Optimal Discrete Fractional Fourier Transform Descriptors for Image Retrieval

This paper proposes to extend the conventional discrete Fourier transform (DFT) descriptor to discrete fractional Fourier transform (DFrFT) descriptors for representing edges in images. The DFrFT descriptors of training images are employed for constructing a dictionary. However, it is required to determine the optimal rotational angles. This problem is formulated as an optimization problem such that the Fisher discriminant is minimized. Nevertheless, this optimization problem is nonconvex. Also, both the intraclass and interclass separations of the DFrFT descriptors are independent of the rotational angles if these separations are defined using the 2-norm operator. To tackle these difficulties, the 1-norm operator is employed instead. However, this reformulated optimization problem is nonsmooth. To solve this problem, the nondifferentiable points of the objective function are found. Then, the stationary points between any two consecutive nondifferentiable points are identified. After that, the objective functional values are evaluated at these nondifferentiable points and stationary points. The smallest L objective functional values are picked up and the corresponding rotational angles are chosen for constructing the dictionary. Here, L is the total number of the rotational angles for constructing the dictionary. Finally, a 1-NN classification rule is applied for performing the image retrieval. Computer numerical simulation results show that our proposed method outperforms the conventional DFT descriptor approach.