advanced(medical(image( analysis(

Advanced Medical Image Analysis

Fall 2012 Lecture 11 – Surface and Volume Meshing

Michel Aude<e, Ph.D. Old Dominion University

Dept. Modeling, SimulaFon and VisualizaFon Engineering

Why this is cool – surface and volume meshing

•  Generally speaking, segmentaFon gives us either a voxel-‐based volume, or its boundary.

•  For biomedical simulaFon, we need to break down Fssues into simple shapes -‐ triangles or tetrahedra.

MR/CT Tissue map Polygonalize tissue boundary

Insert inner vertices Simulation

BACKGROUND – MEDICAL SIMULATION

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InteracFve vs predicFve simulaFon

•  Surgical simulaAon: synthesize Fssue response to surgical acFon

• Predic)ve: high-‐quality offline (FE) simulaFons for experts. • Interac)ve: real-‐Fme response to hapFc gesture for residents.

• Research: reconcile predic've with interac've biomechanics.

Images reproduced from M. Chabanas, Université de Grenoble, S. Cotin, INRIA, Wikimedia Commons, and ixbtlabs.com

InteracFve simulaFon – h/w architecture

•  HapFc interacFon: 2-‐way device – 200 Hz+ – Angles to posiFon by forward kinemaFcs.

–  Force to joint torques by inverse Jacobian transformaFon.

•  Visual interacFon: monitor

InteracFve simulaFon – s/w architecture

•  Anatomical model, typically composed of tetrahedra, encapsulaFng the anatomy over simulated Fme;

•  Biomechanics engine that synthesizes a Fssue response to a virtual gesture;

•  Collision detecAon algorithm that efficiently determines where the interacFon takes place;

•  Engines dedicated to hapAc and visual rendering.

Surgery simulaFon – biomechanics

•  Trade-‐off between speed vs cons)tu)ve fidelity.

•  InteracFve simulaFon tradiFonally emphasizes interac)vity at expense of fidelity to Fssue biomechanics.

•  Predic)ve simula)on for expert surgeons places emphasis on fidelity – need not be interacFve.

speed fidelity

Mass-springs Finite elements

TLED FE MJED FE

Multi-grid FE

Surgery simulaFon – biomechanics (2)

• 


• 


•  Emerging soluFons for interacFve simulaFon. •  PrecomputaFons: derivaFves via undeformed coordinates:

–  Total Lagrangian Explicit Dynamics (explicit) – MulFplicaFve Jacobian Energy DecomposiFon (implicit)

•  MulF-‐grid: coarse FE helping medium and fine FE systems to converge more quickly.

Surgery simulaFon – cubng mechanics

•  New methods for simulaFon cubng based on both finite element formalisms and point-‐cloud-‐based methods.

– Extended finite element method (XFEM).

– Meshless methods: obviate dynamic remeshing

Sub-domain

Domain of definition of point

x

IMPLICATIONS FOR MESHING

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Tetrahedral meshing implicaFons -‐ resoluFon

•  Importance of mesh resoluAon: – We care about resolu)on, especially for interacFve simulaFon: solving 2K tets quicker than 20K, 200K tets.

– Mul)-‐grid FE impossible w/o coarse/medium/fine tets.

– Also, no need to mesh whole volume at medium & fine resoluFons – target and path to target only.

Tet. meshing implicaFons – Quality (1)

•  Importance of mesh quality: – We care about element quality, for fast convergence: slivers (flat tets) lead to slow/unstable soluFon.

– Quality of an element: absence of…

•  edges of small size (vs. other edges), and •  small dihedral angles (angle between two planes).

Sliver: near-complanar tet

High-quality tetrahedron

Various near-degenerate tets.

Jane Tournois1, Rahul Srinivasan2, and Pierre Alliez1 - Perturbing Slivers in 3D Delaunay Meshes


•  FE convergence – based on condi)on number of sFffness matrix K;

•  K composed of elemental sFffnesses Ke

•  Ke is a funcFon of dihedral angles and edge lengths


• 


•  Measure of tet mesh quality: histogram of dihedral angles •  B. Klingner (Berkley): Aggressive Tet Mesh Improvement

Tet. meshing implicaFons – ResoluFon and Quality

• 

What kind of element to use?

•  Q: Is tet meshing always desirable? E.g: cranial nerve •  Consider 20mm segment between points A and B.

•  Volume = 20 mm x Π x (0.5 mm)2 = 15.7 mm3 ≈ 1056 tets (0.5mm edges)

•  Or… piecewise-‐linear beam element ≈ 6 elements.

