Worst-case Analysis of Set Union Algorithms
Journal of the ACM (JACM)
Three-dimensional object recognition
ACM Computing Surveys (CSUR) - Annals of discrete mathematics, 24
Geometric modeling
Introduction to Solid Modeling
Introduction to Solid Modeling
Solid shape
Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
Graph-directed modelling from serial sections
The Visual Computer: International Journal of Computer Graphics
Computational complexity of combinatorial surfaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations
IEEE Transactions on Pattern Analysis and Machine Intelligence
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
ACM Transactions on Graphics (TOG)
Thinning Methodologies-A Comprehensive Survey
IEEE Transactions on Pattern Analysis and Machine Intelligence
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Topographic distance and watershed lines
Signal Processing - Special issue on mathematical morphology and its applications to signal processing
A review of algorithms for shape analysis
Document image analysis
An incremental algorithm for Betti numbers of simplicial complexes on the 3-spheres
Computer Aided Geometric Design - Special issue on grid generation, finite elements, and geometric design
Geodesic Saliency of Watershed Contours and Hierarchical Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Attribute openings, thinnings, and granulometries
Computer Vision and Image Understanding
Computing Size Functions from Edge Maps
International Journal of Computer Vision
VIS '97 Proceedings of the 8th conference on Visualization '97
Contour trees and small seed sets for isosurface traversal
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Construction of contour trees in 3D in O(n log n) steps
Proceedings of the fourteenth annual symposium on Computational geometry
Primitives for the manipulation of general subdivisions and the computation of Voronoi
ACM Transactions on Graphics (TOG)
Handbook of discrete and computational geometry
Visualization of scalar topology for structural enhancement
Proceedings of the conference on Visualization '98
A connected component approach to the watershed segmentation
ISMM '98 Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing
On the definition and the construction of pockets in macromolecules
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
Level set diagrams of polyhedral objects
Proceedings of the fifth ACM symposium on Solid modeling and applications
Computing Morse functions on triangulated manifolds
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Isosurface extraction in time-varying fields using a temporal branch-on-need tree (T-BON)
VIS '99 Proceedings of the conference on Visualization '99: celebrating ten years
Collapsing flow topology using area metrics
VIS '99 Proceedings of the conference on Visualization '99: celebrating ten years
VIS '99 Proceedings of the conference on Visualization '99: celebrating ten years
Connected filtering and segmentation using component trees
Computer Vision and Image Understanding
Computing contour trees in all dimensions
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
The watershed transform: definitions, algorithms and parallelization strategies
Fundamenta Informaticae - Special issue on mathematical morphology
Representations for Rigid Solids: Theory, Methods, and Systems
ACM Computing Surveys (CSUR)
Optimal surface reconstruction from planar contours
Communications of the ACM
Representation of contours and regions for efficient computer search
Communications of the ACM
State of the art in shape matching
Principles of visual information retrieval
Extracting regions of interest applying a local watershed transformation
Proceedings of the conference on Visualization '00
Topology preserving and controlled topology simplifying multiresolution isosurface extraction
Proceedings of the conference on Visualization '00
Semi-regular mesh extraction from volumes
Proceedings of the conference on Visualization '00
Hierarchical morse complexes for piecewise linear 2-manifolds
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Topology matching for fully automatic similarity estimation of 3D shapes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Shape abstraction using computational topology techniques
From geometric modeling to shape modeling
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Continuous topology simplification of planar vector fields
Proceedings of the conference on Visualization '01
Exploring scalar fields using critical isovalues
Proceedings of the conference on Visualization '02
Efficient computation of the topology of level sets
Proceedings of the conference on Visualization '02
Topological Zone Organization of Scalar Volume Data
Journal of Mathematical Imaging and Vision
Automatic Isosurface Propagation Using an Extrema Graph and Sorted Boundary Cell Lists
IEEE Transactions on Visualization and Computer Graphics
Partitioning 3D Surface Meshes Using Watershed Segmentation
IEEE Transactions on Visualization and Computer Graphics
Fast Isosurface Generation Using the Volume Thinning Algorithm
IEEE Transactions on Visualization and Computer Graphics
Surface Coding Based on Morse Theory
IEEE Computer Graphics and Applications
Constructing a Reeb graph automatically from cross sections
IEEE Computer Graphics and Applications
Volume Data Mining Using 3D Field Topology Analysis
IEEE Computer Graphics and Applications
Computing contour trees in all dimensions
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
The flow complex: a data structure for geometric modeling
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the Second Joint European - US Workshop on Applications of Invariance in Computer Vision
Path seeds and flexible isosurfaces using topology for exploratory visualization
VISSYM '03 Proceedings of the symposium on Data visualisation 2003
