Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group

In this paper, {the goal is to design deterministic sampling patterns on the sphere and the rotation group} and, thereby, construct sensing matrices for sparse recovery of band-limited functions. It is first shown that random sensing matrices, which consists of random samples of Wigner D-functions, satisfy the Restricted Isometry Property (RIP) with proper preconditioning and can be used for sparse recovery on the rotation group. The mutual coherence, however, is used to assess the performance of deterministic and regular sensing matrices. We show that many of widely used regular sampling patterns yield sensing matrices with the worst possible mutual coherence, and therefore are undesirable for sparse recovery. Using tools from angular momentum analysis in quantum mechanics, we provide a new expression for the mutual coherence, which encourages the use of regular elevation samples. We construct low coherence deterministic matrices by fixing the regular samples on the elevation and minimizing the mutual coherence over the azimuth-polarization choice. It is shown that once the elevation sampling is fixed, the mutual coherence has a lower bound that depends only on the elevation samples. This lower bound, however, can be achieved for spherical harmonics, which leads to new sensing matrices with better coherence than other representative regular sampling patterns. This is reflected as well in our numerical experiments where our proposed sampling patterns perfectly match the phase transition of random sampling patterns.Comment: IEEE Trans. on Signal Processin

Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids

Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics

Wigner Distribution Deconvolution Adaptation for Live Ptychography Reconstruction

We propose a modification of Wigner Distribution Deconvolution (WDD) to support live processing ptychography. Live processing allows to reconstruct and display the specimen transfer function gradually while diffraction patterns are acquired. For this purpose we reformulate WDD and apply a dimensionality reduction technique that reduces memory consumption and increases processing speed. We show numerically that this approach maintains the reconstruction quality of specimen transfer functions as well as reduces computational complexity during acquisition processes. Although we only present the reconstruction for Scanning Transmission Electron Microscopy (STEM) datasets, in general, the live processing algorithm we present in this paper can be applied to real-time ptychographic reconstruction for different fields of application

Signal recovery on the sphere from compressive and phaseless measurements

In this thesis, we investigate the possibility of reducing the number of measurements and using only their magnitudes to reconstruct spherical signals and their three-dimensional rotations. These problems appear, for instance, in spherical near-field (SNF) antenna measurements, where the number of samples to acquire a signal implies a long measurement time and occasionally unreliable phase information. We tailor the compressed sensing (CS) approach to reconstruct spherical signals from fewer measurements. Instead of considering fully random sensing matrices like in the typical CS, we restrict the randomness to follow certain structures derived from the properties of spherical harmonics and Wigner D-functions. In this setting, we provide a bound on the number of measurements required to allow stable and robust signal recovery as well as numerical evaluations to verify these results. Although we limit the randomness, applying random sampling on the sphere is still cumbersome since collecting samples requires the use of mechanical devices that move smoothly over the spherical surface, such as robots. Hence, a deterministic sampling pattern to construct sensing matrices is still desired in many applications and the notion of coherence is used to measure the sensing matrices’ quality. First, we show a class of commonly used deterministic sampling patterns on the sphere that produces the worst coherence sensing matrices and, thereby, is unsuitable for CS. Subsequently, we propose a sampling strategy on the sphere to construct low coherence deterministic sensing matrices from spherical harmonics and Wigner D-functions, and derive a coherence bound for both cases. Apart from dealing with compressive measurements, we also study signal recovery on the sphere from phase less measurements and identify some potential ambiguities under this setting. We analyse ambiguities which arise from the properties of complex and real spherical harmonics, as well as the ambiguity caused by the implementation of inappropriate sampling patterns. As a further contribution, numerical evaluations are conducted to compare several reconstruction algorithms as well as the effects of different sampling patterns on the sphere. Finally, these results are implemented using SNF data, where we numerically show that our proposed sampling pattern requires significantly less number of measurements to provide high-quality reconstructions of far-field patterns when compared to classical approaches

A Circular Harmonic Oscillator Basis

Polar coordinates are frequently used to transform 2D images appearing in 4D scanning transmission electron microscopy (4D-STEM) as the dominant feature of the ronchigram is a central spot where the undeflected electron beam hits the detector. The information of interest resides in the deviations from a circular shape of the spot. The function basis of the quantum mechanical harmonic osciallator consists of Hermite polynomials and a Gaussian envelope function for the one-dimensional problem. For the two-dimensional isotropic problem, the basis can be represented either as a Cartesian product of two 1D basis functions or in polar coordinates.A unitary transformation connects both representations. To allow fast and affordable compression of STEM images, we incorporate the Cartesian product representation as it leads to two successive matrix-matrix multiplications. This compression method is particularly suitable for single-side-band (SSB) ptychography. We present the explicit shape of the associated radial functions of a circular harmonic oscillator and compression factors in relation to computational costs for a typical SSB ptychography application