Master Thesis: Conformal multi-material mesh generation from labelled medical volumes
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An important step in orthopaedic pre-operative planning is the generation of accurate volume meshes out of segmented volume image. These meshes are used in patient-specific, bio-mechanical finite element simulations to optimize positioning and design of implants. The development of accurate, multi-material volume meshing methods for medical applications is an active and interdisciplinary field of research. Several methods in the field that were proposed in recent years claim to accurately perform the task, each concept with its advantages and disadvantages. The approaches to the task are diverse. The question is: Which approach is the most suitable one? How do we evaluate the excellence of such methods ? What criteria can be applied to measure the quality of a multi-labelled volume mesh ? And which ones have the most impact on the subsequent simulation, so that stress calculations on the implant are realistic and correct ? These are the basic research questions that are discussed in this work.
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Master Thesis: Conformal multi-material mesh generation from labelled medical volumes
- 1. 1Challenge the future Meshing multi-labelled medical volumes
- 2. 3Challenge the future Introduction • Generation of volume meshes for FEA • Particular use case: hip prostheses analysis • Typical Pipeline:
- 3. 4Challenge the future • Input: segmented volume image • Expected output: labelled volume mesh • Restrictions to final mesh • Algorithm: robust and fast Introduction
- 4. 5Challenge the future Introduction • Challenge: bio-medial structures • Cavities and holes • Non-manifold structures • Junctions of multiple structures Images Meshes
- 5. 6Challenge the future Research Questions Which meshing concept is most suitable for extracting accurate, multi-material volume meshes from medical volume images ? How can the performance of a chosen concept be evaluate ? What criteria can be applied to measure the quality of a multi- labelled volume mesh ?
- 6. 7Challenge the future Related Work Zhang2007 Ju2002
- 7. 8Challenge the future Related Work Dey2011
- 8. 11Challenge the future Related Work Dynamic Particle Systems Meyer2008
- 9. 12Challenge the future Related Work Dynamic Particle Systems • Feature-adaptive sampling and meshing • Label interface protection
- 10. 13Challenge the future Our starting point
- 11. 14Challenge the future Our starting point • BioMesh3D: dynamic particle system Advantages: • Feature adaptive mesh • optimal vertex distribution • very good mesh quality • Meshing of multi-genus, non-manifold structures
- 12. 16Challenge the future Issues • Multi-labelled junctions • Structural conformity • Minimal tetrahedral element count • Medial Axis Transform runtime • High MAT accuracy required
- 13. 17Challenge the future Issues • Mathematical Explanation of Meshing Problem • Delaunay Triangulation • Sampling Requirement • Local feature size • Medial Axis Transform • Discrete image space
- 14. 18Challenge the future • Meshing algorithms for sparse particles: • Delaunay & Alpha Shapes Alternative Meshing Algorithms Fischer2004 ETH Zürich / Applied Geometry Group
- 15. 19Challenge the future • Meshing algorithms for sparse particles: • Voronoi Diagrams Alternative Meshing Algorithms Amenta2001
- 16. 20Challenge the future • Meshing algorithms for sparse particles: • Implicit Surfaces Alternative Meshing Algorithms
- 17. 21Challenge the future • Meshing algorithms for sparse particles: • Local Triangulation • Spherical Parametrization Alternative Meshing Algorithms Brink2005
- 18. 22Challenge the future • Meshing algorithms for sparse particles: • Local Triangulation • Poisson Surface Reconstruction Alternative Meshing Algorithms Kazhdan2006
- 19. 23Challenge the future • Meshing algorithms for sparse particles: • Local Triangulation • 2D generic Meshing Algorithms (Graham, TwoPeasant) Alternative Meshing Algorithms created with FastSurfaceReconstruction – PCL / Marton2009
- 20. 24Challenge the future • Meshing algorithms for sparse particles: • Local Triangulation • Local Delaunay Triangulation Alternative Meshing Algorithms
- 21. 25Challenge the future Contribution • Fast medial axis transform for faster execution • Local Meshing for multi-labelled volumes • Using known isosurface properties: normals and principal curvatures • Enabling sparser particle sampling, or • More detail with equal sampling
- 22. 26Challenge the future Our Approach: Integer Medial Axis
- 23. 27Challenge the future Our Approach: Integer Medial Axis • Motivation: much faster computation • BioMesh3D uses: Center of inscribed spheres • We use: Shortest path in feature transform [Hesselink and Roerdink 2008]
- 24. 29Challenge the future Our Approach: Surface Meshing • Parameter Extraction from multi-labelled iso-surfaces (Normals, Curvatures, Feature Size) • Particles + parameters local reconstruction
