Corresponding to optical Fresnel transformation characteristic of a ray transfer matrix (A,B,C,D),AD-BC=1, there exists Fresnel operator F(A,B,C,D) in quantum optics, we show that under the Fresnel transformation the pure-state position density ∣x〉〈x∣ becomes density operator ∣x〉s,rs,r〈x∣, which is just the Radon transform of the Wigner operator, i.e.,F∣x〉〈x∣F†=∣x〉s,rs,r〈x∣=∫∫-∞∞dp′dx′δx-Dx′-Bp′Δx′,p′,where s,r are the complex-value expression of (A,B,C,D). So the probability distribution for the Fresnel quadrature phase is the tomography (Radon transform of the Wigner function), correspondingly, s,r〈x∣.ψ〉=〈x∣F†∣ψ〉. Similarly, we findF∣p〉〈p∣F†=∣p〉s,rs,rp=∫∫-∞∞dp′dx′δp-Ax′-Cp′Δx′,p′,where ∣p〉〈p∣ is the pure-state momentum density.