Kernel estimator
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We describe now the direct estimation method for the Wigner function. For the problem of estimating a probability density $f:\mathbb{R}^{2}\to \mathbb{R}$ directly from data $(X_\ell,
\Phi_\ell)$ with density $\mathcal{R}[f]$ we refer to the literature on X-ray tomography and PET, studied by and many other references therein. In the context of tomography of bounded objects with noisy observations solved the problem of estimating the borders of the object (the support). The estimation of a quadratic functional of the Wigner function has been treated in [@Meziani]. For the problem of Wigner function estimation when no noise is present, we mention the work by . They use a kernel estimator and compute sharp minimax results over a class of Wigner functions characterised by their smoothness. In a more recent paper , Butucea, Guţă and Artiles treated the noisy problem for the pointwise estimation of $W_{\rho}$; however the functions needed to prove minimax optimality there do not belong to the class of Wigner functions that we consider here.

In this chapter, as in , we modify the usual tomography kernel in order to take into account the additive noise on the observations and construct a kernel $K_h^\eta$ which performs both deconvolution 