Summary: In this paper, we prove that the Riesz wavelet type basis in \(L^2(\mathbb{R}^d)\) is image of an orthonormal wavelet basis under a bounded invertible operator and prove that an orthonormal wavelet basis and an orthonormal wavelet type basis in \(L^2(\mathbb{R}^d)\) are unitarily equivalent. Also, it is proved that the image of an orthonormal wavelet basis under unitary operator is an orthonormal wavelet type basis. Further, we prove that dual of Riesz wavelet basis always exists and is biorthogonal to it. Finally, we study dual wavelet frames for a given wavelet frame.