12,169 research outputs found
SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels
We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant
of deep neural networks for irregular structured and geometric input, e.g.,
graphs or meshes. Our main contribution is a novel convolution operator based
on B-splines, that makes the computation time independent from the kernel size
due to the local support property of the B-spline basis functions. As a result,
we obtain a generalization of the traditional CNN convolution operator by using
continuous kernel functions parametrized by a fixed number of trainable
weights. In contrast to related approaches that filter in the spectral domain,
the proposed method aggregates features purely in the spatial domain. In
addition, SplineCNN allows entire end-to-end training of deep architectures,
using only the geometric structure as input, instead of handcrafted feature
descriptors. For validation, we apply our method on tasks from the fields of
image graph classification, shape correspondence and graph node classification,
and show that it outperforms or pars state-of-the-art approaches while being
significantly faster and having favorable properties like domain-independence.Comment: Presented at CVPR 201
Mapping carbon nanotube orientation by fast fourier transform of scanning electron micrographs
A novel method of applying a two-dimensional Fourier transform (2D-FFT) to SEM was developed to map the CNT orientation in pre-formed arrays. Local 2D-FFTs were integrated azimuthally to determine an orientation distribution function and the associated Herman parameter. This approach provides data rapidly and over a wide range of lengthscales. Although likely to be applicable to a wide range of anisotropic nanoscale structures, the method was specifically developed to study CNT veils, a system in which orientation critically controls mechanical properties. Using this system as a model, key parameters for the 2D-FFT analysis were optimised, including magnification and domain size; a model set of CNT veils were pre-strained to 5%, 10% and 15%, to vary the alignment degree. The algorithm confirmed a narrower orientation distribution function and increasing Herman parameter, with increasing pre-strain. To validate the algorithm, the local orientation was compared to that derived from a common polarised Raman spectroscopy. Orientation maps of the Herman parameter, derived by both methods, showed good agreement. Quantitatively, the mean Herman parameter calculated using the polarised Raman spectroscopy was 0.42 ± 0.004 compared to 0.32 ± 0.002 for the 2D-FFT method, with a correlation coefficient of 0.73. Possible reasons for the modest and systematic discrepancy were discussed
3D time series analysis of cell shape using Laplacian approaches
Background:
Fundamental cellular processes such as cell movement, division or food uptake critically depend on cells being able to change shape. Fast acquisition of three-dimensional image time series has now become possible, but we lack efficient tools for analysing shape deformations in order to understand the real three-dimensional nature of shape changes.
Results:
We present a framework for 3D+time cell shape analysis. The main contribution is three-fold: First, we develop a fast, automatic random walker method for cell segmentation. Second, a novel topology fixing method is proposed to fix segmented binary volumes without spherical topology. Third, we show that algorithms used for each individual step of the analysis pipeline (cell segmentation, topology fixing, spherical parameterization, and shape representation) are closely related to the Laplacian operator. The framework is applied to the shape analysis of neutrophil cells.
Conclusions:
The method we propose for cell segmentation is faster than the traditional random walker method or the level set method, and performs better on 3D time-series of neutrophil cells, which are comparatively noisy as stacks have to be acquired fast enough to account for cell motion. Our method for topology fixing outperforms the tools provided by SPHARM-MAT and SPHARM-PDM in terms of their successful fixing rates. The different tasks in the presented pipeline for 3D+time shape analysis of cells can be solved using Laplacian approaches, opening the possibility of eventually combining individual steps in order to speed up computations
A novel sampling theorem on the rotation group
We develop a novel sampling theorem for functions defined on the
three-dimensional rotation group SO(3) by connecting the rotation group to the
three-torus through a periodic extension. Our sampling theorem requires
samples to capture all of the information content of a signal band-limited at
, reducing the number of required samples by a factor of two compared to
other equiangular sampling theorems. We present fast algorithms to compute the
associated Fourier transform on the rotation group, the so-called Wigner
transform, which scale as , compared to the naive scaling of .
