2,388 research outputs found
A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)
We present a new flexible wavefront propagation algorithm for the boundary
value problem for sub-Riemannian (SR) geodesics in the roto-translation group
with a metric tensor depending on a smooth
external cost , , computed from
image data. The method consists of a first step where a SR-distance map is
computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system
derived via Pontryagin's Maximum Principle (PMP). Subsequent backward
integration, again relying on PMP, gives the SR-geodesics. For
we show that our method produces the global minimizers. Comparison with exact
solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics.
We present numerical computations of Maxwell points and cusp points, which we
again verify for the uniform cost case . Regarding image
analysis applications, tracking of elongated structures in retinal and
synthetic images show that our line tracking generically deals with crossings.
We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven
Sub-Riemannian Geodesics in SE(2)
Extracting 3D parametric curves from 2D images of Helical objects
Helical objects occur in medicine, biology, cosmetics, nanotechnology, and engineering. Extracting a 3D parametric curve from a 2D image of a helical object has many practical applications, in particular being able to extract metrics such as tortuosity, frequency, and pitch. We present a method that is able to straighten the image object and derive a robust 3D helical curve from peaks in the object boundary. The algorithm has a small number of stable parameters that require little tuning, and the curve is validated against both synthetic and real-world data. The results show that the extracted 3D curve comes within close Hausdorff distance to the ground truth, and has near identical tortuosity for helical objects with a circular profile. Parameter insensitivity and robustness against high levels of image noise are demonstrated thoroughly and quantitatively
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, based on 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 (i) parameterizing the
intermediate temporal scale levels, (ii) analysing the resulting temporal
dynamics, (iii) transferring the theory to a discrete implementation, (iv)
computing scale-normalized spatio-temporal derivative expressions for
spatio-temporal feature detection and (v) computational modelling of receptive
fields in the lateral geniculate nucleus (LGN) and the primary visual cortex
(V1) in biological vision.
We show that by distributing the intermediate temporal scale levels according
to a logarithmic distribution, we obtain much faster temporal response
properties (shorter temporal delays) compared to a uniform distribution.
Specifically, these kernels converge very rapidly to a limit kernel possessing
true self-similar scale-invariant properties over temporal scales, thereby
allowing for true scale invariance over variations in the temporal scale,
although the underlying temporal scale-space representation is based on a
discretized temporal scale parameter.
We show how scale-normalized temporal derivatives can be defined for these
time-causal scale-space kernels and how the composed theory can be used for
computing basic types of scale-normalized spatio-temporal derivative
expressions in a computationally efficient manner.Comment: 39 pages, 12 figures, 5 tables in Journal of Mathematical Imaging and
Vision, published online Dec 201
The formation of spiral arms and rings in barred galaxies
In this and in a previous paper (Romero-Gomez et al. 2006) we propose a
theory to explain the formation of both spirals and rings in barred galaxies
using a common dynamical framework. It is based on the orbital motion driven by
the unstable equilibrium points of the rotating bar potential. Thus, spirals,
rings and pseudo-rings are related to the invariant manifolds associated to the
periodic orbits around these equilibrium points. We examine the parameter space
of three barred galaxy models and discuss the formation of the different
morphological structures according to the properties of the bar model. We also
study the influence of the shape of the rotation curve in the outer parts, by
making families of models with rising, flat, or falling rotation curves in the
outer parts. The differences between spiral and ringed structures arise from
differences in the dynamical parameters of the host galaxies. The results
presented here will be discussed and compared with observations in a
forthcoming paper.Comment: 16 pages, 13 figures, accepted in A&A. High resolution version
available at http://www.oamp.fr/dynamique/pap/merce.htm
Active Vision With Multiresolution Wavelets
A wavelet decomposition for multiscale edge detection is used to separate border edges from texture in an image, toward the goal of a complete segmentation by Active Perception for robotic exploration of a scene. The physical limitations of the image acquisition system and the robotic system provide the limitations on the range of scales which we consider. We link edges through scale space, using the characteristics of these wavelets for guidance. The linked zero crossings are used to remove texture and preserve borders, then the scene can be reconstructed without texture
Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging
Locally adaptive differential frames (gauge frames) are a well-known
effective tool in image analysis, used in differential invariants and
PDE-flows. However, at complex structures such as crossings or junctions, these
frames are not well-defined. Therefore, we generalize the notion of gauge
frames on images to gauge frames on data representations defined on the extended space of positions and
orientations, which we relate to data on the roto-translation group ,
. This allows to define multiple frames per position, one per
orientation. We compute these frames via exponential curve fits in the extended
data representations in . These curve fits minimize first or second
order variational problems which are solved by spectral decomposition of,
respectively, a structure tensor or Hessian of data on . We include
these gauge frames in differential invariants and crossing preserving PDE-flows
acting on extended data representation and we show their advantage compared
to the standard left-invariant frame on . Applications include
crossing-preserving filtering and improved segmentations of the vascular tree
in retinal images, and new 3D extensions of coherence-enhancing diffusion via
invertible orientation scores
The circular SiZer, inferred persistence of shape parameters and application to early stem cell differentiation
We generalize the SiZer of Chaudhuri and Marron (J. Amer. Statist. Assoc. 94
