维格纳函数小波：天文图像的辛表示
Wigner function shapelets: Symplectic representation of astronomical images
AI总结：
该研究将小波扩展到相空间得到维格纳函数小波（WFSs），在四维相空间用辛群Sp(4,R)表示图像，通过拉盖尔 - 高斯模式交叉维格纳函数双线性形式构成基，利用维格纳函数特性分析图像，汇集相关公式并进行形式化组织。
摘要：
我们将用于天文图像分析的小波扩展到相空间，引入了“维格纳函数小波”（WFSs）。传统小波在构型或傅里叶空间中使用厄米 - 高斯或拉盖尔 - 高斯模式分别展开图像，而WFSs在具有辛群Sp(4,R)的四维相空间中直接表示图像，通过相空间单元2πλbar进行量化，该单元决定了望远镜的分辨率极限。WFSs由拉盖尔 - 高斯模式的交叉维格纳函数的双线性形式组成，作为图像维格纳函数的正交完备基，通过霍普夫环面进行相空间的SU(2)不可约表示。我们引入一个标量函数Wkl(Q0,Q2)，从U(1)×U(1)协变环面到二维运动常数空间(Q0,Q2)——谐波能量和轴向角动量，从而产生自然的相空间“能带结构”，给定一对整数缠绕数(k,ℓ)∈Z2。WFSs利用维格纳函数的关键特性进行图像分析：它以对称保持的方式编码图像的全部信息；其输运方程在λbar→0时自然涉及刘维尔方程；它在(Q0,Q2)平面上允许正负振荡模式，可敏感反映星系形态和宇宙学印记的空间相干结构；系统误差和噪声可作为量子通道操作进行处理。本文旨在汇集天体物理学和宇宙学背景下与维格纳函数相关的所有公式，用天文学和量子信息论的术语对其进行形式化组织。
英文摘要：
We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing \textit{Wigner function shapelets}(WFSs). Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFSs represent images directly in the four-dimensional phase space with symplectic group $\mathrm{Sp}(4,\mathbb{R})$, which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out $\mathrm{SU}(2)$ irreducible representations of the phase space with the Hopf tori. We introduce a scalar function $\mathcal{W}_{k\ell} (Q_0,Q_2)$ from the $\mathrm{U}(1)\times \mathrm{U}(1)$ - covariant tori to a two-dimensional space of constants of motion $(Q_0,Q_2)$ -- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number $(k,\ell) \in \mathbb{Z}^2$. % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on $(Q_0,Q_2)$ plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.
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