For the common case of a low directional band-limit , complexity is reduced
to . Our fast algorithms will be of direct use in speeding up the
computation of directional wavelet transforms on the sphere. We make our SO3
code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for
publication. Code available at http://www.sothree.or
A Fast and Accurate Algorithm for Spherical Harmonic Analysis on HEALPix Grids with Applications to the Cosmic Microwave Background Radiation
The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used
extensively in astrophysics for data collection and analysis on the sphere. The
scheme was originally designed for studying the Cosmic Microwave Background
(CMB) radiation, which represents the first light to travel during the early
stages of the universe's development and gives the strongest evidence for the
Big Bang theory to date. Refined analysis of the CMB angular power spectrum can
lead to revolutionary developments in understanding the nature of dark matter
and dark energy. In this paper, we present a new method for performing
spherical harmonic analysis for HEALPix data, which is a central component to
computing and analyzing the angular power spectrum of the massive CMB data
sets. The method uses a novel combination of a non-uniform fast Fourier
transform, the double Fourier sphere method, and Slevinsky's fast spherical
harmonic transform (Slevinsky, 2019). For a HEALPix grid with pixels
(points), the computational complexity of the method is , with an initial set-up cost of . This compares
favorably with runtime complexity of the current methods
available in the HEALPix software when multiple maps need to be analyzed at the
same time. Using numerical experiments, we demonstrate that the new method also
appears to provide better accuracy over the entire angular power spectrum of
synthetic data when compared to the current methods, with a convergence rate at
least two times higher
Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm
Practical applications of thermoacoustic tomography require numerical
inversion of the spherical mean Radon transform with the centers of integration
spheres occupying an open surface. Solution of this problem is needed (both in
2-D and 3-D) because frequently the region of interest cannot be completely
surrounded by the detectors, as it happens, for example, in breast imaging. We
present an efficient numerical algorithm for solving this problem in 2-D
(similar methods are applicable in the 3-D case). Our method is based on the
numerical approximation of plane waves by certain single layer potentials
related to the acquisition geometry. After the densities of these potentials
have been precomputed, each subsequent image reconstruction has the complexity
of the regular filtration backprojection algorithm for the classical Radon
transform. The peformance of the method is demonstrated in several numerical
examples: one can see that the algorithm produces very accurate reconstructions
if the data are accurate and sufficiently well sampled, on the other hand, it
is sufficiently stable with respect to noise in the data
Electron tomography at 2.4 {\AA} resolution
Transmission electron microscopy (TEM) is a powerful imaging tool that has
found broad application in materials science, nanoscience and biology(1-3).
With the introduction of aberration-corrected electron lenses, both the spatial
resolution and image quality in TEM have been significantly improved(4,5) and
resolution below 0.5 {\AA} has been demonstrated(6). To reveal the 3D structure
of thin samples, electron tomography is the method of choice(7-11), with
resolutions of ~1 nm^3 currently achievable(10,11). Recently, discrete
tomography has been used to generate a 3D atomic reconstruction of a silver
nanoparticle 2-3 nm in diameter(12), but this statistical method assumes prior
knowledge of the particle's lattice structure and requires that the atoms fit
rigidly on that lattice. Here we report the experimental demonstration of a
general electron tomography method that achieves atomic scale resolution
without initial assumptions about the sample structure. By combining a novel
projection alignment and tomographic reconstruction method with scanning
transmission electron microscopy, we have determined the 3D structure of a ~10
nm gold nanoparticle at 2.4 {\AA} resolution. While we cannot definitively
locate all of the atoms inside the nanoparticle, individual atoms are observed
in some regions of the particle and several grains are identified at three
dimensions. The 3D surface morphology and internal lattice structure revealed
are consistent with a distorted icosahedral multiply-twinned particle. We
anticipate that this general method can be applied not only to determine the 3D
structure of nanomaterials at atomic scale resolution(13-15), but also to
improve the spatial resolution and image quality in other tomography
fields(7,9,16-20).Comment: 27 pages, 17 figure