- 25. 30Challenge the future Our Approach: Surface Meshing • Idea: replace 3D Delaunay tetrahedralization with local 2D Delaunay triangulation • 3D sampling contraints 2D surface constraints • E.g.: project to local tangent plane • Locally watertight • Issue: We need guaranteed closed watertight 3D surface
- 26. 32Challenge the future Visualization • Toolbox: VTK • Visualization of multiple information and multi-labelled objects • Separate control for: • All labelled objects of one information group • All information of one Label • Volume Mesh clipping • Edge highlighting for visual element size estimation
- 28. 34Challenge the future Conclusion • Initial analysis revelead • MAT bottleneck • Sampling limitation dependencies • Alternative approaches • Integer Medial axis speed-up • Local Delaunay Triangulation less restrictive reconstruction • Prototype implementation with: • Multi-material surface mesh visualization • Multi-material volume mesh visualization • quality measurement
- 29. 35Challenge the future Future Work • Watertight local reconstruction algorithm • Modified sampling criterion • Mesh quality criteria consensus needed
- 30. 36Challenge the future Propositions 1. Discrete Medial Axis Transform schemes are more prone to errors in comparison to continuous schemes 2. feature-preserving surface reconstruction by 3D Delaunay Tetrahedralization is not possible 3. the spatial 1-Ring neighbourhood of a point cloud is dependent on the meshing scheme 4. Local Delaunay Triangulation enables shape- and feature- conform surface reconstruction of loose samples 5. scientific work requires a methodological perspective as well as a systematic perspective
Editor's Notes
- Today, I will present parts of my master thesis. The related computer graphics field of my thesis topic is computational geometry. Therefore today’s title of my talk is ...
- Introduction: What are we trying to do? (very) roughly Initial Conditions / pre-requirements General guideline Final goal Related Work: Recent development in that area Different approaches Basis of my work (with pipeline sketch) Criticism & Contribution general issues of dynamic particle systems day-to-day trouble consequently arising our contribution to the field Our Approach [/comment for later] Integer Medial Axis Normal Extraction from Multi-Label Iso-Surface Fast Surface Reconstruktion Filtering On-going work Quality Measurments Appropriate Volume Visualization (Distance Fields) Runtime + Quality Comparison
- General idea/guideline: Start: CT scan data of the patient => Segmentation into separate regions => Conversion into a separate surface descption => Generation of volume mesh based on particular geometric primitives => fusion of separate models into one model of separate labels with interfaces
- Restrictions to final mesh: topological conformity with focus on label interfaces reasonable amount of geometric elements Adaptive to features preferably features are preserved
- Advantages Classical Method: Fast Easy Tessellation (element number control) Drawbacks -not topology conformant -sometimes bad quality -inordinary amount of elements (bad for FEA) -stepping artifacts with Marching Cubes -no feature adaptive meshing
- Advantages Classical Method: Fast Easy Tessellation (element number control) Drawbacks -not topology conformant -sometimes bad quality -inordinary amount of elements (bad for FEA) -stepping artifacts with Marching Cubes -no feature adaptive meshing
- With example pictures, refer to author and year (e.g. Dey2011) Delaunay: Feature preserving non-linear filter methods Interface extraction as cells Interfaces cell protection with protecting balls during Tetrahedralization Sliver removal Advantages DT: Good runtime behavior Conformant tetrahedra and good quality Feature adaptive meshing Watertight meshes Disadvantages: Topology correctness for vertices and edges, interface planar boundaries not guaranteed to be correct (protecting balls doesn’t work for planes) High dependency on pre-processing Long time no full solution existent (now in Amira)
- Point-based Registration Framework: Extracts fine grid-resolution volume mesh and Feature points (based on image properties) refine grid-volume to feature adaptive, precise volume mesh representation improved constrained Delaunay Tetrahedralization refinement improvements on geometric quality guarantees for interface surfaces by blacklisting [Kahnt2011] Delaunay refinement on piecewiese-smooth complexes [Dey2011] interface-conform, multi-tissue constrained Delaunay refinement with focus on ideal Dihedral Angles [Foteinos2011] extension of Meyer’s particle-based meshing approach on single-material CAD-surfaces with improved particle system and local triangulation scheme -optimal shape via LDA a-shape and conformal a-shape [Presley2011] introduction Local Density-Adaptive a-shape [Maillot2010] realization of 3D LDA a-shapes [Chevallier2011] modelling ambiguous 3D structures with compoundly weighted a-shapes [Chazals2011] notation of extremal surfaces [Lui2010] introduction of multi-level partitions of unity implicit surfaces [d’Otreppe2011] smooth surface description by modified Marching Cubes interpolator [Manson2010]
