Nuclear magnetic resonance (NMR) provides a powerful tool to describe local nuclear environments. In this work, unique structural information on kaolinite and on kaolinite dimethylsulfoxide (DMSO) intercalate were provided by solid-state 1H and 27Al magic-angle spinning (MAS) NMR. The interlayer chemistry of kaolinite (K) was examined by intercalating a select group of highly polar organic molecules or salts into kaolinite as a first step. Once the interlayer space is expanded, the intercalated compounds can be replaced in a second step. Intercalating DMSO into kaolinite to form the DMSO-K intercalate is, thus, a particularly useful first step toward the intercalation of a large variety of molecules, including polymers and ionic liquids. Well developed characterization methods are essential to define the structural modifications of kaolinite, and MAS NMR is a useful complement to other techniques. The use of 1H and 27Al MAS NMR for this purpose has been relatively rare. 1H NMR, nevertheless, can give unique information about kaolinite hydroxyls. Because quadrupolar interactions are sensitive to the local octahedral Al(III) geometry, solid-state 27Al NMR can follow subtle structural modifications in the octahedral sheet. In the present work, the 1H MAS NMR chemical shifts of KGa-1b were unambiguously attributed to the internal surface hydroxyls at 2.7 ppm and to the internal hydroxyls at 1.7 ppm. The 1H MAS NMR chemical shifts of the two methyl groups in DMSO-K are not equivalent and can be attributed to the 2.9 and 4.2 ppm peaks. The 27Al MAS NMR spectra of KGa-1b obtained under different magnetic fields revealed that most of the quadrupolar effects were highly reduced at 21.1 T, whereas the spectra at lower field, 4.7 T, were dominated by quadrupolar effects. The two octahedral Al(III) sites are not equivalent and can be distinguished in the low-field spectral simulation. Increased quadrupolar constants were observed and showed the major perturbations of the local Al symmetry that resulted from DMSO intercalation. Both the 1H and 27Al MAS NMR studies at different magnetic fields afforded important information about the local environments of the kaolinite hydroxyl groups and structural Al(III).

Three-dimensional fluorescence microscopy is a key technology for inspecting biological samples, ranging from single cells to entire organisms. We recently proposed a novel approach called spatially modulated Selective Volume Illumination Microscopy (smSVIM) to suppress illumination artifacts and to reduce the required number of measurements using an LED source. Here, we discuss a new strategy based on smSVIM for imaging large transparent specimens or voluminous chemically cleared tissues. The strategy permits steady mounting of the sample, achieving uniform resolution over a large field of view thanks to the synchronized motion of the illumination lens and the camera rolling shutter. Aided by a tailored deconvolution method for image reconstruction, we demonstrate significant improvement of the resolution at different magnification using samples of varying sizes and spatial features.

Today, researchers have a number of powerful image processing methods at hand that have the potential to make comparably simple and inexpensive widefield microscope systems more powerful and versatile. These techniques can enable new insights into biological samples and allow new discoveries, so long as their limitations and pitfalls are known.

A deconvolutional method for preprocessing powder diffraction data has been improved. The cumulants of instrumental aberration functions of Bragg-Brentano (Parrish) diffractometer calculated up to the fourth order are presented. The treatments of axial-divergence aberration and the effective spectroscopic profile of the source X-ray have been simplified from those used in previous methods. The current method has been applied to powder diffraction data collected with a Cu-target X-ray tube, used over 20 years, and a Ni-foil Kβ filter.

Point spread function (PSF) deconvolution is an attractive software-based technique for resolution improvement in the scanning electron microscope (SEM) because it can restore information which has been blurred by challenging operating conditions. In Part 1, we studied a modern PSF determination method for SEM and explored how various parameters affected the method's ability to accurately estimate the PSF. In Part 2, we extend this exploration to PSF deconvolution for image restoration. The parameters include reference particle size, PSF smoothing (K), background correction, and restoration denoising (λ). Image quality was assessed by visual inspection and Fourier analysis. Overall, PSF deconvolution improved image quality. Low λ enhanced image sharpness at the cost of noise, while high λ created smoother restorations with less detail. λ should be chosen to balance feature preservation and denoising based on the application. Reference particle size within ±0.9 nm and K within a reasonable range had little effect on restoration quality. Restorations using background-corrected PSFs had superior quality compared with using no background correction, but if the correction was too high, the PSF was cut off causing blurrier restorations. Future efforts to automatically determine parameters would remove user guesswork, improve this method's consistency, and maximize interpretability of outputs.