Signal processing with Fourier analysis, novel algorithms and applications
Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions, also analogously known as sinusoidal modeling. The original idea of Fourier had a profound impact on mathematical analysis, physics and engineering because it diagonalizes time-invariant convolution operators. In the past signal processing was a topic that stayed almost exclusively in electrical engineering, where only the experts could cancel noise, compress and reconstruct signals. Nowadays it is almost ubiquitous, as everyone now deals with modern digital signals. Medical imaging, wireless communications and power systems of the future will experience more data processing conditions and wider range of applications requirements than the systems of today. Such systems will require more powerful, efficient and flexible signal processing algorithms that are well designed to handle such needs. No matter how advanced our hardware technology becomes we will still need intelligent and efficient algorithms to address the growing demands in signal processing. In this thesis, we investigate novel techniques to solve a suite of four fundamental problems in signal processing that have a wide range of applications. The relevant equations, literature of signal processing applications, analysis and final numerical algorithms/methods to solve them using Fourier analysis are discussed for different applications in the electrical engineering/computer science. The first four chapters cover the following topics of central importance in the field of signal processing: • Fast Phasor Estimation using Adaptive Signal Processing (Chapter 2) • Frequency Estimation from Nonuniform Samples (Chapter 3) • 2D Polar and 3D Spherical Polar Nonuniform Discrete Fourier Transform (Chapter 4) • Robust 3D registration using Spherical Polar Discrete Fourier Transform and Spherical Harmonics (Chapter 5) Even though each of these four methods discussed may seem completely disparate, the underlying motivation for more efficient processing by exploiting the Fourier domain signal structure remains the same. The main contribution of this thesis is the innovation in the analysis, synthesis, discretization of certain well known problems like phasor estimation, frequency estimation, computations of a particular non-uniform Fourier transform and signal registration on the transformed domain. We conduct propositions and evaluations of certain applications relevant algorithms such as, frequency estimation algorithm using non-uniform sampling, polar and spherical polar Fourier transform. The techniques proposed are also useful in the field of computer vision and medical imaging. From a practical perspective, the proposed algorithms are shown to improve the existing solutions in the respective fields where they are applied/evaluated. The formulation and final proposition is shown to have a variety of benefits. Future work with potentials in medical imaging, directional wavelets, volume rendering, video/3D object classifications, high dimensional registration are also discussed in the final chapter. Finally, in the spirit of reproducible research we release the implementation of these algorithms to the public using Github
Time-Dependent Tomographic Reconstruction of the Solar Corona
Solar rotational tomography (SRT) applied to white-light coronal images
observed at multiple aspect angles has been the preferred approach for
determining the three-dimensional (3D) electron density structure of the solar
corona. However, it is seriously hampered by the restrictive assumption that
the corona is time-invariant which introduces significant errors in the
reconstruction. We first explore several methods to mitigate the temporal
variation of the corona by decoupling the "fast-varying" inner corona from the
"slow-moving" outer corona using multiple masking (either by juxtaposition or
recursive combination) and radial weighting. Weighting with a radial
exponential profile provides some improvement over a classical reconstruction
but only beyond 3 Rsun. We next consider a full time-dependent tomographic
reconstruction involving spatio-temporal regularization and further introduce a
co-rotating regularization aimed at preventing concentration of reconstructed
density in the plane of the sky. Crucial to testing our procedure and properly
tuning the regularization parameters is the introduction of a time-dependent
MHD model of the corona based on observed magnetograms to build a time-series
of synthetic images of the corona. Our procedure, which successfully reproduces
the time-varying model corona, is finally applied to a set of of 53 LASCO-C2 pB
images roughly evenly spaced in time from 15 to 29 March 2009. Our procedure
paves the way to a time-dependent tomographic reconstruction of the coronal
electron density to the whole set of LASCO-C2 images presently spanning 20
years.Comment: 24 pages, 18 figure
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