Lmj Luc Florack

On the Axioms of Scale Space Theory

We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all “reasonable” axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half plane {(x, s) ∈ ℝd × ℝ | s > 0}. We investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.

An efficient method for tensor voting using steerable filters

In many image analysis applications there is a need to extract curves in noisy images. To achieve a more robust extraction, one can exploit correlations of oriented features over a spatial context in the image. Tensor voting is an existing technique to extract features in this way. In this paper, we present a new computational scheme for tensor voting on a dense field of rank-2 tensors. Using steerable filter theory, it is possible to rewrite the tensor voting operation as a linear combination of complex-valued convolutions. This approach has computational advantages since convolutions can be implemented efficiently. We provide speed measurements to indicate the gain in speed, and illustrate the use of steerable tensor voting on medical applications.

A General Framework for Geometry-Driven Evolution Equations

This paper presents a general framework to generate multi-scale representations of image data. The process is considered as an initial value problem with an acquired image as initial condition and a geometrical invariant as “driving force” of an evolutionary process. The geometrical invariants are extracted using the family of Gaussian derivative operators. These operators naturally deal with scale as a free parameter and solve the ill-posedness problem of differentiation. Stability requirements for numerical approximation of evolution schemes using Gaussian derivative operators are derived and establish an intuitive connection between the allowed time-step and scale. This approach has been used to generalize and implement a variety of nonlinear diffusion schemes. Results on test images and medical images are shown.

Multiplicative Calculus in Biomedical Image Analysis

We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. It provides a natural framework in problems in which positive images or positive definite matrix fields and positivity preserving operators are of interest. Indeed, its merit lies in the fact that preservation of positivity under basic but important operations, such as differentiation, is manifest. In the case of positive scalar functions, or in general any set of positive definite functions with a commutative codomain, it is a convenient, albeit arguably redundant framework. However, in the increasingly important non-commutative case, such as encountered in diffusion tensor imaging and strain tensor analysis, multiplicative calculus complements standard calculus in a truly nontrivial way. The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis.

Multiscale Approach to Image Sequence Analysis

In optic flow based velocity estimation the image brightness constraint equation is used. However, for measurements performed at a certain scale, the brightness constraint equation does not apply. We therefore use a recently developed approach which reconciles optic flow and scale space theory. It specifically incorporates the scale (aperture) of image measurements, leading to a scheme which is essentially different from existing approaches. To obtain a unique velocity field, the data-derived information has to be augmented with physical knowledge. By keeping a strict separation between data-derived and external information, we can locally adapt or modify the user-supplied information without affecting the image-derived information. The two free scale parameters in time and space can be used for attentive vision (selecting particular velocities or objects) and to improve the reliability of velocity estimates.

Measures for pathway analysis in brain white matter using diffusion tensor images

In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesics on the manifold. Here we do not discuss methods to compute these geodesics, nor do we rely on the availability of geodesics. Instead we propose local measures which are directly computable from the local DTI data, and which enable us to preselect viable or exclude uninteresting seed points for the potentially time consuming extraction of geodesics. If geodesics are available, our measures can be readily applied to these as well. We consider two types of geodesic measures. One pertains to the connectivity saliency of a geodesic, the second to its stability with respect to local spatial perturbations. For the first type of measure we consider both differential as well as integral measures for characterizing a geodesics saliency either locally or globally. (In the latter case one needs to be in possession of the geodesic curve, in the former case a single tangent vector suffices.) The second type of measure is intrinsically local, and turns out to be related to a well known tensor in Riemannian geometry.

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.

Content Based Image Retrieval Using Multiscale Top Points

A feasibility study for a new method for content based image retrieval is presented. First, an image representation using multiscale top points is introduced. This representation is validated using a minimal variance reconstruction algorithm. The image retrieval problem can now be translated into comparing distances between point sets. For this purpose the proportional transportation distance (PTD) is used. A method is proposed using multiscale top points and their reconstruction coefficients in the PTD to define these distances between images. We present some experiments with promising results on a database with face images.

Image reconstruction from multiscale critical points

A minimal variance reconstruction scheme is derived using derivatives of the Gaussian as filters. A closed form mixed correlation matrix for reconstructions from multiscale points and their local derivatives up to the second order is presented. With the inverse of this mixed correlation matrix, a reconstruction of the image can be easily calculated. Some interesting results of reconstructions from multiscale critical points are presented. The influence of limited calculation precision is considered, using the condition number of the mixed correlation matrix.

Modeling foveal vision

A geometric model is proposed for an artificial foveal vision system, and its plausibility in the context of biological vision is explored. The model is based on an isotropic, scale invariant two-form that describes the spatial layout of receptive fields in the the visual sensorium (in the biological context roughly corresponding to retina, LGN, and V1). It overcomes the limitation of the familiar log-polar model by handling its singularity in a graceful way. The log-polar singularity arises as a result of ignoring the physical resolution limitation inherent in any real (artificial or biological) visual system. The incorporation of such a limitation requires the introduction of a physical constant, measuring the radius of the geometric foveola (a central region characterized by maximal resolving power). The proposed model admits a description in singularity-free canonical coordinates that generalize the well-established log-polar coordinates, and that reduce to these in the asymptotic case of negligibly sized geometric foveola (or, equivalently, at peripheral locations in the visual field). Biological plausibility of the model is demonstrated by comparison with known facts on human vision.