2,408 research outputs found
Morphology for matrix data : ordering versus PDE-based approach
Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs).
We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data.
These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data
PDE-based morphology for matrix fields : numerical solution schemes
Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrix-valued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets
Improving Fiber Alignment in HARDI by Combining Contextual PDE Flow with Constrained Spherical Deconvolution
We propose two strategies to improve the quality of tractography results
computed from diffusion weighted magnetic resonance imaging (DW-MRI) data. Both
methods are based on the same PDE framework, defined in the coupled space of
positions and orientations, associated with a stochastic process describing the
enhancement of elongated structures while preserving crossing structures. In
the first method we use the enhancement PDE for contextual regularization of a
fiber orientation distribution (FOD) that is obtained on individual voxels from
high angular resolution diffusion imaging (HARDI) data via constrained
spherical deconvolution (CSD). Thereby we improve the FOD as input for
subsequent tractography. Secondly, we introduce the fiber to bundle coherence
(FBC), a measure for quantification of fiber alignment. The FBC is computed
from a tractography result using the same PDE framework and provides a
criterion for removing the spurious fibers. We validate the proposed
combination of CSD and enhancement on phantom data and on human data, acquired
with different scanning protocols. On the phantom data we find that PDE
enhancements improve both local metrics and global metrics of tractography
results, compared to CSD without enhancements. On the human data we show that
the enhancements allow for a better reconstruction of crossing fiber bundles
and they reduce the variability of the tractography output with respect to the
acquisition parameters. Finally, we show that both the enhancement of the FODs
and the use of the FBC measure on the tractography improve the stability with
respect to different stochastic realizations of probabilistic tractography.
This is shown in a clinical application: the reconstruction of the optic
radiation for epilepsy surgery planning
Controlling cell-matrix traction forces by extracellular geometry
We present a minimal continuum model of strongly adhering cells as active
contractile isotropic media and use the model to study the effect of the
geometry of the adhesion patch in controlling the spatial distribution of
traction and cellular stresses. Activity is introduced as a contractile, hence
negative, spatially homogeneous contribution to the pressure. The model shows
that patterning of adhesion regions can be used to control traction stress
distribution and yields several results consistent with experimental
observations. Specifically, the cell spread area is found to increase with
substrate stiffness and an analytic expression for the dependence is obtained
for circular cells. The correlation between the magnitude of traction stresses
and cell boundary curvature is also demonstrated and analyzed.Comment: 12 pages, 4 figure
Improved texture image classification through the use of a corrosion-inspired cellular automaton
In this paper, the problem of classifying synthetic and natural texture
images is addressed. To tackle this problem, an innovative method is proposed
that combines concepts from corrosion modeling and cellular automata to
generate a texture descriptor. The core processes of metal (pitting) corrosion
are identified and applied to texture images by incorporating the basic
mechanisms of corrosion in the transition function of the cellular automaton.
The surface morphology of the image is analyzed before and during the
application of the transition function of the cellular automaton. In each
iteration the cumulative mass of corroded product is obtained to construct each
of the attributes of the texture descriptor. In a final step, this texture
descriptor is used for image classification by applying Linear Discriminant
Analysis. The method was tested on the well-known Brodatz and Vistex databases.
In addition, in order to verify the robustness of the method, its invariance to
noise and rotation were tested. To that end, different variants of the original
two databases were obtained through addition of noise to and rotation of the
images. The results showed that the method is effective for texture
classification according to the high success rates obtained in all cases. This
indicates the potential of employing methods inspired on natural phenomena in
other fields.Comment: 13 pages, 14 figure
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