• The importance of optimal parameters in the fractional-order moments is discussed. • The proposed algorithm searches the optimal fractional parameter. • The optimal parameter found is invariant to changes in scale and rotation. • The experimental analysis demonstrates the superiority of the optimal parameter. • The proposed method performs superbly in rotation-invariant object recognition. In this paper, we briefly review the fractional-order circular moments, such as fractional-order Zernike moments, fractional-order Fourier–Mellin moments, fractional-order Legendre–Fourier moments, and fractional-order Chebyshev–Fourier moments, which can characterize, analyze, and manipulate the information contained in an image with minimal redundancy. Also, they depend on an α parameter for better feature extraction. Therefore, we propose a procedure to find the optimal α in terms of image reconstruction error and classification. We validate the search for the best rotation-invariant features using the MNIST and MNIST-R datasets. Finally, we present the study results and conclusions.