International Journal of Wavelets, Multiresolution and Information Processing

Since the introduction of quaternion by Hamilton in 1843, quaternions have been used in a lot of applications. One of the most interesting qualities is that we can use quaternions to carry out rotations and operate on other quaternions; this characteristic of the quaternions inspired us to investigate how the quantum states and quantum operator work in the field of quaternions and how we can use it to construct a quantum neural network. This new type of quantum neural network (QNN) is developed in the quaternion algebra framework that is isomorphic to the rotor algebra $G_3^{+}$ of the geometric algebra and is based on the so-called qubit neuron model. The quaternion quantum neural network (QQNN) is tested and shows robust performance.

This paper summarizes our results and findings made throughout the past years on the way towards a one millimeter thin, flexible, and scalable foil camera. The sensor part of the camera consists of multiple thin-film luminescent concentrator (LC) layers, each of which is sensitive to a different band of the light spectrum. A special optical micro-structure cut into the edges of the LC layers multiplexes the transported light signal into a variant of the Radon transform of the image focused on the LC surface. For increasing the camera’s depth of field beyond the sensor surface, various thin-film imaging layers, such as optical Söller collimators realized by means of X-ray lithography on a PMMA wafer, have been investigated. The flexibility and scalability of our thin-film camera has the potential to lead to new human–computer interfaces that are unconstrained in shape and sensing-distance. Applications such as contact-less sensing, smart skin sensors for autonomous robots and industrial machines, and environment sensing for vehicles are imaginable.

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed L^p-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This different perspective on these spaces enables the application of embedding results that allow the complete characterization of embeddings for certain integrability exponents, and thus provides access to a deeper understanding of the smoothness properties of coorbit spaces, and of the influence of the choice of shearlet groups on these properties. We give a detailed analysis, identifying which features of the dilation groups have an influence on the embedding behavior, and which do not. Our results also allow to comment on the validity of the interpretation of shearlet coorbit spaces as smoothness spaces.

In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form $G=\frac{1}{2}{\sum }_1^m{(E{y}_i-C_i)}^2$, which often appear in the calibration of stochastic models. We demonstrate that it allows a perfect SIMD parallelization and provides its relative computational cost. In addition, we demonstrate that this theoretical result is in concordance with numerical experiments.

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly=0$ where $L$ is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in practical applications. We develop an iterative projected-gradient-type algorithm that recovers the signal subject to ensuring consistency with the measurement, meaning that the recovered signal, when encoded, must yield the same set of measurements that one started with. The algorithm involves rank-1 projection, which stems from the idea of lifting, originally proposed in the context of PhaseLift. The consistency criterion is enforced using a one-sided quadratic cost. We also determine the probability with which different vectors lead to the same set of quantized measurements, which makes it impossible to resolve them. Naturally, this probability depends on how correlated such vectors are, and how coarsely/finely the measurements are quantized. The proposed algorithm is also capable of incorporating a sparsity constraint on the signal. An analysis of the cost function reveals that it is bounded probabilistically, both above and below, by functions that are dependent on how well-correlated the estimate is with the ground-truth. We also derive the Cramér-Rao lower bound (CRB) on the achievable reconstruction accuracy. A comparison with the state-of-the-art algorithms shows that the proposed algorithm has a higher reconstruction accuracy and is about 2 to 3 dB away from the CRB. The edge, in terms of the reconstruction signal-to-noise ratio, over the competing algorithms is higher (about 5 to 6 dB) when the quantization is coarse, thereby making the proposed scheme particularly attractive in such scenarios. We also demonstrate a concrete application of the proposed method to frequency-domain optical-coherence tomography (FDOCT).

In this paper, we unify the Riesz transform with the Teager–Kaiser energy operator (TKEO) and monogenic multi-resolution analysis in comparison to conventional gradient and Riesz transform-based approaches. We show that benefits of our approach maintain the following aspects of different known estimators: the stability of mixed (higher) order of derivatives, like given in TKEO, the all-pass filter characteristics featuring the Riesz transform and the scale-based analysis. Furthermore, we demonstrate the ability of the considered technique for extracting structural defects at different scales, estimating orientation or being used for 2D demodulation of amplitude- and phase modulated signals, like interferometric fringe patterns. Additionally, the abilities of orientation estimation of 3D data is demonstrated. From the viewpoint of mathematical analysis, we will emphasize the connection to monogenic higher dimensional signals, Clifford frames, and monogenic multi-resolution signal analysis. Furthermore, we will discuss higher order Riesz transform.

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.