We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing 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 $2\pi ƛ$ 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 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 numbers $(k,\ell )\in {\mathbb{Z}}^2$. The WFS leverages key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way, (ii) its transport equation naturally evolves with a Liouville equation at $ƛ\to 0$, (iii) it admits positive/negative oscillatory patterns on the $(Q_0,Q_2)$ plane that can be a 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 quantum information theory.

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• Image processing