The Gaussian scale-space paradigm and the multiscale local jet

A representation of local image structure is proposed which takes into account both the image's spatial structure at a given location, as well as its deep structure, that is, its local behaviour as a function of scale or resolution (scale-space). This is of interest for several low-level image tasks. The proposed basis of scale-space, for example, enables a precise local study of interactions of neighbouring image intensities in the course of the blurring process. It also provides an extrapolation scheme for local image data, obtained at a given spatial location and resolution, to a finite scale-space neighbourhood. This is especially useful for the determination of sampling rates and for interpolation algorithms in a multilocal context. Another, particularly straightforward application is image enhancement or deblurring, which is an instance of data extrapolation in the high-resolution direction.A potentially interesting feature of the proposed local image parametrisation is that it captures a trade-off between spatial and scale extrapolations from a given interior point that do not exceed a given tolerance. This (rade-off suggests the possibility of a fairly coarse scale sampling at the expense of a dense spatial sampling large relative spatial overlap of scale-space kernels).The central concept developed in this paper is an equivalence class called the multiscale local jet, which is a hierarchical, local characterisation of the image in a full scale-space neighbourhood. For this local jet, a basis of fundamental polynomials is constructed that captures the scale-space paradigm at the local level up to any given order.     

Non-Linear Scale-Spaces Isomorphic to the Linear Case with Applications to Scalar,Vector and Multispectral Images

A basic requirement of scale-space representations in general is that of scale causality, which states that local extrema in the image should not be enhanced when resolution is diminished. We consider a special class of nonlinear scale-spaces consistent with this constraint, which can be linearised by a suitable isomorphism in the grey-scale domain so as to reproduce the familiar Gaussian scale-space. We consider instances in which nonlinear representations may be the preferred choice, as well as instances in which they enter by necessity. We also establish their relation to morphological scale-space representations based on a quadratic structuring function.     

A Linear Image Reconstruction Framework Based on Sobolev Type Inner Products

Exploration of information content of features that are present in images has led to the development of several reconstruction algorithms. These algorithms aim for a reconstruction from the features that is visually close to the image from which the features are extracted. Degrees of freedom that are not fixed by the constraints are disambiguated with the help of a so-called prior (i.e. a user defined model). We propose a linear reconstruction framework that generalizes a previously proposed scheme. The algorithm greatly reduces the complexity of the reconstruction process compared to non-linear methods. As an example we propose a specific prior and apply it to the reconstruction from singular points. The reconstruction is visually more attractive and has a smaller 핃₂-error than the reconstructions obtained by previously proposed linear methods. Bart Jansen, Frans Kanters and Remco Duits are joint main authors of this article.     

