In this work we demonstrate how Finsler geometry---and specifically the related geodesic tracto-graphy---can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. Using data from the Human Connectome Project, as well as population data from an autism spectrum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity measures, results in connectivity maps that are much closer to known tract anatomy compared to previous geodesic connectivity methods. Our implementation can be used to compute geodesic distance and connectivity maps for segmented areas and is publicly available.

We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358–1372, 2004) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus.In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data.

In this paper we address the problem of object retrieval based on scale-space interest points, namely top-points. The original retrieval algorithm can only cope with scale-Euclidean transformations. We extend the algorithm to the case of non-rigid transformations like affine and perspective transformations and investigate its robustness. The proposed algorithm is proven to be highly robust under various degrading factors, such as noise, occlusion, rendering artifacts, etc. and can deal with multiple occurrences of the object.

We propose an operational method to extract the left ventricle (LV) systole dynamics using harmonic phase (HARP) images extracted from tagged cardiac MR sequences. Established techniques to generate HARP sequences provide independent evidence for motion extraction, in the sense that the combined linear system for scalar brightness conservation, applied to the HARP images, can be uniquely solved for a dense field of motion parameters without the need for regularization. In contrast to some of the previously proposed popular methods, no segmentation or tracking of tags over time, nor interpolation of a sparse motion field explicitly coupled to the tag pattern is required, and the problem of tag fading is bypassed. An important novelty is the incorporation of automatic local scale selection so as to obtain a robust solution, which not only yields a stable, but also a smoothly varying motion field of the (healthy) LV myocardial wall. The scheme relies on an integer parameter representing order of approximation, and allows one to simultaneously obtain a dense field of differential tensors capturing the low order differential structure of the motion field, which is useful for the computation of relevant local quantities such as strain rates and material acceleration fields. The methodology is generic and straightforward to implement, and can be generalized to 3D and, in principle, to account for higher order differential structure.

It is well-known that scale space theory and Tikhonov regularization are close-knit. In previous studies, qualitative analogies and formal relations have already been found, but none of these established both an exact as well as operational connection. We establish such a connection for the case of a Gaussian scale space representation and a first order Tikhonov scheme. The free parameter of the latter turns out to be related to a particular attenuation of scales in a procedure whereby one "collapses" the scale space image along the scale axis via the Laplace transform. The result provides a physical interpretation of first order Tikhonov regularization and its associated control parameter.

We discuss the feasibility of using singular points in a scale space representation (referred to as top points) for image matching purposes. These points are easily extracted from the scale space of an image and they form a compact description of the image. The image matching problem thus becomes a point cloud matching problem. This is related to the transportation problem known from linear optimization and we solve it by using an earth mover's distance algorithm. To match points in scale space, a distance measure is needed, as Euclidean distance no longer applies. We suggest a metric that can be used in scale space and show that it indeed performs better than a Euclidean distance measure. To distinguish between stable and unstable top points, we derive a stability norm based on the total variation norm which only depends on the second order derivatives at the top point. To improve matching results further, we show that other features at the top points can also increase the accuracy of matching.

Using a Gaussian scale space, one can use the extra dimension, viz. scale, for investigation of "built-in" properties of the image in scale space. We show that one of such induced properties is the nesting of special iso-intensity manifolds, which yield an implicitly present hierarchy of the critical points and regions of their influence, in the original image. Its very nature allows one not only to segment the original image automatically, but also to apply "logical filters" to it, obtaining simplified images. We give an algorithm deriving this hierarchy and show its effectiveness on two different kinds of images, both with respect to segmentation and simplification.

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.

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.

Despite the fact that images constitute the main objects in machine vision, there is remarkably little concern about their actual definition. An operational definition is proposed beyond machine-specific technicalities. Such a definition is of interest in the design of applications for which the details of the storage or processing medium are not crucially important (the usual case). It provides a unified conceptual framework for various existing theories and techniques, such as scale-space theory and mathematical morphology, and possibly other, nonlinear image representations.

In recent years several nonlinear diffusion schemes have been introduced. We discuss the numerical implementation of a number of current nonlinear evolution schemes, using the notion of well-posed differentiation by Gaussian kernels. The infinitesimal change of an image when increasing scale depends on the local differential invariants evaluated at the scale of the image considered, i.e. on terms of the local jet (the set of all spatial partial derivatives at that point). All these differential terms can be obtained in a well-posed fashion by a convolution of the original image with the family of the Gaussian and its derivatives. The nonlinear partial differential evaluation can thus be numerically approximated by an iterative calculation of the appropriate terms in the local jet. Examples are given for medical images.>

Differentiation is known to be ill-posed in the sense of Hadamard. The theory of regular tempered distributions and the concept of Gaussian convolution filters open the way to a well-posed differentiation process, thereby introducing the notion of scale (or: inverse resolution). There is no a priori fundamental limit to the order of differentiation of images provided they are calculated on a sufficiently high scale (relative to pixel scale and noise correlation width), and provided we have a sufficient dynamic range of intensity values. Constraints in resolution (both in the spatial and in the intensity domain) enforce a scale-dependent restriction to the accuracy with which Gaussian kernels G/sub n/(x; /spl sigma/) can be represented in a physical sense. So at a given scale /spl sigma/ (e.g. in units of the sampling scale) and a given measure of inaccuracy /spl alpha/ there is a maximal order n above which the margin /spl alpha/ is exceeded. In this paper we quantify this relation.>

Presents a method of multi-resolution image analysis that gives the most concise set of local orthogonally invariant features of 2-dimensional input images. Solving the equivalence problem corresponding to a local jet and the group of orthogonal transformations of the cartesian coordinate frame the authors find a complete and irreducible set of local algebraic invariants that may describe any local orthogonally invariant feature of the jet.>