We present a novel problem formulation for model-free data-driven fault diagnosis, in which possible faults are diagnosed simultaneously to identifying the linear time-invariant system. This problem is practically relevant for systems whose model cannot be identified reliably prior to diagnosing possible faults, for instance when operating conditions change over time, when a fault is already present before system identification is carried out, or when the system dynamics change due to the presence of the fault. A computationally attractive solution is proposed by solving the problem using unconstrained convex optimization, where the objective function consists of three terms of which two are non-differentiable. An additional recursive implementation based on a proximal algorithm is presented in order to solve the optimization problem online. The numerical results on a buck converter show the application of the proposed solution both offline and online.

We propose to use the State Estimation by Sum-of-Norms Regularisation (STATESON-)algorithm for recovering the tip-sample interaction in high-speed tapping mode atomic force microscopy (AFM). This approach enables accurate sample height estimation for each independent cantilever oscillation period, provided that the tip-sample interaction dominates the noise. The entire course of the cantilever deflection signal is compared to a modelled counterpart in subsequent convex minimisations, such that the sparse tip-sample interaction can be recovered. Afterwards, the sample height is determined using the minimum smoothed cantilever deflection per cantilever oscillation period. Results from simulation experiments are in favour of the proposed approach as it consistently reveals sharp edges in sample height, as opposed to both the conventional and a closely related existing approach. However, the non-processed cantilever deflection provided most accurate sample height estimation. It is recommended to implement the STATESON-algorithm in the form of a filter to use it in feedback control of the scanner and cantilever excitation.

Model-based fault diagnosis for dynamical systems is a sophisticated task due to model inaccuracies, measurement noise and many possible fault scenarios. By presenting faults in terms of a dictionary, the latter obstacle is recently addressed using well-known techniques for recovering sparse information (e.g. lasso). However, current state-of-the-art methods still require accurate models and measurements for adequate diagnosis. In our contribution we address the problem of data-driven fault diagnosis in the sense that the model of the linear time-invariant (LTI) system is unknown in addition to the fault. Moreover, our aim is to diagnose (concurrent) faults while only having input/output data and the fault dictionary. This implies the user simply plugs in the data and specifies the set of possible faults in order to know the active faults together with an estimate of the dynamic model. The problem is formulated within a blind system identification context resulting in computationally efficient solutions based on convex optimization.

We have applied a combination of blind deconvolution and deep learning to the processing of Shack-Hartmann images.By using the intensity information contained in spot positions, and the fine structure of the separate images created by the lenslets,we have increased the sensitivity and resolution of the sensor over the limit defined by standard processing of spot displacements only.We also have demonstrated the applicability of the method to wavefront sensing using extended objects as a reference.

We demonstrate a novel closed-loop input design technique on the detection of particles in an imaging system such as a fluorescence microscope. The probability of misdiagnosis is minimized while constraining the input energy such that for instance phototoxicity is reduced. The key novelty of the closed-loop design is that each next input is designed based on the most recent information. Using updated hypothesis probabilities, the input energy distribution is optimized for detection such that unresolved pixels have increased illumination next image acquisition. As compared to conventional open-loop, the results show that (regions of) particles are diagnosed using less energy in the closed-loop approach. Besides the closed-loop approach being viable for particle detection in fluorescence microscopy measurements, it can be developed further to apply in different areas such as sequential object segmentation for reliable and efficient product inspection in Industry 4.0.

Some applications require high level of image-based classification certainty while keeping the total illumination energy as low as possible. Examples are minimally invasive visual inspection in Industry 4.0, and medical imaging systems such as computed tomography, in which the radiation dose should be kept “as low as is reasonably achievable”. We introduce a sequential object recognition scheme aimed at minimizing phototoxicity or bleaching while achieving a predefined level of decision accuracy. The novel online procedure relies on approximate weighted Bhattacharyya coefficients for determination of future inputs. Simulation results on the MNIST handwritten digit database show how the total illumination energy is decreased with respect to a detection scheme using constant illumination.

High-NA light sheet illumination can improve the resolution of single-molecule localization microscopy (SMLM) by reducing the background fluorescence. These approaches currently require custom-made sample holders or additional specialized objectives, which makes the sample mounting or the optical system complex and therefore reduces the usability of these approaches. Here, we developed a single-objective lens-inclined light sheet microscope (SOLEIL) that is capable of 2D and 3D SMLM in thick samples. SOLEIL combines oblique illumination with point spread function PSF engineering to enable dSTORM imaging in a wide variety of samples. SOLEIL is compatible with standard sample holders and off-the-shelve optics and standard high NA objectives. To accomplish optimal optical sectioning we show that there is an ideal oblique angle and sheet thickness. Furthermore, to show what optical sectioning delivers for SMLM we benchmark SOLEIL against widefield and HILO microscopy with several biological samples. SOLEIL delivers in 15 μm thick Caco2-BBE cells a 374% higher intensity to background ratio and a 54% improvement in the estimated CRLB compared to widefield illumination, and a 184% higher intensity to background ratio and a 20% improvement in the estimated CRLB compared to HILO illumination.