ScaleSpaceViz: α-Scale spaces in practice

Kernels of the so-called α-scale space have the undesirable property of having no closed-form representation in the spatial domain, despite their simple closed-form expression in the Fourier domain. This obstructs spatial convolution or recursive implementation. For this reason an approximation of the 2D α-kernel in the spatial domain is presented using the well-known Gaussian kernel and the Poisson kernel. Experiments show good results, with maximum relative errors of less than 2.4%. The approximation has been successfully implemented in a program for visualizing α-scale spaces. Some examples of practical applications with scale space feature points using the proposed approximation are given. The text was submitted by the authors in English. Frans Kanters received his MSc degree in Electrical Engineering in 2002 from the Eindhoven University of Technology in the Netherlands. Currently he is working on his PhD at the Biomedical Imaging and Informatics group at the Eindhoven University of Technology. His PhD work is part of the “Deep Structure, Singularities, and Computer Vision (DSSCV)” project sponsored by the European Union. His research interests include scale space theory, image reconstruction, image processing algorithms, and hardware implementations thereof. Luc Florack received his MSc degree in theoretical physics in 1989 and his PhD degree cum laude in 1993 with a thesis on image structure, both from Utrecht University, the Netherlands. During the period from 1994 to 1995, he was an ERCIM/HCM research fellow at INRIA Sophia-Antipolis, France, and IN-ESC Aveiro, Portugal. In 1996 he was an assistant research professor at DIKU, Copenhagen, Denmark, on a grant from the Danish Research Council. From 1997 to June 2001, he was an assistant research professor at Utrecht University in the Department of Mathematics and Computer Science. Since June 1, 2001, he has been working as an assistant professor and, then, as an associate professor at Eindhoven University of Technology, Department of Biomedical Engineering. His interest includes all multiscale structural aspects of signals, images, and movies and their applications to imaging and vision. Remco Duits received his MSc degree (cum laude) in Mathematics in 2001 from the Eindhoven University of Technology, the Netherlands. Today he is a PhD student at the Department of Biomedical Engineering at the Eindhoven University of Technology on the subject of multiscale perceptual organization. His interest subtends functional analysis, group theory, partial differential equations, multiscale representations and their applications to biomedical imaging and vision, perceptual grouping. Currently, he is finishing his thesis “Perceptual Organization in Image Analysis (A Mathematical Approach Based on Scale, Orientation and Curvature).” During his PhD work, several of his submissions at conferences were chosen as selected or best papers—in particular, at the PRIA 2004 conference on pattern recognition and image analysis in St. Petersburg, where he received a best paper award (second place) for his work on invertible orientation scores. Bram Platel received his Masters Degree cum laude in biomedical engineering from the Eindhoven University of Technology in 2002. His research interests include image matching, scale space theory, catastrophe theory, and image-describing graph constructions. Currently he is working on his PhD in the Biomedical Imaging and Informatics group at the Eindhoven University of Technology. Bart M. ter Haar Romany is full professor in Biomedical Image Analysis at the Department of Biomedical Engineering at Eindhoven University of Technology. He has been in this position since 2001. He received a MSc in Applied Physics from Delft University of Technology in 1978, and a PhD on neuromuscular nonlinearities from Utrecht University in 1983. After being the principal physicist of the Utrecht University Hospital Radiology Department, in 1989 he joined the department of Medical Imaging at Utrecht University as an associate professor. His interests are mathematical aspects of visual perception, in particular linear and non-linear scale-space theory, computer vision applications, and all aspects of medical imaging. He is author of numerous papers and book chapters on these issues; he edited a book on non-linear diffusion theory and is author of an interactive tutorial book on scale-space theory in computer vision. He has initiated a number of international collaborations on these subjects. He is an active teacher in international courses, a senior member of IEEE, and IEEE Chapter Tutorial Speaker. He is chairman of the Dutch Biophysical Society.     

A New Tensorial Framework for Single-Shell High Angular Resolution Diffusion Imaging

Single-shell high angular resolution diffusion imaging data (HARDI) may be decomposed into a sum of eigenpolynomials of the Laplace-Beltrami operator on the unit sphere. The resulting representation combines the strengths hitherto offered by higher order tensor decomposition in a tensorial framework and spherical harmonic expansion in an analytical framework, but removes some of the conceptual weaknesses of either. In particular it admits analytically closed form expressions for Tikhonov regularization schemes and estimation of an orientation distribution function via the Funk-Radon Transform in tensorial form, which previously required recourse to spherical harmonic decomposition. As such it provides a natural point of departure for a Riemann-Finsler extension of the geometric approach towards tractography and connectivity analysis as has been stipulated in the context of diffusion tensor imaging (DTI), while at the same time retaining the natural coarse-to-fine hierarchy intrinsic to spherical harmonic decomposition.     