- Pictures Dynamic Particle Systems: good Delaunay Tetrahedralization requires good vertices sample criteria Amenta et. al Idea: distribute particles along the material’s surface to generate “optimal” Delaunay vertices Interface adaptive and feature adaptive sampling Delaunay Tetrahedralization forms volume mesh; surfaces at multi-material interfaces form boundary mesh Advantages: watertight mesh high-quality mesh elements Copes with interfaces vertices, edges and faces Tessellation adapted to interfaces and junctions
- Improvements: conform meshing for more than 2-material junctions [Quadrants picture] capture sharp features in sampling correct boundary- and volume description robust implementation
- Citicism: Tightening filters volume Non-feature preserving (feature reducing) operation [CLICK FOR PICTURE] Low curvature -> high radius of curvature and v.v. High radius of curvature goes along with smoother surfaces which are coarser tessellated Therefore: small number of elements gradually reduces features [CLICK FOR PICTURE] -Limiting cuvature alters model and topology Hard features (corners, edges) are eroded Curvature limitation -> lost features -> features are of topological significance in non-manifold structures [CLICK FOR PICTURE] Feature size detected by medial axis depending on radius of curvature Medial axis well defined in continious space; correct medial axis hard to compute in discrete space (i.e. Volume images) Therefore correct feature size extraction (with required level of detail) from volume images is not guaranteed, but required by the particle system Unprecise feature size leads to mathematical convergence problems during the iterative refinement of the particle system Boundary extraction can be done by other methods than with 3D Delaunay Triangulation, which requires additional surface information Surface information present by unused Because meshing output significantly dependent on the curvature parameter, quality control hard to establish
- Local Feature Size = distance from boundary point to the medial axis Distance medial axis – boundary at corners = radius of curvature = local feature size Sharp corners: feature size = 0 Sampling criterium for Delaunay Triangulation: d(p1,p2) = e* lfs; e -> practical boundary 0.6 Lfs @ corners = 0 => d(p1,p2)=0 =>number of particles at sharp corners = ∞ [CLICK FOR PICTURE] Detection of Medial Axis in continious space defined by partial differential equations (Solving Eikonal Equation of the distance field’s gradient and detecting ridges with evolving front; MA = outward normal flux (negative divergence) of the evolving front) Not directly convertable to discrete space Discrete Approximation create lots of branches; Distance/Feature Size not precise enough For particle system continious space-similar medial axis necessary That’s the reason for Meyer’s new MA-approach based on isosurface description. But incredibly slow and computationally unstable -Introduction of new Meshing Algorithm will solve most problems Delaunay and Alpha shapes inherit sampling criterium limitation Power Crust Algorithm based on Voronoi diagrams inherit the same sampling criterium Implicit Surfaces not restricted by sampling criterium, but implicit surface needs to be converted to triangle mesh including again a resampling Local Meshing Schemes include projection of neighbour points into local tangent plane; variety of local meshing algorithms can be used
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- -new Medial axis approach to improve runtime improvement -modifiy particle system to capture sharp features and not be limited to epsilon-criteria -choice of a suitable Local Triangulation Algorithm
- Abbreviation: CMS = center of maximal spheres
- - Base: smoothed marching cubes model; option to take Surfaces Nets [de Bruin et al. 2000] - averaging of parameters from particle attributes Compute principal curvature and consitently oriented normals Nearest surface points to particles are extracted parameter averaging and vector normalization
- Issue: Neighbourhood computation based on sample criterium [Gopi2000] replaced with local feature size TBN matrix at each point is computed Neighbourhood projected on tangent plane 2D Delaunay on tangent plane (watertight) For closed surface, projection position of connected neighbourhoods needs to be similar (otherwise wrong topology) -> sampling criterium ensures small gradual changes by high sampling of high-curvature regions Consistent, correct surface normals are required
- Curvature limitation lowered, but not erased Problem: Interplay Features – Feature Size – Curvature base problem here: Particle System [future idea: curvature-independent sampling criteria] Coupling Tessellation and Curvature still exists [ideas to overcome this: use precomputed curvatures instead of everywhere the local feature size subsequent triangle reduction (fusion) operation] lowered curvature limitation and meshing limitation can improve topological correctness because all things based on the local feature size, dependency on the medial axis is inevitable Local Delaunay Triangulation much more robust scheme as 3D Delaunay; is guaranteed to converge Isosurface parameters are now effectively used in new meshing schemes to imrpove stability quality control stays hard