We investigate the general adjustment of projection-based phase retrieval algorithms for use with saturated data. In the phase retrieval problem, model fidelity of experimental data containing a non-zero background level, fixed pattern noise, or overexposure, often presents a serious obstacle for standard algorithms. Recently, it was shown that overexposure can help to increase the signal-to-noise ratio in AI applications. We present our first results in exploring this direction in the phase retrieval problem, using as an example the Gerchberg-Saxton algorithm with simulated data. The proposed method can find application in microscopy, characterisation of precise optical instruments, and machine vision applications of Industry4.0.

This paper presents a computationally efficient framework in which a single focal-plane image is used to obtain a high-resolution reconstruction of dynamic aberrations. Assuming small-phase aberrations, a non-linear Kalman filter implementation is developed whose computational complexity scales close to linearly with the number of pixels of the focal-plane camera. The performance of themethod is tested in a simulation of an adaptive optics system, where the small-phase assumption is enforced by considering a closed-loop system that uses a low-resolution wavefront sensor to control a deformable mirror. The results confirmthe computational efficiency of the algorithm and showa large robustness against noise and model uncertainties.

We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant projection operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.

We present the convergence analysis of convex combination of the alternating projection and Douglas–Rachford operators for solving the phase retrieval problem. New convergence criteria for iterations generated by the algorithm are established by applying various schemes of numerical analysis and exploring both physical and mathematical characteristics of the phase retrieval problem. Numerical results demonstrate the advantages of the algorithm over the other widely known projection methods in practically relevant simulations.

This paper discusses various practical problems arising in the design and simulation of predictive control methods for adaptive optics. Although there has been increased attention towards optimal prediction and control methods for AO systems, they are often tested in simplified simulation environments. The use of advanced AO simulators however, is a valuable alternative to the use of real data or laboratory experiments, as they provide both a flexible environment which is ideal for testing a new algorithm and are more accessible to non-experts. Topics that are often not explicitly discussed, such as the identification of a turbulence dynamics model from data, the use of matrix structures in AO systems to decrease the computational complexity and the implementation of Kalman filters to optimally deal with realistic noise conditions are examined. All topics discussed are illustrated by an accompanying Matlab code, which is based on the existing Matlab AO toolbox OOMAO.

In this Letter, we report on an algorithm and its implementation to reconstruct the wavefront as a continuous function from a bitmap image of the Hartmann–Shack pattern. The approach works with arbitrary raster geometry and does not require explicit spot definition and phase unwrapping. The system matrix, defining the coefficients of wavefront decomposition in the system of basis functions, is obtained as a result of a series of convolutions and thresholding operations on the reference and sample images.

Inhomogeneities in the refractive index of a biological microscopy sample can introduce phase aberrations, severely impairing the quality of images. Adaptive optics can be employed to correct for phase aberrations and improve image quality. However, conventional adaptive optics can only correct a single phase aberration for the whole field of view (isoplanatic correction) while, due to the highly heterogeneous nature of biological tissues, the sample induced aberrations in microscopy often vary throughout the field of view (anisoplanatic aberration), limiting significantly the effectiveness of adaptive optics. This paper reports on a new approach for aberration correction in laser scanning confocal microscopy, in which a spatial light modulator is used to generate multiple excitation points in the sample to simultaneously scan different portions of the field of view with completely independent correction, achieving anisoplanatic compensation of sample induced aberrations, in a significantly shorter time compared to sequential isoplanatic correction of multiple image subregions. The method was tested in whole Drosophila brains and in larval Zebrafish, each showing a dramatic improvement in resolution and sharpness when compared to conventional isoplanatic adaptive optics.

We present an efficient phase retrieval approach for imaging systems with high numerical aperture based on the vectorial model of the point spread function. The algorithm is in the class of alternating minimization methods and can be adjusted for applications with either known or unknown amplitude of the field in the pupil. The algorithm outperforms existing solutions for high-numerical-aperture phase retrieval: (1) the generalization of the method of Hanser et al., based on extension of the scalar diffraction theory by representing the out-of-focus diversity applied to the image by a spherical cap, and (2) the method of Braat et al., which assumes through the use of extended Nijboer–Zernike expansion the phase to be smooth. The former is limited in terms of accuracy due to model deviations, while the latter is of high computational complexity and excludes phase retrieval problems where the phase is discontinuous or sparse. Extensive numerical results demonstrate the efficiency, robustness, and practicability of the proposed algorithm in various practically relevant simulations.

We consider the extension of the traditional projection-based phase retrieval algorithms by increasing the problem dimensionality and introducing novel projection operators. The approach is demonstrated on an example of phase retrieval for the high-NA case.

In theory of optical aberrations, an aberrated wavefront is represented by its coefficients in some orthogonal basis, for instance by Zernike polynomials. However, many wavefront measurement techniques implicitly approximate the gradient of the wavefront by the gradients of the basis functions. For a finite number of approximation terms, the transition from a basis to its gradient might introduce an aliasing error. To simplify the measurements, another set of functions, an "optimal basis"with orthogonal gradients, is often introduced, for instance Lukosz-Braat polynomials. This paper first shows that such bases do not necessarily eliminate the aliasing error and secondly considers the problem of finding an alias-free basis on example of second-moment-based indirect wavefront sensing methods. It demonstrates that for these methods any alias-free basis should be formed by functions simultaneously orthogonal in two dot-products and be composed of the eigenfunctions of the Laplace operator. The fitness of such alias-free basis for optical applications is analyzed by means of numerical simulations on typical aberrations occurring in microscopy and astronomy.