Atomic force microscopy (AFM) is typically used for analysis of relatively flat surfaces with topographic features smaller than the height of the AFM tip. On flat surfaces, it is relatively easy to find the object of interest and deconvolute imaging artifacts resulting from the finite size of the AFM tip. In contrast, AFM imaging of three-dimensional objects much larger than the AFM tip height is rarely attempted although it could provide topographic information that is not readily available from two-dimensional imaging, such as scanning electron microscopy. In this paper, we report AFM measurements of a vertically-mounted razor blade, which is taller and sharper than the AFM tip. In this case, the AFM height data, except for the data collected around the cutting edge of the blade, reflect the shape of the AFM tip. The height data around the apex area are effectively the convolution of the AFM tip and the blade cutting edge. Based on computer simulations mimicking an AFM tip scanning across a round sample, a simple algorithm is proposed to deconvolute the AFM height data of an object taller and sharper than the AFM tip and estimate its effective curvature.

Light sheet fluorescence microscopy (LSFM) allows for high-resolution three-dimensional imaging with minimal photo-damage. By viewing the sample from different directions, different regions of large specimens can be imaged optimally. Moreover, owing to their good spatial resolution and high signal-to-noise ratio, LSFM data are well suited for image deconvolution. Here we present the Huygens Fusion and Deconvolution Wizard, a unique integrated solution for restoring LSFM images, and show that improvements in signal and resolution of 1.5 times and higher are feasible.

A method to remove small CuKβ peaks and step structures caused by NiK-edge absorption as well as CuKα2 sub-peaks from powder diffraction intensity data measured with Cu-target X-ray source and Ni-foil filter is proposed. The method is based on deconvolution–convolution treatment applying scale transform of abscissa, Fourier transform, and a realistic spectroscopic model for the source X-ray. The validity of the method has been tested by analysis of the powder diffraction data of a standard LaB6 powder (NIST SRM660a) sample, collected with the combination of CuKα X-ray source, Ni-foil Kβ filter, flat powder specimen and one-dimensional Si strip detector. The diffraction intensity data treated with the method have certainly shown background intensity profile without CuKβ peaks and NiK-edge step structures.

Four series of small parasite peaks observed in powder diffraction data recorded with a Cu-target X-ray tube and a Ni filter on the diffracted beam side in Bragg–Brentano geometry are investigated. One series of the parasite peaks is assigned to the tungsten Lα-emission. Other three types of the parasite peak series are likely to be caused by the K-emissions of Ni, but the peak locations are deviated from those predicted by the Bragg's law. An empirical formula to locate the parasite peaks and a method to remove them from observed powder diffraction data are proposed. The method is based on the whole-pattern deconvolution–convolution treatment on the transformed scale of abscissa. The parameters optimized for the diffraction data measured for Si powder has been applied on treatment of the data of LaB6 powder recorded under the same experimental conditions. It has been confirmed that the parasite peaks in the observed data can effectively be removed by the deconvolution treatment with parameters determined by a reference measurement.

An improved method to correct observed shift and asymmetric deformation of diffraction peak profile caused by the axial-divergence aberration in Bragg–Brentano geometry is proposed. The method is based on deconvolution–convolution treatment applying scale transform of abscissa, Fourier transform, and cumulant analysis of an analytical model for the axial-divergence aberration. The method has been applied to the powder diffraction data of a standard LaB6 powder (NIST SRM660a) sample, collected with a one-dimensional Si strip detector. The locations, widths and shape of the peaks in the deconvolved–convolved powder diffraction data have been analyzed. The finally obtained whole powder diffraction pattern ranging from 10° to 145° in diffraction angle has been simulated by the Pawley method applying a symmetric Pearson VII peak profile model to each peak with ten background, two peak-shift, three line-width, and two peak-shape parameters, and the Rp value of the best fit has been estimated at 4.4%.

Light microscopy is a powerful tool that allows for many types of samples to be examined in a rapid, easy, and nondestructive manner. Subsequent image analysis, however, is compromised by distortion of signal by instrument optics. Deconvolution of images prior to analysis allows for the recovery of lost information by procedures that utilize either a theoretically or experimentally calculated point spread function (PSF). Using a laser scanning confocal microscope (LSCM), we have imaged whole impact tracks of comet particles captured in silica aerogel, a low density, porous SiO2 solid, by the NASA Stardust mission. In order to understand the dynamical interactions between the particles and the aerogel, precise grain location and track volume measurement are required. We report a method for measuring an experimental PSF suitable for three-dimensional deconvolution of imaged particles in aerogel. Using fluorescent beads manufactured into Stardust flight-grade aerogel, we have applied a deconvolution technique standard in the biological sciences to confocal images of whole Stardust tracks. The incorporation of an experimentally measured PSF allows for better quantitative measurements of the size and location of single grains in aerogel and more accurate measurements of track morphology.