Detecting critical regions in scalar fields
VISSYM '03 Proceedings of the symposium on Data visualisation 2003
Morphological Image Analysis: Principles and Applications
Morphological Image Analysis: Principles and Applications
Computational Geometry: Theory and Applications - Special issue: The European workshop on computational geometry -- CG01
Loops in reeb graphs of 2-manifolds
Proceedings of the nineteenth annual symposium on Computational geometry
Molecular shape analysis based upon the morse-smale complex and the connolly function
Proceedings of the nineteenth annual symposium on Computational geometry
Morse-smale complexes for piecewise linear 3-manifolds
Proceedings of the nineteenth annual symposium on Computational geometry
Topological persistence and simplification
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Topology Recognition of 3D Closed Freeform Objects Based on Topological Graphs
PG '01 Proceedings of the 9th Pacific Conference on Computer Graphics and Applications
Re-Meshing Techniques for Topological Analysis
SMI '01 Proceedings of the International Conference on Shape Modeling & Applications
Topological quadrangulations of closed triangulated surfaces using the Reeb graph
Graphical Models - Special issue: Discrete topology and geometry for image and object representation
Isosurface generation by using extrema graphs
VIS '94 Proceedings of the conference on Visualization '94
Morphology-driven simplification and multiresolution modeling of terrains
GIS '03 Proceedings of the 11th ACM international symposium on Advances in geographic information systems
Discrete exterior calculus
Part decomposition of 3d surfaces
Part decomposition of 3d surfaces
A Formal Classification of 3D Medial Axis Points and Their Local Geometry
IEEE Transactions on Pattern Analysis and Machine Intelligence
Topological Volume Skeletonization Using Adaptive Tetrahedralization
GMP '04 Proceedings of the Geometric Modeling and Processing 2004
Digital Geometry: Geometric Methods for Digital Picture Analysis
Digital Geometry: Geometric Methods for Digital Picture Analysis
Removing excess topology from isosurfaces
ACM Transactions on Graphics (TOG)
Parallel Computation of the Topology of Level Sets
Algorithmica
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Extreme elevation on a 2-manifold
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Time-varying reeb graphs for continuous space-time data
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
A Survey of Content Based 3D Shape Retrieval Methods
SMI '04 Proceedings of the Shape Modeling International 2004
Augmented Reeb Graphs for Content-Based Retrieval of 3D Mesh Models
SMI '04 Proceedings of the Shape Modeling International 2004
Reeb Graph Representation of Surfaces with Boundary
SMI '04 Proceedings of the Shape Modeling International 2004
Applications of Forman's Discrete Morse Theory to Topology Visualization and Mesh Compression
IEEE Transactions on Visualization and Computer Graphics
The Use of Size Functions for Comparison of Shapes Through Differential Invariants
Journal of Mathematical Imaging and Vision
Fair morse functions for extracting the topological structure of a surface mesh
ACM SIGGRAPH 2004 Papers
Morse Homology Descriptor for Shape Characterization
ICPR '04 Proceedings of the Pattern Recognition, 17th International Conference on (ICPR'04) Volume 4 - Volume 04
Topological manipulation of isosurfaces
Topological manipulation of isosurfaces
Simplifying Flexible Isosurfaces Using Local Geometric Measures
VIS '04 Proceedings of the conference on Visualization '04
Feature-based surface parameterization and texture mapping
ACM Transactions on Graphics (TOG)
Discrete & Computational Geometry
Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)
Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)
Simple and optimal output-sensitive construction of contour trees using monotone paths
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
Persistence barcodes for shapes
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
A Multi-resolution Data Structure for Two-dimensional Morse-Smale Functions
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
Subdomain Aware Contour Trees and Contour Evolution in Time-Dependent Scalar Fields
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Maximizing Adaptivity in Hierarchical Topological Models
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Volumetric Data Analysis using Morse-Smale Complexes
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
IEEE Transactions on Visualization and Computer Graphics
Feature-based similarity search in 3D object databases
ACM Computing Surveys (CSUR)
Content-based retrieval of 3D models
ACM Transactions on Multimedia Computing, Communications, and Applications (TOMCCAP)
Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry
SMI '06 Proceedings of the IEEE International Conference on Shape Modeling and Applications 2006
A Topological Approach to Simplification of Three-Dimensional Scalar Functions
IEEE Transactions on Visualization and Computer Graphics
3D Distance Fields: A Survey of Techniques and Applications
IEEE Transactions on Visualization and Computer Graphics
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Persistence-sensitive simplification functions on 2-manifolds
Proceedings of the twenty-second annual symposium on Computational geometry
Spectral surface quadrangulation
ACM SIGGRAPH 2006 Papers
Computer Aided Geometric Design - Special issue: Applications of geometric modeling in the life sciences
Partitioning of 3D Meshes using Reeb Gra
ICPR '06 Proceedings of the 18th International Conference on Pattern Recognition - Volume 01
Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities
IEEE Transactions on Visualization and Computer Graphics
