Image Retrieval Based on Discrete Fractional Fourier Transform Via Fisher Discriminant

Abstract

Discrete fractional Fourier transform (DFrFT) is a powerful signal processing tool. This paper proposes a method for DFrFT-based image retrieval via Fisher discriminant and 1-NN classification rule. First, this paper proposes to extend the conventional discrete Fourier transform (DFT) descriptors to the DFrFT descriptors to be used for representing the edges of images. The DFrFT descriptors extracted from the training images are employed to construct a dictionary, for which the corresponding optimal rotational angles of the DFrFTs are required to be determined. This dictionary design problem is formulated as an optimization problem, where the Fisher discriminant is the objective function to be minimized. This optimization problem is nonconvex (Guan et al. in IEEE Trans Image Process 20(7):2030---2048, 2011; Ho et al. in IEEE Trans Signal Process 58(8):4436---4441, 2010). Furthermore, both the intraclass separation and interclass separation of the DFrFT descriptors are independent of the rotational angles if these separations are defined in terms of the 2-norm operator. To tackle these difficulties, the 1-norm operator is employed. 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. The objective function values are evaluated at these nondifferentiable points and these stationary points. The smallest L objective function values are picked up and the corresponding rotational angles are determined, which are then used to construct the dictionary. Here, L is the total number of the rotational angles of the DFrFTs used to construct the dictionary. Finally, an 1-NN classification rule is applied to perform the image retrieval. Application examples and experimental results show that our proposed method outperforms the conventional DFT approach.