The mapping from a potential \(q(x)\) on \(\mathbb{R}^ n\) to the backscattering amplitude \(a(\theta, -\theta, k)\) associated with the Hamiltonian \(- \Delta+ q(x)\) is studied (the backscattering amplitude is a restriction of the scattering amplitude \(a(\theta, \omega, k))\). It is shown that the backscattering map \(S\) is locally an isomorphism. The map preserves the weighted Hölder spaces \(H_{\alpha, N}\). The treatment concerns \(n\geq 3\). The case \(n=2\) is complicated by the presence of a singularity and it is dealt with separately. The results were already described previously [\textit{G. Eskin} and \textit{J. Ralston}, Commun. Math. Phys. 124, 169-215 (1989; Zbl 0706.35136) and ibid. 138, 451-486 (1991; Zbl 0728.35146)].