results

• Future work

3D Scalar FieldFunction defined within a 3D volume

(x,y,z) h=f(x,y,z)

Examples:

• Pressure, density temperature…

• Geological data, atmospheric data…

Understanding 3D Fields

Function values are known at a finite

set of points within the volume

• A tetrahedral mesh with vertices at those points

• Linear interpolation inside each tetrahedronFIGURES IN 2D

Understanding 3D Fields

Difficult, we cannot see all data at once

• False colors cannot see inside• Graph should draw it in 4D• Isosurfaces cannot see many togetherFIGURES IN 2D AND 3D WHERE

POSSIBLE

Understanding 3D Fields

Detect features• Critical points (maxima, minima…)

Segmentation of the 3D domain• 3D cells with uniform behavior

(e.g., decreasing from a maximum)• 1D and 2D boundaries where behavior changes

Understanding 3D Fields

Segmentation of the 3D domain based on

• the field function

• another function computed from it and able to enhance features– E.g., Discrete distortion

Outline

• Motivations

• Background notions

• Discrete distortion

• Experimental results

• Future work

Critical Points

Point p within the 3D domain• Maximum = field decreases towards p• Minimum = field increases towards p• Saddle = field increases in some directions and

decreses in other directions– 1-saddle– 2-saddle

low high

Critical Points

minimum maximum

Critical Points

Field v =f(x,y,z)Function f continuous and differentiable

Mathematical definition in terms of• Gradient vector = the 3 first derivatives of f• Hessian matrix = the 3x3 second derivatives of fVECTOR AND MATRIX AS FIGURES

Critical Points

• Gradient vector is (0,0,0) at critical points

• If the eigenvalues of the Hessian matrix are non-zero at critical points– Function f is called a Morse function– Critical points are isolated

Critical Points

Sign of eigenvalues Feature type

- - - maximum + + + minimum - + + 1-saddle - - + 2-saddle

FIGURES OF MAX MIN SADDLES…

Volume Segmentation

• Isosurface = locus of points with a given field value

• Integral line = follow direction of the negative gradient

• Mutually perpendicular

FIGURES

Volume Segmentation

Integral lines• Start from maxima• Converge to minima• Pass through saddles

FIGURES

Volume Segmentation

Stable cell of a critical point p• Union of all integral lines

converging to pUnstable cell of a critical point p• Union of all integral lines

emanating from p

FIGURES

Understanding 3D Fields

Point type Stable cell Unstable cell maximum point volume minimum volume point 1-saddle surface line 2-saddle line surface

Volume Segmentation

Two segmentations• Stable Morse decomposition =

Collection of all stable cells of minima

• Unstable Morse decomposition =Collection of all unstable cells of maxima

Background

Discrete distortion for 3D fields (graph is a tetrahedral mesh in 4D)

Generalizes

Concentrated curvature 2D fields(graph is a triangle mesh in 3D)

Concentrated Curvature

2D scalar field defined on a triangle meshGraph is a triangle mesh in 3D Vertex p and its incident trianglesSum of all angles incident in p• In the 2D domain (flat) the sum is 2• In the 3D graph it is an angle p

Concentrated curvature K(p)= 2 – p

• K(p)=0 p flat• K(p)>0 p convex/concave• K(p)>0 p saddle

Outline

• Motivations

• Background notions

• Discrete distortion

• Experimental results

• Future work

Discrete Distortion

3D scalar field defined on a tetrahedral meshGraph is a tetrahedral mesh in 4D Vertex p and its incident tetrahedraSum of all trihedral angles incident in p• In the 3D domain (flat) the sum is 4• In the 4D graph it is an angle p

Discrete distortion D(p)= 4 – p

• D(p)=0 p flat• D(p)>0 p convex/concave• D(p)>0 p saddle

Distortion: Idea

• Field function h=f(x,y,z)

• Vertex in 3D Vertex in 4D(x,y,z) (x,y,z,h)

• Tetrahedron in 3D tetrahedron in 4D

• The shape of tetrahedra may change

• Measure how much the tetrahedra around a vertex p are distorted from 3D to 4D

Computing Morse Decompositions

• Distortion can be seen as another field defined on the same mesh

• We compute morse decomposition based on original field and based on distortion

• The decomposition algorithm is a 3D extension of the 2D algorithm in [De Floriani, Mesmoudi, Danovaro, ICPR 2002]

Computing Morse Decompositions

• Consider the unstable Morse decomposition• (volumes associated with maxima)

• Construct unstable cell in order of decreasing field value

• Progressively classify tetrahedra into some cell…

Computing Morse Decompositions

Step 1• Take vertex v = maximum of the unclassified

part of the mesh• Classify tetrahedra belonging to its cell

– Its incident tetrahedra– Those tetrahedra that can be recursively reached by

moving along faces towards a vertex with smalled field value

– Consider the unstable Morse decomposition

• Repeat until all tetrahedra are classified…

Step 1

Step 1

Step 1

Step 1

Step 1

Step 1

Step 1

Step 1

Computing Morse Decompositions

Step 2• Now some cells are associated with a non-

maximum v• Such v1 lies on the boundary of the cell of

some other vertex v• Merge the cell of v1 into that of v• Repeat as long as we have some v1 in that

condition…

Step 2

Step 2

Merging

Morse decompositions are often over-segmented

Merge pair of cells such that• Field difference is small • Size (number of tetrahedra) is small• Common boundary surface is large

Saliency = weighted combination of such criteria• Iterative merging process• At each step merge the pair of cells with minimum saliency

Outline

• Motivations

• Background notions

• Discrete distortion

• Experimental results

• Future work

Experimental results

• Data set in San Fernando Valley (CA) • Field is underground density• Earthquake simulation• Generated by a parallel algorithm using data

partition

Density vs Distortion

• Density field and its distortion field in false colors• Distortion reveals regular patterns in the data

(due to the parallel algorithm used to generate them)

• Distortion also highlights features• FIGURES FROM PAPER

Density vs DistortionDensity Distortion

Distortion reveals regular patterns in the data

(due to the parallel algorithm)

Density Distortion

Distortion also highlights features

Density vs Distortion

Morse Decomposition

Number of cells in the decompositions

Stable Unstable

Density 19 32 no merge

Distortion 255 606 merged to 20

Morse Decomposition

We visualize Morse decompositions by plotting• The seed of each region in red• The boundaries between cells in blue• The interior of each cell in yellow

• FIGURES FROM PAPER

Density Distortion

Stable Decomposition

Stable DecompositionDensity Distortion

Distortion gives a more complicated segmentation

(revealing complexity of the data)

Unstable Decomposition

OTHER FIGURES FROM PAPER….

Density Distortion

Unstable DecompositionDensity Distortion

Distortion is less sensitive to the regular patterns

(due to the parallel algorithm)

Outline

• Motivations

• Background notions

• Discrete distortion

• Experimental results

• Future work

Future Work

• Extension of discrete distortion to multiple fields defined on the same volume (mutual interactions)

• Optimization of tetrahedral meshes discretizing the field volume, based on discrete distortion

• Extension to 4D (time-varying) scalar fields

Acnowledgements

This work has been partially supported by:

• National Science Foundation

• MIUR-FIRB Project Shalom

End of the talk

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