Acoustic Seafloor Classification Using the Weyl Transform of Multibeam Echosounder Backscatter Mosaic

The use of multibeam echosounder systems (MBES) for detailed seafloor mapping is increasing at a fast pace. Due to their design, enabling continuous high-density measurements and the coregistration of seafloor’s depth and reflectivity, MBES has become a fundamental instrument in the advancing field of acoustic seafloor classification (ASC). With these data becoming available, recent seafloor mapping research focuses on the interpretation of the hydroacoustic data and automated predictive modeling of seafloor composition. While a methodological consensus on which seafloor sediment classification algorithm and routine does not exist in the scientific community, it is expected that progress will occur through the refinement of each stage of the ASC pipeline: ranging from the data acquisition to the modeling phase. This research focuses on the stage of the feature extraction; the stage wherein the spatial variables used for the classification are, in this case, derived from the MBES backscatter data. This contribution explored the sediment classification potential of a textural feature based on the recently introduced Weyl transform of 300 kHz MBES backscatter imagery acquired over a nearshore study site in Belgian Waters. The goodness of the Weyl transform textural feature for seafloor sediment classification was assessed in terms of cluster separation of Folk’s sedimentological categories (4-class scheme). Class separation potential was quantified at multiple spatial scales by cluster silhouette coefficients. Weyl features derived from MBES backscatter data were found to exhibit superior thematic class separation compared to other well-established textural features, namely: (1) First-order Statistics, (2) Gray Level Co-occurrence Matrices (GLCM), (3) Wavelet Transform and (4) Local Binary Pattern (LBP). Finally, by employing a Random Forest (RF) categorical classifier, the value of the proposed textural feature for seafloor sediment mapping was confirmed in terms of global and by-class classification accuracies, highest for models based on the backscatter Weyl features. Further tests on different backscatter datasets and sediment classification schemes are required to further elucidate the use of the Weyl transform of MBES backscatter imagery in the context of seafloor mapping.

Quantum eigenstates from classical Gibbs distributions

We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schr"odinger equation follows from the Liouville equation, with \hbarℏ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner’s quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schr"odinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, Berry phases, Landau levels, level statistics and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including \hbarℏ) on the order of unity.

Symbol Correspondence for Euclidean Systems

The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\star$-product of phase space functions. In this article, the $\star$-product of functions on the Euclidean motion group of rank three, $\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\star_s$ on $\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.

Central Splitting of A2 Discrete Fourier–Weyl Transforms

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group: مبرهنات بالي _ وينر لتحويل فورييه اعتماداً على زمرة هايزنبرغ

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we discuss the representation theory of this group and the relationship between the representation theory of the Heisenberg group and the position and momentum operators and momentum operators relationship between the representation theory of the Heisenberg group and the position and momentum that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schrodinger picture. That is, all the representations we considered are realized in the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator-valued function, and other facts and properties. In our research, we depended on new formulas for some mathematical concepts such as Fourier Transform and Weyl transform. The main aim of our research is to introduce the Paley_ Wiener theorem for the Fourier transform on the Heisenberg group. We will show that the classical Paley_ Wiener theorem for the Euclidean Fourier transform characterizes compactly supported functions in terms of the behaviour of their Fourier transforms and Weyl transform. And we are interested in establishing results for the group Fourier transform and the Weyl transform.