(1999) 807-823, Ann. Statist. 28 (2000) 408-428) for the detection of shape
parameters of densities on the real line to the case of circular data. It turns
out that only the wrapped Gaussian kernel gives a symmetric, strongly Lipschitz
semi-group satisfying "circular" causality, that is, not introducing possibly
artificial modes with increasing levels of smoothing. Some notable differences
between Euclidean and circular scale space theory are highlighted. Based on
this, we provide an asymptotic theory to make inference about the persistence
of shape features. The resulting circular mode persistence diagram is applied
to the analysis of early mechanically-induced differentiation in adult human
stem cells from their actin-myosin filament structure. As a consequence, the
circular SiZer based on the wrapped Gaussian kernel (WiZer) allows the
verification at a controlled error level of the observation reported by Zemel
et al. (Nat. Phys. 6 (2010) 468-473): Within early stem cell differentiation,
polarizations of stem cells exhibit preferred directions in three different
micro-environments.Comment: Published at http://dx.doi.org/10.3150/15-BEJ722 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Multi-scale active shape description in medical imaging
Shape description in medical imaging has become an increasingly important research field in recent years. Fast and high-resolution image acquisition methods like Magnetic Resonance (MR) imaging produce very detailed cross-sectional images of the human body - shape description is then a post-processing operation which abstracts quantitative descriptions of anatomically relevant object shapes. This task is usually performed by clinicians and other experts by first segmenting the shapes of interest, and then making volumetric and other quantitative measurements. High demand on expert time and inter- and intra-observer variability impose a clinical need of automating this process. Furthermore, recent studies in clinical neurology on the correspondence between disease status and degree of shape deformations necessitate the use of more sophisticated, higher-level shape description techniques. In this work a new hierarchical tool for shape description has been developed, combining two recently developed and powerful techniques in image processing: differential invariants in scale-space, and active contour models. This tool enables quantitative and qualitative shape studies at multiple levels of image detail, exploring the extra image scale degree of freedom. Using scale-space continuity, the global object shape can be detected at a coarse level of image detail, and finer shape characteristics can be found at higher levels of detail or scales. New methods for active shape evolution and focusing have been developed for the extraction of shapes at a large set of scales using an active contour model whose energy function is regularized with respect to scale and geometric differential image invariants. The resulting set of shapes is formulated as a multiscale shape stack which is analysed and described for each scale level with a large set of shape descriptors to obtain and analyse shape changes across scales. This shape stack leads naturally to several questions in regard to variable sampling and appropriate levels of detail to investigate an image. The relationship between active contour sampling precision and scale-space is addressed. After a thorough review of modem shape description, multi-scale image processing and active contour model techniques, the novel framework for multi-scale active shape description is presented and tested on synthetic images and medical images. An interesting result is the recovery of the fractal dimension of a known fractal boundary using this framework. Medical applications addressed are grey-matter deformations occurring for patients with epilepsy, spinal cord atrophy for patients with Multiple Sclerosis, and cortical impairment for neonates. Extensions to non-linear scale-spaces, comparisons to binary curve and curvature evolution schemes as well as other hierarchical shape descriptors are discussed
Large amplitude behavior of the Grinfeld instability: a variational approach
In previous work, we have performed amplitude expansions of the continuum
equations for the Grinfeld instability and carried them to high orders.
Nevertheless, the approach turned out to be restricted to relatively small
amplitudes. In this article, we use a variational approach in terms of
multi-cycloid curves instead. Besides its higher precision at given order, the
method has the advantages of giving a transparent physical meaning to the
appearance of cusp singularities and of not being restricted to interfaces
representable as single-valued functions. Using a single cycloid as ansatz
function, the entire calculation can be performed analytically, which gives a
good qualitative overview of the system. Taking into account several but few
cycloid modes, we obtain remarkably good quantitative agreement with previous
numerical calculations. With a few more modes taken into consideration, we
improve on the accuracy of those calculations. Our approach extends them to
situations involving gravity effects. Results on the shape of steady-state
solutions are presented at both large stresses and amplitudes. In addition,
their stability is investigated.Comment: subm. to EPJ
Unsupervised Color Image Segmentation: with Application to Skin Tumor Borders
The images used in this research were digitized from 35mm color photographic slides obtained from a private dermatology practice and from New York University. The authors compared 6 color segmentation methods and their effectiveness as part of an overall border-finding algorithm. The PCT/median cut and adaptive thresholding algorithms provided the lowest average error and show the most promise for further individual algorithm development. Combining the different methods resulted in further improvement in the number of correctly identified tumor borders, and by incorporating additional heuristics in merging the segmented object information, one could potentially further increase the success rate. The algorithm is broad-based and suggests several areas for further research. One possible area of exploration is to incorporate an intelligent decision making process as to the number of colors that should be used for segmentation in the PCT/median cut and adaptive thresholding algorithms. For comparison purposes, the number of colors was kept constant at three in the authors\u27\u27 application. Other areas that can be explored are noise removal and object classification to determine the correct tumor objec
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