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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
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Outline
1. Motivations
2. Background notions
3. Discrete distortion
4. Experimental results
5. Future work
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Outline
• Motivations
• Background notions
• Discrete distortion
• Experimental results
• Future work
![Page 4: Morphological Analysis of 3D Scalar Fields based on Morse Theory and Discrete Distortion Mohammed Mostefa Mesmoudi Leila De Floriani Paola Magillo Dept](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649dea5503460f94ae4fb9/html5/thumbnails/4.jpg)
3D Scalar FieldFunction defined within a 3D volume
(x,y,z) h=f(x,y,z)
Examples:
• Pressure, density temperature…
• Geological data, atmospheric data…
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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
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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
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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
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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
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Outline
• Motivations
• Background notions
• Discrete distortion
• Experimental results
• Future work
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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
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low high
Critical Points
minimum maximum
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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
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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
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Critical Points
Sign of eigenvalues Feature type
- - - maximum + + + minimum - + + 1-saddle - - + 2-saddle
FIGURES OF MAX MIN SADDLES…
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Volume Segmentation
• Isosurface = locus of points with a given field value
• Integral line = follow direction of the negative gradient
• Mutually perpendicular
FIGURES
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Volume Segmentation
Integral lines• Start from maxima• Converge to minima• Pass through saddles
FIGURES
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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
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Understanding 3D Fields
Point type Stable cell Unstable cell maximum point volume minimum volume point 1-saddle surface line 2-saddle line surface
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Volume Segmentation
Two segmentations• Stable Morse decomposition =
Collection of all stable cells of minima
• Unstable Morse decomposition =Collection of all unstable cells of maxima
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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)
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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
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Outline
• Motivations
• Background notions
• Discrete distortion
• Experimental results
• Future work
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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
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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
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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]
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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…
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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…
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Step 1
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Step 1
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Step 1
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Step 1
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Step 1
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Step 1
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Step 1
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Step 1
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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…
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Step 2
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Step 2
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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
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Outline
• Motivations
• Background notions
• Discrete distortion
• Experimental results
• Future work
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Experimental results
• Data set in San Fernando Valley (CA) • Field is underground density• Earthquake simulation• Generated by a parallel algorithm using data
partition
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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
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Density vs DistortionDensity Distortion
Distortion reveals regular patterns in the data
(due to the parallel algorithm)
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Density Distortion
Distortion also highlights features
Density vs Distortion
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Morse Decomposition
Number of cells in the decompositions
Stable Unstable
Density 19 32 no merge
Distortion 255 606 merged to 20
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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
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Density Distortion
Stable Decomposition
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Stable DecompositionDensity Distortion
Distortion gives a more complicated segmentation
(revealing complexity of the data)
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Unstable Decomposition
OTHER FIGURES FROM PAPER….
Density Distortion
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Unstable DecompositionDensity Distortion
Distortion is less sensitive to the regular patterns
(due to the parallel algorithm)
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Outline
• Motivations
• Background notions
• Discrete distortion
• Experimental results
• Future work
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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
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Acnowledgements
This work has been partially supported by:
• National Science Foundation
• MIUR-FIRB Project Shalom
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End of the talk
• Thank you!
• Question?