Inversion formulas for \(k\)-plane transforms of functions \(f \in L^2({\mathbb R}^n)\) are obtained in terms of continuous wavelet transforms generated by a wavelet measure. The admissibility conditions for a wavelet measure \(\mu\) are formulated in terms of the Fourier transform of \(\mu\) and also without using the Fourier transform. The work is based on the wavelet type representation of Riesz fractional derivatives developed in, e. g., the author's book [Fractional integrals and potentials (1996; Zbl 0864.26004)].