Risk aggregation is a popular method used to estimate the sum of a collection of financial assets or events, where each asset or event is modelled as a random variable. Applications include insurance, operational risk, stress testing and sensitivity analysis. In practice, the sum of a set of random variables involves the use of two well-known mathematical operations: n-fold convolution (for a fixed number n) and N-fold convolution, defined as the compound sum of a frequency distribution N and a severity distribution, where the number of constant n-fold convolutions is determined by N, where the severity and frequency variables are independent, and continuous, currently numerical solutions such as, Panjer’s recursion, fast Fourier transforms and Monte Carlo simulation produce acceptable results. However, they have not been designed to cope with new modelling challenges that require hybrid models containing discrete explanatory (regime switching) variables or where discrete and continuous variables are inter-dependent and may influence the severity and frequency in complex, non-linear, ways. This paper describes a Bayesian Factorisation and Elimination (BFE) algorithm that performs convolution on the hybrid models required to aggregate risk in the presence of causal dependencies. This algorithm exploits a number of advances from the field of Bayesian Networks, covering methods to approximate statistical and conditionally deterministic functions to factorise multivariate distributions for efficient computation. Experiments show that BFE is as accurate on conventional problems as competing methods. For more difficult hybrid problems BFE can provide a more general solution that the others cannot offer. In addition, the BFE approach can be easily extended to perform deconvolution for the purposes of stress testing and sensitivity analysis in a way that competing methods do not.

Consider an autoregressive model with measurement error: we observe Zi =Xi +εi, where theunobserved Xi is a stationarysolution of the autoregressive equation Xi =gθ0(Xi− 1) + ξi. Theregression function gθ0 isknown up to a finite dimensional parameter θ0 to be estimated. The distributions ofξ1 and X0 are unknownand gθ belongs to a largeclass of parametric regression functions. The distribution of ε0 is completelyknown. We propose an estimation procedure with a new criterion computed as the Fouriertransform of a weighted least square contrast. This procedure provides an asymptoticallynormal estimator \hbox{$\hat \theta$}θ̂ of θ0, for a large class of regressionfunctions and various noise distributions.

A method is presented for determining the point spread function (PSF) of an electron beam in a scanning electron microscope for the examination of near planar samples. Once measured, PSFs can be used with two or more low-resolution images of a selected area to create a high-resolution reconstructed image of that area. As an example, a 4× improvement in resolution for images is demonstrated for a fine gold particle sample. Since thermionic source instruments have high beam currents associated with large probe sizes, use of this approach implies that high-resolution images can be produced rapidly if the probe diameter is less of a limiting factor. Additionally, very accurate determination of the PSFs can lead to a better understanding of instrument performance as exemplified by very accurate measurement of the beam shape and therefore the degree of astigmatism.

With the advent of in vivo laser scanning fluorescence microscopy techniques, time-series and three-dimensional volumes of living tissue and vessels at micron scales can be acquired to firmly analyze vessel architecture and blood flow. Analysis of a large number of image stacks to extract architecture and track blood flow manually is cumbersome and prone to observer bias. Thus, an automated framework to accomplish these analytical tasks is imperative. The first initiative toward such a framework is to compensate for motion artifacts manifest in these microscopy images. Motion artifacts in in vivo microscopy images are caused by respiratory motion, heart beats, and other motions from the specimen. Consequently, the amount of motion present in these images can be large and hinders further analysis of these images. In this article, an algorithmic framework for the correction of time-series images is presented. The automated algorithm is comprised of a rigid and a nonrigid registration step based on shape contexts. The framework performs considerably well on time-series image sequences of the islets of Langerhans and provides for the pivotal step of motion correction in the further automatic analysis of microscopy images.

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

Scanning confocal electron microscopy (SCEM) is a new imaging technique that is capable of depth sectioning with nanometer-scale depth resolution. However, the depth resolution in the optical axis direction (Z) is worse than might be expected on the basis of the vertical electron probe size calculated with the existence of spherical aberration. To investigate the origin of the degradation, the effects of electron energy loss and chromatic aberration on the depth resolution of annular dark-field SCEM were studied through both experiments and computational simulations. The simulation results obtained by taking these two factors into consideration coincided well with those obtained by experiments, which proved that electron energy loss and chromatic aberration cause blurs at the overfocus sides of the Z-direction intensity profiles rather than degrade the depth resolution much. In addition, a deconvolution method using a simulated point spread function, which combined two Gaussian functions, was adopted to process the XZ-slice images obtained both from experiments and simulations. As a result, the blurs induced by energy loss and chromatic aberration were successfully removed, and there was also about 30% improvement in the depth resolution in deconvoluting the experimental XZ-slice image.

Aberration-corrected scanning transmission electron microscopes (STEMs) provide sub-Angstrom lateral resolution; however, the large convergence angle greatly reduces the depth of field. For microscopes with a small depth of field, information outside of the focal plane quickly becomes blurred and less defined. It may not be possible to image some samples entirely in focus. Extended depth-of-field techniques, however, allow a single image, with all areas in focus, to be extracted from a series of images focused at a range of depths. In recent years, a variety of algorithmic approaches have been employed for bright-field optical microscopy. Here, we demonstrate that some established optical microscopy methods can also be applied to extend the ∼6 nm depth of focus of a 100 kV 5th-order aberration-corrected STEM (αmax = 33 mrad) to image Pt-Co nanoparticles on a thick vulcanized carbon support. These techniques allow us to automatically obtain a single image with all the particles in focus as well as a complimentary topography map.