Retrieval of trademark images by means of size functions
Graphical Models - Special issue on the vision, video and graphics conference 2005
Stability of Persistence Diagrams
Discrete & Computational Geometry
Topological analysis of shapes using Morse theory
Computer Vision and Image Understanding
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Topology-Controlled Volume Rendering
IEEE Transactions on Visualization and Computer Graphics
SMI '07 Proceedings of the IEEE International Conference on Shape Modeling and Applications 2007
Robust on-line computation of Reeb graphs: simplicity and speed
ACM SIGGRAPH 2007 papers
Defining and computing curve-skeletons with medial geodesic function
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Size functions for 3D shape retrieval
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Surface Segmentation through Concentrated Curvature
ICIAP '07 Proceedings of the 14th International Conference on Image Analysis and Processing
k-dimensional Size Functions for Shape Description and Comparison
ICIAP '07 Proceedings of the 14th International Conference on Image Analysis and Processing
Multi-resolution Morse-Smale Complexes for Terrain Modeling
ICIAP '07 Proceedings of the 14th International Conference on Image Analysis and Processing
Stability of Critical Points with Interval Persistence
Discrete & Computational Geometry
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Recursive geometry of the flow complex and topology of the flow complex filtration
Computational Geometry: Theory and Applications
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
Structural Shape Prototypes for the Automatic Classification of 3D Objects
IEEE Computer Graphics and Applications
Hybrid techniques for real-time radar simulation
AFIPS '63 (Fall) Proceedings of the November 12-14, 1963, fall joint computer conference
Extending Persistence Using Poincaré and Lefschetz Duality
Foundations of Computational Mathematics
Medial Representations: Mathematics, Algorithms and Applications
Medial Representations: Mathematics, Algorithms and Applications
Three-dimensional shape searching: state-of-the-art review and future trends
Computer-Aided Design
A barcode shape descriptor for curve point cloud data
Computers and Graphics
Topological analysis and characterization of discrete scalar fields
Proceedings of the 11th international conference on Theoretical foundations of computer vision
Comparing sets of 3D digital shapes through topological structures
GbRPR'07 Proceedings of the 6th IAPR-TC-15 international conference on Graph-based representations in pattern recognition
Three-Dimensional Shape Description Using the Symmetric Axis Transform I: Theory
IEEE Transactions on Pattern Analysis and Machine Intelligence
A complete keypics experiment with size functions
CIVR'05 Proceedings of the 4th international conference on Image and Video Retrieval
Description of digital images by region-based contour trees
ICIAR'05 Proceedings of the Second international conference on Image Analysis and Recognition
A multi-resolution representation for terrain morphology
GIScience'06 Proceedings of the 4th international conference on Geographic Information Science
Secondary and tertiary structural fold elucidation from 3d EM maps of macromolecules
ICVGIP'06 Proceedings of the 5th Indian conference on Computer Vision, Graphics and Image Processing
A functional-based segmentation of human body scans in arbitrary postures
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
A topological hierarchy for functions on triangulated surfaces
IEEE Transactions on Visualization and Computer Graphics
Antiextensive connected operators for image and sequence processing
IEEE Transactions on Image Processing
Fast computation of a contrast-invariant image representation
IEEE Transactions on Image Processing
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
A barcode shape descriptor for curve point cloud data
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
I/O-Efficient flow modeling on fat terrains
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Size functions for comparing 3D models
Pattern Recognition
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
Technical Section: Discrete Laplace-Beltrami operators for shape analysis and segmentation
Computers and Graphics
Technical Section: Computing smooth approximations of scalar functions with constraints
Computers and Graphics
Recognition of Occluded Shapes Using Size Functions
ICIAP '09 Proceedings of the 15th International Conference on Image Analysis and Processing
A Critical Assessment of 2D and 3D Face Recognition Algorithms
AVSS '09 Proceedings of the 2009 Sixth IEEE International Conference on Advanced Video and Signal Based Surveillance
Topology- and error-driven extension of scalar functions from surfaces to volumes
ACM Transactions on Graphics (TOG)
Morphology analysis of 3D scalar fields based on morse theory and discrete distortion
Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Technical Section: Shape approximation by differential properties of scalar functions
Computers and Graphics
International Journal of Computer Vision
Multiresolution morse triangulations
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Shape comparison through mutual distances of real functions
Proceedings of the ACM workshop on 3D object retrieval
The Global-Local transformation for noise resistant shape representation
Computer Vision and Image Understanding
A global method for reducing multidimensional size graphs
GbRPR'11 Proceedings of the 8th international conference on Graph-based representations in pattern recognition
Robustness and modularity of 2-dimensional size functions: an experimental study
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part I
Persistent betti numbers for a noise tolerant shape-based approach to image retrieval
