45 research outputs found
On stable reconstructions from nonuniform Fourier measurements
We consider the problem of recovering a compactly-supported function from a
finite collection of pointwise samples of its Fourier transform taking
nonuniformly. First, we show that under suitable conditions on the sampling
frequencies - specifically, their density and bandwidth - it is possible to
recover any such function in a stable and accurate manner in any given
finite-dimensional subspace; in particular, one which is well suited for
approximating . In practice, this is carried out using so-called nonuniform
generalized sampling (NUGS). Second, we consider approximation spaces in one
dimension consisting of compactly supported wavelets. We prove that a linear
scaling of the dimension of the space with the sampling bandwidth is both
necessary and sufficient for stable and accurate recovery. Thus wavelets are up
to constant factors optimal spaces for reconstruction
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
In this paper, we consider the problem of recovering a compactly supported
multivariate function from a collection of pointwise samples of its Fourier
transform taken nonuniformly. We do this by using the concept of weighted
Fourier frames. A seminal result of Beurling shows that sample points give rise
to a classical Fourier frame provided they are relatively separated and of
sufficient density. However, this result does not allow for arbitrary
clustering of sample points, as is often the case in practice. Whilst keeping
the density condition sharp and dimension independent, our first result removes
the separation condition and shows that density alone suffices. However, this
result does not lead to estimates for the frame bounds. A known result of
Groechenig provides explicit estimates, but only subject to a density condition
that deteriorates linearly with dimension. In our second result we improve
these bounds by reducing the dimension dependence. In particular, we provide
explicit frame bounds which are dimensionless for functions having compact
support contained in a sphere. Next, we demonstrate how our two main results
give new insight into a reconstruction algorithm---based on the existing
generalized sampling framework---that allows for stable and quasi-optimal
reconstruction in any particular basis from a finite collection of samples.
Finally, we construct sufficiently dense sampling schemes that are often used
in practice---jittered, radial and spiral sampling schemes---and provide
several examples illustrating the effectiveness of our approach when tested on
these schemes
Sparse principal component analysis via axis-aligned random projections
We introduce a new method for sparse principal component analysis, based on
the aggregation of eigenvector information from carefully-selected axis-aligned
random projections of the sample covariance matrix. Unlike most alternative
approaches, our algorithm is non-iterative, so is not vulnerable to a bad
choice of initialisation. We provide theoretical guarantees under which our
principal subspace estimator can attain the minimax optimal rate of convergence
in polynomial time. In addition, our theory provides a more refined
understanding of the statistical and computational trade-off in the problem of
sparse principal component estimation, revealing a subtle interplay between the
effective sample size and the number of random projections that are required to
achieve the minimax optimal rate. Numerical studies provide further insight
into the procedure and confirm its highly competitive finite-sample
performance.The research of the first and third authors was supported by an Engineering and Physical Sciences Research Council (EPSRC) grant EP/N014588/1 for the centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging. The second and third authors were supported by EPSRC Fellowship EP/J017213/1 and EP/P031447/1, and grant RG81761 from the Leverhulme Trust
Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates
We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example.Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries.The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum.As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data
Sharp-SSL: Selective high-dimensional axis-aligned random projections for semi-supervised learning
We propose a new method for high-dimensional semi-supervised learning
problems based on the careful aggregation of the results of a low-dimensional
procedure applied to many axis-aligned random projections of the data. Our
primary goal is to identify important variables for distinguishing between the
classes; existing low-dimensional methods can then be applied for final class
assignment. Motivated by a generalized Rayleigh quotient, we score projections
according to the traces of the estimated whitened between-class covariance
matrices on the projected data. This enables us to assign an importance weight
to each variable for a given projection, and to select our signal variables by
aggregating these weights over high-scoring projections. Our theory shows that
the resulting Sharp-SSL algorithm is able to recover the signal coordinates
with high probability when we aggregate over sufficiently many random
projections and when the base procedure estimates the whitened between-class
covariance matrix sufficiently well. The Gaussian EM algorithm is a natural
choice as a base procedure, and we provide a new analysis of its performance in
semi-supervised settings that controls the parameter estimation error in terms
of the proportion of labeled data in the sample. Numerical results on both
simulated data and a real colon tumor dataset support the excellent empirical
performance of the method.Comment: 49 pages, 4 figure
Reconstruction of optical vector-fields with applications in endoscopic imaging
We introduce a framework for the reconstruction of the amplitude, phase and polarisation of an optical vector-field using measurements acquired by an imaging device characterised by an integral transform with an unknown spatially-variant kernel. By incorporating effective regularisation terms, this new approach is able to recover an optical vector-field with respect to an arbitrary representation system, which may be different from the one used for device calibration. In particular, it enables the recovery of an optical vector-field with respect to a Fourier basis, which is shown to yield indicative features of increased scattering associated with tissue abnormalities. We demonstrate the effectiveness of our approach using synthetic holographic images as well as biological tissue samples in an experimental setting where measurements of an optical vector-field are acquired by a multicore fibre (MCF) endoscope, and observe that indeed the recovered Fourier coefficients are useful in distinguishing healthy tissues from tumours in early stages of oesophageal cancer.M. Gataric and S. E. Bohndiek were supported by an EPSRC grant EP/N014588/1 for the centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging. G. S. D. Gordon and S. E. Bohndiek received funding from CRUK (C47594/A16267, C14303/A17197, C47594/A21102) and a pump-priming award from the Cancer Research UK Cambridge Centre Early Detection Programme (A20976). The work of F. Renna was funded in part by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 655282 and in part by the FCT grant SFRH/BPD/118714/2016