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Geospatial co-registration is a mandatory prerequisite when dealing with remote sensing data. Inter- or intra-sensoral misregistration will negatively affect any subsequent image analysis, specifically when processing multi-sensoral or multi-temporal data. In recent decades, many algorithms have been developed to enable manual, semi- or fully automatic displacement correction. Especially in the context of big data processing and the development of automated processing chains that aim to be applicable to different remote sensing systems, there is a strong need for efficient, accurate and generally usable co-registration. Here, we present AROSICS (Automated and Robust Open-Source Image Co-Registration Software), a Python-based open-source software including an easy-to-use user interface for automatic detection and correction of sub-pixel misalignments between various remote sensing datasets. It is independent of spatial or spectral characteristics and robust against high degrees of cloud coverage and spectral and temporal land cover dynamics. The co-registration is based on phase correlation for sub-pixel shift estimation in the frequency domain utilizing the Fourier shift theorem in a moving-window manner. A dense grid of spatial shift vectors can be created and automatically filtered by combining various validation and quality estimation metrics. Additionally, the software supports the masking of, e.g., clouds and cloud shadows to exclude such areas from spatial shift detection. The software has been tested on more than 9000 satellite images acquired by different sensors. The results are evaluated exemplarily for two inter-sensoral and two intra-sensoral use cases and show registration results in the sub-pixel range with root mean square error fits around 0.3 pixels and better.

The concept of coding metasurface makes a link between physically metamaterial particles and digital codes, and hence it is possible to perform digital signal processing on the coding metasurface to realize unusual physical phenomena. Here, this study presents to perform Fourier operations on coding metasurfaces and proposes a principle called as scattering‐pattern shift using the convolution theorem, which allows steering of the scattering pattern to an arbitrarily predesigned direction. Owing to the constant reflection amplitude of coding particles, the required coding pattern can be simply achieved by the modulus of two coding matrices. This study demonstrates that the scattering patterns that are directly calculated from the coding pattern using the Fourier transform have excellent agreements to the numerical simulations based on realistic coding structures, providing an efficient method in optimizing coding patterns to achieve predesigned scattering beams. The most important advantage of this approach over the previous schemes in producing anomalous single‐beam scattering is its flexible and continuous controls to arbitrary directions. This work opens a new route to study metamaterial from a fully digital perspective, predicting the possibility of combining conventional theorems in digital signal processing with the coding metasurface to realize more powerful manipulations of electromagnetic waves. Convolutions are operated on 2‐bit coding metasurfaces to reach the steering of scattering pattern to an arbitrarily predesigned direction. The radiation angle can be continuously designed in the entire upper‐half space by simply combining two or multiple gradient coding sequences from a 2‐bit coding metasurface which has only four different coding digits.

The main objective of this work is to investigate among other things the validity of the classical Titchmarsh’s theorem ( 1937 , Theorem 84) and Younis’ theorem (Internat. J. Math. Math. Sci. 9 (2), 301–312, 1986 , Theorem 3.3) on the unit sphere Σ m - 1 , m ≥ 3 . More precisely, we prove those theorems on the image under the discrete Fourier-Laplace transform of a set of functions satisfying a generalized Lipschitz and Dini-Lipschitz condition in the space L p ( Σ m - 1 ) , 1 ≤ p ≤ 2 . For this purpose, we use a generalized spherical shift defined by Rudin (Trans. Amer. Math. Soc. 68 , 287–303, 1950 ).

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples.

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.

Let and let denote the indicator function of the segment . We obtain new two-radii theorems for the Bessel convolution operator that are related to quasi-analytic classes of functions. We establish a local analog of the two-radii theorem for functions that satisfy the convolution inequalities and . We also present applications of these results to uniqueness theorems for solutions of the Cauchy problem for the generalized Euler–Poisson–Darboux equation and to closure theorems for generalized shifts.

We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space MAr,s in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with A-translation, then it is a A-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.

Gabor systems are used in fields ranging from audio processing to digital communication. Such a Gabor system ( g , Λ ) consists of all time-frequency shifts π ( λ ) g of a window function g ∈ L 2 ( R ) along a lattice Λ ⊂ R 2 . We focus on Gabor systems that are also Riesz sequences, meaning that one can stably reconstruct the coefficients c = ( c λ ) λ ∈ Λ from the function ∑ λ ∈ Λ c λ π ( λ ) g . In digital communication, a function of this form is used to transmit the digital sequence c . It is desirable for g to be well localized in time and frequency, since the transmitted signal will then be almost compactly supported in time and frequency if the sequence c has finite support. In this paper, we study what additional structural properties the signal space G ( g , Λ ) , i.e., the span of the Gabor system, satisfies in addition to being a closed subspace of L 2 ( R ) . The most well-known result in this direction—the Balian–Low theorem—states that if g is well localized in time and frequency and if ( g , Λ ) is a Riesz sequence, then G ( g , Λ ) is necessarily a proper subspace of L 2 ( R ) . We prove a generalization of this result related to the invariance of G ( g , Λ ) under time-frequency shifts. Precisely, we show that if ( g , Λ ) is a Riesz sequence with g being well localized in time and frequency (precisely, g should belong to the so-called Feichtinger algebra), then π ( μ ) G ( g , Λ ) ⊂ G ( g , Λ ) holds if and only if μ ∈ Λ . For lattices of rational density , this was already known, with the proof based on Zak transform techniques. These methods do not generalize to arbitrary lattices, however. Instead, our proof for lattices of irrational density relies on combining methods from time-frequency analysis with properties of a special C ∗ -algebra, the so-called irrational rotation algebra.

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

We study the behavior at infinity of the solutions of the convolution equations related to the Bessel generalized shift operator. We consider the case where one of the convolutors of the equation is either the characteristic function of the interval or the Dirac measure with support at a given point. Recent results of the authors of the present paper are applied for finding sharp characteristics of the decay rate of nonzero solutions of these equations in terms of the behavior of their integral means. As a corollary, we establish analogs of the well-known theorems by John, Smith, Volchkov, and Thangavelu on injectivity of the spherical mean-value operator on . In addition, in some cases, we strengthen Selmi and Nessibi’s theorem on spectral analysis on the Bessel–Kingman hypergroup; we also prove a new uniqueness theorem for the generalized Euler–Poisson–Darboux equation.