A B

MESH GENERATION

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Structured vs Unstructured Mesh GeneraFon

•  Structured meshing: quadrilateral and hexahedral meshing in 2D and 3D respecFvely;

–  Interior nodes: constant number of incident elements.

– Manual interacFon of user to produce a template &

procedure for warping this template to pa)ent data.

Orthopedic applications of structured meshing, courtesy of N. Grosland (Univ. Iowa)/

Structured vs Unstructured Mesh GeneraFon (2)

•  Unstructured mesh genera)on:

–  relaxes node valence requirement: any number of elements can meet at a given node.

– Minimally supervised methods feasible.

–  Four categories: • Octree-‐based • Delaunay • Advancing Front • OpAmizaAon-‐based

Unstructured Mesh GeneraFon – Basic idea

•  Workflow: i) inserFon of nodes w or w/o boundaries ii) link neighboring nodes by edges

iii) assemble edges into faces, faces into tets.

Octree-‐based Unstructured Mesh GeneraFon

•  Recursive subdivision of Fssue volume into conFguous cubes fully or parFally overlapped by this Fssue.

•  Irregular cells are created where the cubes intersect the boundary of the volume, which typically requires a large number of surface intersecFon computaFons.

Octree-‐based Unstructured Mesh GeneraFon (2)

•  Octree approach leads to a tetrahedralizaFon inserFng 1 or more verFces at each inner octree node, and at intersec'ons between octree edges & 'ssue boundary. – Can insert one node per octree cube, or – Divide each cube into tetrahedra.

•  To prevent dramaFc changes, max. difference of one level between adjacent cells of the octree subdivision.

Octree-‐based Unstructured Mesh GeneraFon (2)

•  Octree-‐based results: Chrisochoides et al.

Delaunay Unstructured Mesh GeneraFon (1)

•  Delaunay parFFon of space: “empty sphere” criterion.

•  2D “empty circle” equivalent: no node can be contained in the circumsphere of a tetrahedron not incident to it.

•  Criterion is not algorithm for genera)ng a mesh, but rather for determining which subset of a point cloud should be connected to form tetrahedra.


•  Conforming Delaunay. A tetrahedraln T conforms to a PLC C if for any face of C, it is a union of faces of T.

– VerFces inserted into mesh, maintaining Delaunay property, unFl it conforms to boundaries: dense mesh!

•  Almost Delaunay: similar to conforming approach, except that vertex inserFon near PLC relaxes Delaunay property. – Can also create large number of tets or short edges.


•  Constrained Delaunay Tetrahedralns, CDTs, not fully Delaunay but have DT qualiFes.

•  Fewer vertex inserFons, resoluFon control, w/o short edges. –  T respects the PLC C: no segment of C cut in two by T, –  There is a circumsphere S of T s.t. there is no vertex v of C that falls inside S & is visible from any point inside of T.

Advancing Front Unstructured Mesh GeneraFon

•  Star)ng from the )ssue boundary, new verFces added by a local heuris)c to ensure that the generated tetrahedra have acceptable shapes and conform to size objecFve. –  2D analogy, where triangles formed at the boundary.

– As the algorithm progresses, the front will advance to fill the remainder of the area with triangles.

OpFmizaFon-‐based Unstructured Tetrahedral Mesh GeneraFon

•  Varia)onal approaches view tet meshing as energy funcFonal, based on calculus of varia)ons.

•  Mesh results from itera)ve minimiza)on of funcFonal, via vertex displacements and connec)vity changes.

•  Stage 1: Identify triangulated surfaces: Marching Cubes –  Basic idea of Marching Cubes in 2D: Marching Squares.

Surface Meshing of Tissues – Marching Cubes (1)

•  3D version: Marching Cubes

Surface Meshing of Tissues – Marching Cubes (2)

•  Problem: Marching Cubes generates many triangles… •  Stage 2: Decimation of Marching Cubes

569K ∆s 142K ∆s

Surface Meshing of Tissues – DecimaFon (1)

•  Alternate approach: Simplex mesh topological operators •  T1-T2 operators (Delingette): add/delete an edge


•  6-edge face insertion or deletion (Gilles). •  Provides both resolution control and produces highly

regular faces, e.g.: dual to high quality triangles.


•  Approximated Centroidal Voronoi Diagrams (ACVD): is varia)onal clustering of triangles.

•  A Voronoi diagram (VD): division of space into regions. –  For each seed point in a set, its region consisFng of all points closer to that seed than to any other.

•  Here, user decides N clusters: N seeds opFmally spread on triangulated surface. Surface-‐based VD; triangulate.


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