The Topological Structure of Scale-Space Images

Authors:
Luc Florack;Arjan Kuijper
Affiliations:
Utrecht University, Department of Mathematics, Budapestlaan 6, NL-3584 CD Utrecht, The Netherlands. Luc.Florack@math.uu.nl;Utrecht University, Department of Computer Science, Padualaan 14, NL-3584 CH Utrecht, The Netherlands. arjan@cs.uu.nl
Venue:
Journal of Mathematical Imaging and Vision
Year:
2000

Citing 15
Cited 53

Dynamic shape

Biological Cybernetics
Representation of local geometry in the visual system

Biological Cybernetics
Edge Focusing

IEEE Transactions on Pattern Analysis and Machine Intelligence
A Hitherto Unnoticed Singularity of Scale-Space

IEEE Transactions on Pattern Analysis and Machine Intelligence
Solid shape

Solid shape
The Gaussian scale-space paradigm and the multiscale local jet

International Journal of Computer Vision
A multiscale approach to image sequence analysis

Computer Vision and Image Understanding - Special issue on physics-based modeling and reasoning in computer vision
Probabilistic Multiscale Image Segmentation

IEEE Transactions on Pattern Analysis and Machine Intelligence
The Intrinsic Structure of Optic Flow Incorporating Measurement Duality

International Journal of Computer Vision
Topological Numbers and Singularities in Scalar Images: Scale-Space Evolution Properties

Journal of Mathematical Imaging and Vision
Feature Detection with Automatic Scale Selection

International Journal of Computer Vision
Edge Detection and Ridge Detection with Automatic Scale Selection

International Journal of Computer Vision
Scale-Space Theory in Computer Vision

Scale-Space Theory in Computer Vision
Gaussian Scale-Space Theory

Gaussian Scale-Space Theory
Linear Scale-Space has First been Proposed in Japan

Journal of Mathematical Imaging and Vision

A Dynamic Scale–Space Paradigm

Journal of Mathematical Imaging and Vision
Scale Space Hierarchy

Journal of Mathematical Imaging and Vision
The Relevance of Non-generic Events in Scale Space Models

ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Understanding and Modeling the Evolution of Critical Points under Gaussian Blurring

ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Calculations on Critical Points under Gaussian Blurring

SCALE-SPACE '99 Proceedings of the Second International Conference on Scale-Space Theories in Computer Vision
Scale-Space Theories for Scalar and Vector Images

Scale-Space '01 Proceedings of the Third International Conference on Scale-Space and Morphology in Computer Vision
The Relevance of Non-Generic Events in Scale Space Models

International Journal of Computer Vision
The Monogenic Scale-Space: A Unifying Approach to Phase-Based Image Processing in Scale-Space

Journal of Mathematical Imaging and Vision
Using Catastrophe Theory to Derive Trees from Images

Journal of Mathematical Imaging and Vision
The Monogenic Scale Space on a Rectangular Domain and its Features

International Journal of Computer Vision
Controlled-topology filtering

Proceedings of the 2006 ACM symposium on Solid and physical modeling
A Linear Image Reconstruction Framework Based on Sobolev Type Inner Products

International Journal of Computer Vision
Constant tangential angle elected interest points

MIR '06 Proceedings of the 8th ACM international workshop on Multimedia information retrieval
Controlled-topology filtering

Computer-Aided Design
Interpretability based interest points detection

Proceedings of the 6th ACM international conference on Image and video retrieval
Gradient Structure of Image in Scale Space

Journal of Mathematical Imaging and Vision
Critical Scale for Unsupervised Cluster Discovery

MLDM '07 Proceedings of the 5th international conference on Machine Learning and Data Mining in Pattern Recognition
Statistically Valid Graph Representations of Scale-Space Geometry

ICISP '08 Proceedings of the 3rd international conference on Image and Signal Processing
Unsupervised cluster discovery using statistics in scale space

Engineering Applications of Artificial Intelligence
Linear Image Reconstruction by Sobolev Norms on the Bounded Domain

International Journal of Computer Vision
Validation of Watershed Regions by Scale-Space Statistics

SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Coarse-to-Fine Image Reconstruction Based on Weighted Differential Features and Background Gauge Fields

SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Multi-scale Stacked Sequential Learning

MCS '09 Proceedings of the 8th International Workshop on Multiple Classifier Systems
The Representation and Matching of Images Using Top Points

Journal of Mathematical Imaging and Vision
A theory of phase singularities for image representation and its applications to object tracking and image matching

IEEE Transactions on Image Processing
Perceptually relevant and piecewise linear matching of silhouettes

Pattern Recognition
On manifolds in Gaussian scale space

Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Content based image retrieval using multiscale top points a feasibility study

Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
The monogenic scale space on a bounded domain and its applications

Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Image reconstruction from multiscale critical points

Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Scale-space clustering with recursive validation

SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
A scale-space Reeb-graph of topological invariants of images and its applications to content identification

SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Combining different types of scale space interest points using canonical sets

SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
On the convergence of planar curves under smoothing

IEEE Transactions on Image Processing
Critical nets and beta-stable features for image matching

ECCV'10 Proceedings of the 11th European conference on computer vision conference on Computer vision: Part III
Optic flow based on multi-scale anchor point movement and discontinuity-preserving regularization

Pattern Recognition
Multiscale extension of the gravitational approach to edge detection

CAEPIA'11 Proceedings of the 14th international conference on Advances in artificial intelligence: spanish association for artificial intelligence
Towards a new paradigm for motion extraction

ICIAR'06 Proceedings of the Third international conference on Image Analysis and Recognition - Volume Part I
Optic flow from multi-scale dynamic anchor point attributes

ICIAR'06 Proceedings of the Third international conference on Image Analysis and Recognition - Volume Part I
Top-points as interest points for image matching

ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part I
Deep structure of images in populations via geometric models in populations

DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
Deep structure from a geometric point of view

DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
Using top-points as interest points for image matching

DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
A comparison of the deep structure of α-scale spaces

DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
A note on local morse theory in scale space and gaussian deformations

DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
Stability of top-points in scale space

Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Discrete representation of top points via scale space tessellation

Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
A linear image reconstruction framework based on sobolev type inner products

Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Tree edit distances from singularity theory

Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Bifurcation of segment edge curves in scale space

SSVM'11 Proceedings of the Third international conference on Scale Space and Variational Methods in Computer Vision
Cardiac motion estimation using covariant derivatives and helmholtz decomposition

STACOM'11 Proceedings of the Second international conference on Statistical Atlases and Computational Models of the Heart: imaging and modelling challenges
Uniqueness Results for Image Reconstruction from Features on Curves in α-Scale Spaces

Journal of Mathematical Imaging and Vision
Multiscale edge detection based on Gaussian smoothing and edge tracking

Knowledge-Based Systems

Quantified Score

Hi-index	0.00

Visualization

Abstract

We investigate the “deep structure” of a scale-space image. The emphasis is on topology, i.e. we concentrateon critical points—points with vanishing gradient—andtop-points—critical points with degenerate Hessian—andmonitor their displacements, respectively generic morsificationsin scale-space. Relevant parts of catastrophe theory in the context of thescale-space paradigm are briefly reviewed, and subsequentlyrewritten into coordinate independent form. This enables one to implement topological descriptors using aconveniently defined coordinate system.