Linear reconstructions and the analysis of the stable sampling rate

The theory of sampling and the reconstruction of data has a wide range of applications and a rich collection of techniques. For many methods a core problem is the estimation of the number of samples needed in order to secure a stable and accurate reconstruction. This can often be controlled by the Stable Sampling Rate (SSR). In this paper we discuss the SSR and how it is crucial for two key linear methods in sampling theory: generalized sampling and the recently developed Parametrized Background Data Weak (PBDW) method. Both of these approaches rely on estimates of the SSR in order to be accurate. In many areas of signal and image processing binary samples are crucial and such samples, which can be modelled by Walsh functions, are the core of our analysis. As we show, the SSR is linear when considering binary sampling with Walsh functions and wavelet reconstruction. Moreover, for certain wavelets it is possible to determine the SSR exactly, allowing sharp estimates for the performance of the methods

On the stable sampling rate for binary measurements and wavelet reconstruction

This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard matrices) and wavelet reconstruction the stable sampling rate is linear. This implies that binary measurements are as efficient as Fourier samples when using wavelets as the reconstruction space. Powerful techniques for reconstructions include generalized sampling and its compressed versions, as well as recent methods based on data assimilation. Common to these methods is that the reconstruction quality depends highly on the subspace angle between the sampling and the reconstruction space, which is dictated by the stable sampling rate. As a result of the theory provided in this paper, these methods can now easily use binary measurements and wavelet reconstruction bases.This work of LT was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. ACH acknowledges support from Royal Society University Research Fellowship as well as the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1

Analysis of new direct sampling indicators for far-field measurements

This article focuses on the analysis of three direct sampling indicators which can be used for recovering scatterers from the far-field pattern of time-harmonic acoustic measurements. These methods fall under the category of sampling methods where an indicator function is constructed using the far-field operator. Motivated by some recent work, we study the standard indicator using the far-field operator and two indicators derived from the factorization method. We show the equivalence of two indicators previously studied as well as propose a new indicator based on the Tikhonov regularization applied to the far-field equation for the factorization method. Finally, we give some numerical examples to show how the reconstructions compare to other direct sampling methods

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 $f$ in a stable and accurate manner in any given finite-dimensional subspace; in particular, one which is well suited for approximating $f$. 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