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part I
A new algorithm for computing the 2-dimensional matching distance between size functions
Pattern Recognition Letters
Computing morse decompositions for triangulated terrains: an analysis and an experimental evaluation
ICIAP'11 Proceedings of the 16th international conference on Image analysis and processing: Part I
Simplifying morphological representations of 2D and 3D scalar fields
Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
SMI 2012: Short Dimension-independent multi-resolution Morse complexes
Computers and Graphics
Certified computation of planar morse-smale complexes
Proceedings of the twenty-eighth annual symposium on Computational geometry
A deterministic o(m log m) time algorithm for the reeb graph
Proceedings of the twenty-eighth annual symposium on Computational geometry
The hitchhiker's guide to the galaxy of mathematical tools for shape analysis
ACM SIGGRAPH 2012 Courses
Multi-scale approximation of the matching distance for shape retrieval
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
Template based shape descriptor
EG 3DOR'09 Proceedings of the 2nd Eurographics conference on 3D Object Retrieval
Multilevel relevance feedback for 3D shape retrieval
EG 3DOR'09 Proceedings of the 2nd Eurographics conference on 3D Object Retrieval
Information-theoretic selection of high-dimensional spectral features for structural recognition
Computer Vision and Image Understanding
Topological maps and robust hierarchical Euclidean skeletons in cubical complexes
Computer Vision and Image Understanding
A spatial approach to morphological feature extraction from irregularly sampled scalar fields
Proceedings of the Third ACM SIGSPATIAL International Workshop on GeoStreaming
Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval
Pattern Recognition Letters
The persistence space in multidimensional persistent homology
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
A study of monodromy in the computation of multidimensional persistence
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
Voronoi-Based extraction of a feature skeleton from noisy triangulated surfaces
ACCV'12 Proceedings of the 11th Asian conference on Computer Vision - Volume Part II
Comparison of persistent homologies for vector functions: From continuous to discrete and back
Computers & Mathematics with Applications
SMI 2013: Grouping real functions defined on 3D surfaces
Computers and Graphics
SMI 2013: Steepest descent paths on simplicial meshes of arbitrary dimensions
Computers and Graphics
Morphologically-aware elimination of flat edges from a TIN
Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Technical Section: Topological saliency
Computers and Graphics
Surface- and volume-based techniques for shape modeling and analysis
SIGGRAPH Asia 2013 Courses
Optimizing polycube domain construction for hexahedral remeshing
Computer-Aided Design
Comparing shapes through multi-scale approximations of the matching distance
Computer Vision and Image Understanding
A primal/dual representation for discrete morse complexes on tetrahedral meshes
EuroVis '13 Proceedings of the 15th Eurographics Conference on Visualization
Learning kernels on extended Reeb graphs for 3d shape classification and retrieval
3DOR '13 Proceedings of the Sixth Eurographics Workshop on 3D Object Retrieval
Hi-index | 0.04 |
Differential topology, and specifically Morse theory, provide a suitable setting for formalizing and solving several problems related to shape analysis. The fundamental idea behind Morse theory is that of combining the topological exploration of a shape with quantitative measurement of geometrical properties provided by a real function defined on the shape. The added value of approaches based on Morse theory is in the possibility of adopting different functions as shape descriptors according to the properties and invariants that one wishes to analyze. In this sense, Morse theory allows one to construct a general framework for shape characterization, parametrized with respect to the mapping function used, and possibly the space associated with the shape. The mapping function plays the role of a lens through which we look at the properties of the shape, and different functions provide different insights. In the last decade, an increasing number of methods that are rooted in Morse theory and make use of properties of real-valued functions for describing shapes have been proposed in the literature. The methods proposed range from approaches which use the configuration of contours for encoding topographic surfaces to more recent work on size theory and persistent homology. All these have been developed over the years with a specific target domain and it is not trivial to systematize this work and understand the links, similarities, and differences among the different methods. Moreover, different terms have been used to denote the same mathematical constructs, which often overwhelm the understanding of the underlying common framework. The aim of this survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner. The term geometrical-topological used in the title is meant to underline that both levels of information content are relevant for the applications of shape descriptions: geometrical, or metrical, properties and attributes are crucial for characterizing specific instances of features, while topological properties are necessary to abstract and classify shapes according to invariant aspects of their geometry. The approaches surveyed will be discussed in detail, with respect to theory, computation, and application. Several properties of the shape descriptors will be analyzed and compared. We believe this is a crucial step to exploit fully the potential of such approaches in many applications, as well as to identify important areas of future research.