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 Field Function 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 tetrahedron FIGURES 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 together FIGURES 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 minimummaximum

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 f VECTOR 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 saddle 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 p Unstable cell of a critical point p Union of all integral lines emanating from p FIGURES

Understanding 3D Fields Point typeStable cellUnstable cell maximumpointvolume minimumvolumepoint 1-saddlesurfaceline 2-saddlelinesurface

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 mesh Graph is a triangle mesh in 3D Vertex p and its incident triangles Sum 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 mesh Graph is a tetrahedral mesh in 4D Vertex p and its incident tetrahedra Sum 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

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

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 Distortion DensityDistortion Distortion reveals regular patterns in the data (due to the parallel algorithm)

DensityDistortion Distortion also highlights features Density vs Distortion

Morse Decomposition Number of cells in the decompositions StableUnstable Density1932no merge Distortion255606merged 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

DensityDistortion Stable Decomposition

DensityDistortion Distortion gives a more complicated segmentation (revealing complexity of the data)

Unstable Decomposition OTHER FIGURES FROM PAPER…. DensityDistortion

Unstable Decomposition DensityDistortion 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

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