Regularization, Scale-Space, and Edge Detection Filters

Computational vision often needs to deal with derivatives ofdigital images. Such derivatives are not intrinsic properties ofdigital data; a paradigm is required to make them well-defined.Normally, a linear filtering is applied. This can be formulated interms of scale-space, functional minimization, or edge detectionfilters. The main emphasis of this paper is to connect these theoriesin order to gain insight in their similarities and differences. We donot want, in this paper, to take part in any discussion of how edgedetection must be performed, but will only link some of the current theories. We take regularization (or functional minimization) as astarting point, and show that it boils down to Gaussian scale-space ifwe require scale invariance and a semi-group constraint to besatisfied. This regularization implies the minimization of afunctional containing terms up to infinite order of differentiation.If the functional is truncated at second order, the Canny-Deriche filter arises. It is also shown that higher dimensional regularizationboils down to a rotated version of the one dimensional case, whenCartesian invariance is imposed and the image is vanishing at theborders. This means that the results from 1D regularization can beeasily generalized to higher dimensions. Finally we show how anefficient implementation of regularization of order n can be made byrecursive filtering using 2n multiplications and additions peroutput element without introducing any approximation.     

The Relevance of Non-Generic Events in Scale Space Models

In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of spatial critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe and model the behaviour of critical point trajectories when various different types of generic events, viz. annihilations and creations of pairs of spatial critical points, (almost) coincide. Although such events are non-generic in mathematical sense, they are not unlikely to be encountered in practice due to numerical limitations. Furthermore, the behaviour of these trajectories leads to the observation that fine-to-coarse tracking of critical points doesn't suffice, since they can form closed loops in scale space. The modelling of the trajectories include these loops. We apply the theory to an artificial image and a simulated MR image and show the occurrence of the described behaviour.     

A spatio-frequency trade-off scale for scale-space filtering

We study implementation issues for spatial convolution filters and their Fourier alternative, with the aim to optimize the accuracy of filter output. We focus on Gaussian scale-space filters and show that there exists a trade-off scale that subdivides the available scale range into two subintervals of equal length. Below this trade-off scale Fourier filtering yields more accurate results than spatial filtering; above it is the other way around. This should be contrasted with demands of computational speed, which show the opposite tenet     

Pseudo-Linear Scale-Space Theory

It has been observed that linear, Gaussian scale-space, and nonlinear, morphological erosion and dilation scale-spaces generated by a quadratic structuring function have a lot in common. Indeed, far-reaching analogies have been reported, which seems to suggest the existence of an underlying isomorphism. However, an actual mapping appears to be missing.In the present work a one-parameter isomorphism is constructed in closed-form, which encompasses linear and both types of morphological scale-spaces as (non-uniform) limiting cases. The unfolding of the one-parameter family provides a means to transfer known results from one domain to the other. Moreover, for any fixed and non-degenerate parameter value one obtains a novel type of pseudo-linear multiscale representation that is, in a precise way, in-between the familiar ones. This is of interest in its own right, as it enables one to balance pros and cons of linear versus morphological scale-space representations in any particular situation.     

Novel Similarity Measures for Differential Invariant Descriptors for Generic Object Retrieval

Local feature matching is an essential component of many image and object retrieval algorithms. Euclidean and Mahalanobis distances are mostly used in order to quantify the similarity of two stipulated feature vectors. The Euclidean distance is inappropriate in the typical case where the components of the feature vector are incommensurable entities, and indeed yields unsatisfactory results in practice. The Mahalanobis distance performs better, but is less generic in the sense that it requires specific training data. In this paper we consider two alternative ways to construct generic distance measures for image and object retrieval, which do not suffer from any of these shortcomings. The first approach aims at obtaining a (image independent) covariance matrix for a Mahalonobis-like distance function without explicit training, and is applicable to feature vectors consisting of partial image derivatives. In the second approach a stability based similarity measure (SBSM) is introduced for feature vectors that are composed of arbitrary algebraic combinations of image derivatives, and likewise requires no explicit training. The strength and novelty of SBSM lies in the fact that the associated covariance matrix exploits local image structure. A performance analysis shows that feature matching based on SBSM outperforms algorithms based on Euclidean and Mahalanobis distances.