Luc Florack

Gaussian Scale-Space Theory

Preface. Scale in Perspective J.J. Koenderink. I: Applications. 1. Applications of Scale-Space Theory B. ter Haar Romeny. 2. Enhancement of Fingerprint Images Using Shape-Adapted Scale-Space Operators A. Almansa, T. Lindeberg. 3. Optic Flow and Stereo W.J. Niessen, R. Maas. II: The Foundation. 4. On The History of Gaussian Scale-Space Axiomatics J. Weickert, et al. 5. Scale-Space and Measurement Duality L. Florack. 6. On The Axiomatic Foundations of Linear Scale-Space T. Lindeberg. 7. Scale-Space Generators and Functionals M. Nielsen. 8. Invariance Theory A. Salden. 9. Stochastic Analysis of Image Acquisition and Scale-Space Smoothing K. Astrom, A. Heyden. III: The Structure. 10. Local Analysis of Image Scale Space P. Johansen. 11. Local Morse Theory for Gaussian Blurred Functions J. Damon. 12. Critical Point Events in Affine Scale-Space L. Griffin. 13. Topological Numbers and Singularities S. Kalitzin. 14. Multi-Scale Watershed Segmentation O.F. Olsen. IV: Non-Linear Extensions. 15. The Morphological Equivalent of Gaussian Scale-Space R. van den Boomgaard, L. Dorst. 16. Nonlinear Diffusion Scale-Spaces J. Weickert. Bibliography. Index.

The Topological Structure of Scale-Space Images

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

Scale-Space: Its Natural Operators and Differential Invariants

Why and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a ‘scale-space representation’, in which a given image is represented by a one-dimensional family of images representing that image on various levels of inner spatial scale. An early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family of scaled partial differential operators: the Gaussian kernel (the lowest order, rescaling operator) and its linear partial derivatives. They enable local image analysis in a robust way, while at the same time capturing global features through the extra scale degree of freedom. The operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale, orthogonal differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles in the mammalian front-end visual system.

The Intrinsic Structure of Optic Flow Incorporating Measurement Duality

The purpose of this article is to define optic flow for scalar and density images without using a priori knowledge other than its defining conservation principle, and to incorporate measurement duality, notably the scale-space paradigm. It is argued that the design of optic flow based applications may benefit from a manifest separation between factual image structure on the one hand, and goal-specific details and hypotheses about image flow formation on the other.The approach is based on a physical symmetry principle known as gauge invariance. Data-independent models can be incorporated by means of admissible gauge conditions, each of which may single out a distinct solution, but all of which must be compatible with the evidence supported by the image data. The theory is illustrated by examples and verified by simulations, and performance is compared to several techniques reported in the literature.

Scale-Space Theory in Computer Vision

A basic problem when deriving information from measured data, such as images, originates from the fact that objects in the world, and hence image structures, exist as meaningful entities only over ...

α scale spaces on a bounded domain

We consider α scale spaces, a parameterized class (α ∈ (0, 1)) of scale space representations beyond the well-established Gaussian scale space, which are generated by the α-th power of the minus Laplace operator on a bounded domain using the Neumann boundary condition. The Neumann boundary condition ensures that there is no grey-value flux through the boundary. Thereby no artificial grey-values from outside the image affect the evolution proces, which is the case for the α scale spaces on an unbounded domain. Moreover, the connection between the α scale spaces which is not trivial in the unbounded domain case, becomes straightforward: The generator of the Gaussian semigroup extends to a compact, self-adjoint operator on the Hilbert space L2(Ω) and therefore it has a complete countable set of eigen functions. Taking the α-th power of the Gaussian generator simply boils down to taking the α-th power of the corresponding eigenvalues. Consequently, all α scale spaces have exactly the same eigen-modes and can be implemented simultaneously as scale dependent Fourier series. The only difference between them is the (relative) contribution of each eigen-mode to the evolution process. By introducing the notion of (non-dimensional) relative scale in each α scale space, we are able to compare the various α scale spaces. The case α = 0.5, where the generator equals the square root of the minus Laplace operator leads to Poisson scale space, which is at least as interesting as Gaussian scale space and can be extended to a (Clifford) analytic scale space.

Deep Structure, Singularities, and Computer Vision

Oral Presentations.- Blurred Correlation Versus Correlation Blur.- A Scale Invariant Covariance Structure on Jet Space.- Essential Loops and Their Relevance for Skeletons and Symmetry Sets.- Pre-symmetry Sets of 3D Shapes.- Deep Structure of Images in Populations Via Geometric Models in Populations.- Estimating the Statistics of Multi-object Anatomic Geometry Using Inter-object Relationships.- Histogram Statistics of Local Model-Relative Image Regions.- The Bessel Scale-Space.- Linear Image Reconstruction from a Sparse Set of ?-Scale Space Features by Means of Inner Products of Sobolev Type.- A Riemannian Framework for the Processing of Tensor-Valued Images.- From Stochastic Completion Fields to Tensor Voting.- Deep Structure from a Geometric Point of View.- Maximum Likely Scale Estimation.- Adaptive Trees and Pose Identification from External Contours of Polyhedra.- Poster Presentations.- Exploiting Deep Structure.- Scale-Space Hierarchy of Singularities.- Computing 3D Symmetry Sets A Case Study.- Irradiation Orientation from Obliquely Viewed Texture.- Using Top-Points as Interest Points for Image Matching.- Transitions of Multi-scale Singularity Trees.- A Comparison of the Deep Structure of ?-Scale Spaces.- A Note on Local Morse Theory in Scale Space and Gaussian Deformations.

A Multiscale Geometric Model of Human Vision

A crucial factor in human perception is that we are able to move around in the three- dimensional world we live in. This induces continuous changes in the structure of the visual world as it is projected onto the retina. Much attention has been paid to the analysis of the “pictorial mode” of perception, the analysis of the retinal images as such. Gibson was one of the pioneers in this field, studying the behavior and perception of aircraft pilots during landing manoeuvres. He coined the term “ecological optics” for the study of the natural inflow of information, in which the deformation of structure due to relative movements of objects and observer (or the observer’s eyes) is studied.

A generic approach to diffusion filtering of matrix-fields

SummaryDiffusion tensor magnetic resonance imaging, is a image acquisition method, that provides matrix- valued data, so-called matrix fields. Hence image processing tools for the filtering and analysis of these data types are in demand. In this article, we propose a generic framework that allows us to find the matrix-valued counterparts of the Perona–Malik PDEs with various diffusivity functions. To this end we extend the notion of derivatives and associated differential operators to matrix fields of symmetric matrices by adopting an operator-algebraic point of view. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our novel matrix-valued Perona–Malik diffusion filters.

Top-points as interest points for image matching

We consider the use of top-points for object retrieval. These points are based on scale-space and catastrophe theory, and are invariant under gray value scaling and offset as well as scale-Euclidean transformations. The differential properties and noise characteristics of these points are mathematically well understood. It is possible to retrieve the exact location of a top-point from any coarse estimation through a closed-form vector equation which only depends on local derivatives in the estimated point. All these properties make top-points highly suitable as anchor points for invariant matching schemes. By means of a set of repeatability experiments and receiver-operator-curves we demonstrate the performance of top-points and differential invariant features as image descriptors.