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 bandwidthit 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.

Genome sequencing of cancers often reveals mosaics of different subclones present in the same tumour1–3. Although these are believed to arise according to the principles of somatic evolution, the exact spatial growth patterns and underlying mechanisms remain elusive4,5. Here, to address this need, we developed a workflow that generates detailed quantitative maps of genetic subclone composition across whole-tumour sections. These provide the basis for studying clonal growth patterns, and the histological characteristics, microanatomy and microenvironmental composition of each clone. The approach rests on whole-genome sequencing, followed by highly multiplexed base-specific in situ sequencing, single-cell resolved transcriptomics and dedicated algorithms to link these layers. Applying the base-specific in situ sequencing workflow to eight tissue sections from two multifocal primary breast cancers revealed intricate subclonal growth patterns that were validated by microdissection. In a case of ductal carcinoma in situ, polyclonal neoplastic expansions occurred at the macroscopic scale but segregated within microanatomical structures. Across the stages of ductal carcinoma in situ, invasive cancer and lymph node metastasis, subclone territories are shown to exhibit distinct transcriptional and histological features and cellular microenvironments. These results provide examples of the benefits afforded by spatial genomics for deciphering the mechanisms underlying cancer evolution and microenvironmental ecology.

The ability to form images through hair-thin optical fibres promises to open up new applications from biomedical imaging to industrial inspection. Unfortunately, their deployment has been limited because small changes in mechanical deformation (e.g. bending) and temperature can completely scramble optical information, which distorts the resulting images. Since such changes are dynamic, correcting them requires measurement of the fibre transmission matrix in situ immediately before imaging. Transmission matrix calibration typically requires access to both the proximal and distal facets of the fibre simultaneously, which is not feasible during most realistic usage scenarios without compromising the thin form factor with bulky distal optics. Here, we introduce a new approach to determine the transmission matrix of multi-mode or multi-core optical fibre in a reflection-mode configuration without requiring access to the distal facet. A thin stack of structured metasurface reflectors is used at the distal facet of the fibre to introduce wavelength-dependent, spatially heterogeneous reflectance profiles. We derive a first-order fibre model that compensates these wavelength-dependent changes in the fibre transmission matrix and show that, consequently, the reflected data at 3 wavelengths can be used to unambiguously reconstruct the full transmission matrix by an iterative optimisation algorithm. We then present a method for sample illumination and imaging following reconstruction of the transmission matrix. Unlike previous approaches, our method does not require the fibre matrix to be unitary making it applicable to physically realistic fibre systems that have non-negligible power loss. We demonstrate the transmission matrix reconstruction and imaging method first using simulated non-unitary fibres and noisy reflection matrices, then using much larger experimentally-measured transmission matrices of a densely-packed multicore fibre. Finally, we demonstrate the method on an experimentally-measured multi-wavelength set of transmission matrices recorded from a step-index multimode fibre. Our findings pave the way for online transmission matrix calibration in situ in hair-thin optical fibres.

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 Gröchenig 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 this 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.