2,232 research outputs found
Efficient computation of partition of unity interpolants through a block-based searching technique
In this paper we propose a new efficient interpolation tool, extremely
suitable for large scattered data sets. The partition of unity method is used
and performed by blending Radial Basis Functions (RBFs) as local approximants
and using locally supported weight functions. In particular we present a new
space-partitioning data structure based on a partition of the underlying
generic domain in blocks. This approach allows us to examine only a reduced
number of blocks in the search process of the nearest neighbour points, leading
to an optimized searching routine. Complexity analysis and numerical
experiments in two- and three-dimensional interpolation support our findings.
Some applications to geometric modelling are also considered. Moreover, the
associated software package written in \textsc{Matlab} is here discussed and
made available to the scientific community
Implicit reconstructions of thin leaf surfaces from large, noisy point clouds
Thin surfaces, such as the leaves of a plant, pose a significant challenge
for implicit surface reconstruction techniques, which typically assume a
closed, orientable surface. We show that by approximately interpolating a point
cloud of the surface (augmented with off-surface points) and restricting the
evaluation of the interpolant to a tight domain around the point cloud, we need
only require an orientable surface for the reconstruction. We use polyharmonic
smoothing splines to fit approximate interpolants to noisy data, and a
partition of unity method with an octree-like strategy for choosing subdomains.
This method enables us to interpolate an N-point dataset in O(N) operations. We
present results for point clouds of capsicum and tomato plants, scanned with a
handheld device. An important outcome of the work is that sufficiently smooth
leaf surfaces are generated that are amenable for droplet spreading
simulations
The curvelet transform for image denoising
We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement
Data fusion for a multi-scale model of a wheat leaf surface: a unifying approach using a radial basis function partition of unity method
Realistic digital models of plant leaves are crucial to fluid dynamics
simulations of droplets for optimising agrochemical spray technologies. The
presence and nature of small features (on the order of 100)
such as ridges and hairs on the surface have been shown to significantly affect
the droplet evaporation, and thus the leaf's potential uptake of active
ingredients. We show that these microstructures can be captured by implicit
radial basis function partition of unity (RBFPU) surface reconstructions from
micro-CT scan datasets. However, scanning a whole leaf () at
micron resolutions is infeasible due to both extremely large data storage
requirements and scanner time constraints. Instead, we micro-CT scan only a
small segment of a wheat leaf (). We fit a RBFPU implicit
surface to this segment, and an explicit RBFPU surface to a lower resolution
laser scan of the whole leaf. Parameterising the leaf using a locally
orthogonal coordinate system, we then replicate the now resolved microstructure
many times across a larger, coarser, representation of the leaf surface that
captures important macroscale features, such as its size, shape, and
orientation. The edge of one segment of the microstructure model is blended
into its neighbour naturally by the partition of unity method. The result is
one implicit surface reconstruction that captures the wheat leaf's features at
both the micro- and macro-scales.Comment: 23 pages, 11 figure
Compact elliptical basis functions for surface reconstruction
In this technical report I present a method to reconstruct a surface representation from a a set of EBF's, and in addition present an efficient top--down method to build an EBF representation from a point cloud representation of a surface. I also discuss the advantages and disadvantages of this approach
Reconstructing triangulated surfaces from unorganized points through local skeletal stars
Surface reconstruction from unorganized points arises in a variety of practical situations such
as range scanning an object from multiple view points, recovery of biological shapes from twodimensional
slices, and interactive surface sketching. [...]Reconstrução da superfície de pontos desorganizados surge em uma variedade de situações práticas,
tais como rastreamento de um objeto a partir de vários pontos de vista, a recuperação de
formas biológicas de fatias bi-dimensionais, e esboçar superfícies interativas. [...
Point-set manifold processing for computational mechanics: thin shells, reduced order modeling, cell motility and molecular conformations
In many applications, one would like to perform calculations on smooth manifolds of dimension d embedded in a high-dimensional space of dimension D. Often, a continuous description of such manifold is not known, and instead it is sampled by a set of scattered points in high dimensions. This poses a serious challenge. In this thesis, we approximate the point-set manifold as an overlapping set of smooth parametric descriptions, whose geometric structure is revealed by statistical learning methods, and then parametrized by meshfree methods. This approach avoids any global parameterization, and hence is applicable to manifolds of any genus and complex geometry. It combines four ingredients: (1) partitioning of the point set into subregions of trivial topology, (2) the automatic detection of the local geometric structure of the manifold by nonlinear dimensionality reduction techniques, (3) the local parameterization of the manifold using smooth meshfree (here local maximum-entropy) approximants, and (4) patching together the local representations by means of a partition of unity.
In this thesis we show the generality, flexibility, and accuracy of the method in four different problems. First, we exercise it in the context of Kirchhoff-Love thin shells, (d=2, D=3). We test our methodology against classical linear and non linear benchmarks in thin-shell analysis, and highlight its ability to handle point-set surfaces of complex topology and geometry. We then tackle problems of much higher dimensionality. We perform reduced order modeling in the context of finite deformation elastodynamics, considering a nonlinear reduced configuration space, in contrast with classical linear approaches based on Principal Component Analysis (d=2, D=10000's). We further quantitatively unveil the geometric structure of the motility strategy of a family of micro-organisms called Euglenids from experimental videos (d=1, D~30000's). Finally, in the context of enhanced sampling in molecular dynamics, we automatically construct collective variables for the molecular conformational dynamics (d=1...6, D~30,1000's)
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