64 research outputs found
Separable time-causal and time-recursive spatio-temporal receptive fields
We present an improved model and theory for time-causal and time-recursive
spatio-temporal receptive fields, obtained by a combination of Gaussian
receptive fields over the spatial domain and first-order integrators or
equivalently truncated exponential filters coupled in cascade over the temporal
domain. Compared to previous spatio-temporal scale-space formulations in terms
of non-enhancement of local extrema or scale invariance, these receptive fields
are based on different scale-space axiomatics over time by ensuring
non-creation of new local extrema or zero-crossings with increasing temporal
scale. Specifically, extensions are presented about parameterizing the
intermediate temporal scale levels, analysing the resulting temporal dynamics
and transferring the theory to a discrete implementation in terms of recursive
filters over time.Comment: 12 pages, 2 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1404.203
Codomain scale space and regularization for high angular resolution diffusion imaging
Regularization is an important aspect in high angular resolution diffusion imaging (HARDI), since, unlike with classical diffusion tensor imaging (DTI), there is no a priori regularity of raw data in the co-domain, i.e. considered as a multispectral signal for fixed spatial position. HARDI preprocessing is therefore a crucial step prior to any subsequent analysis, and some insight in regularization paradigms and their interrelations is compulsory. In this paper we posit a codomain scale space regularization paradigm that has hitherto not been applied in the context of HARDI. Unlike previous (first and second order) schemes it is based on infinite order regularization, yet can be fully operationalized. We furthermore establish a closed-form relation with first order Tikhonov regularization via the Laplace transform
Deep structure from a geometric point of view
The geometry of empty scale space is investigated. Byvirtue of the proposed geometric axioms the generating PDE, the linearisotropic heat equation, can be presented in covariant, or geometricalform. The postulate of a metric for scale space cannot be upheld, asit is incompatible with the generating equation. Two familiar instancesof scale spaces consistent with the geometric axioms are considered byway of example, viz. classical, homogeneous scale space, and foveal scalespace
Scale space representations locally adapted to the geometry of base and target manifold
We generalize the Gaussian multi-resolution image paradigm for a Euclidean domain to general Riemannian base manifolds and also account for the codomain by considering the extension into a fibre bundle structure. We elaborate on aspects of parametrization and gauge, as these are important in practical applications. We subsequently scrutinize two examples that are of interest in bio-mathematical modeling, viz. scale space on the unit sphere, used among others for codomain regularization in the context of high angular resolution diffusion imaging (HARDI), and retino-cortical scale space, proposed as a biologically plausible model of the human visual pathway from retina to striate cortex
Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging
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) [24] 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
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