Adjustable Jacobi–Fourier Moment for Image Representation

The widely adopted Jacobi-Fourier moment (JFM) is limited by its inability to effectively capture spatial information. Although fractional-order JFM (FOJFM) introduces spatial information through a fractional-order parameter, the control of spatial information remains inadequate. This limitation stems from the insufficient control over zeros distribution associated with the used moment's radial kernel. To address this issue, we generalize both JFM and FOJFM into a transformed JFM. A transformed function with four parameters is designed, and adjustable JFM (AJFM) is proposed. Two parameters correlate to increasing velocities on the left and right parts of the transformed functions, enabling zeros quantities of radial kernel fall in the left and right parts of the interval. The other two parameters segment the transformed function, adjusting regions where different quantities of zeros fall in. This refined control over the radial kernel's zero distribution enhances the versatility of feature extraction by the AJFM, governed by the introduced parameters. Experimental results demonstrate that AJFM, with properly chosen parameters, can emphasize specific regions within an image more effectively.