A Discussion of The Generalization of The Central Projection Theorem in Fractional Fourier Transform

With the development of science and technology, electron microscope 3D reconstruction has gradually become the mainstream technology of structural biology. Fourier transform central projection theorem is an important tool in electron microscope 3D reconstruction. However, due to the mechanics of electron microscopy, it faces the challenge of what is known as a 'missing wedge.' In this paper, we focus on discussing the 'missing wedge' problem of 3D reconstruction from the field of image processing. A new mathematical tool, Fractional Fourier transform, is introduced into the 3D reconstruction technique of electron microscope. Combined with most existing methods, the difficulty of this idea is whether the central projection theorem can be generalized under fractional Fourier transform. The validity of the central projection theorem in Fractional Fourier transform is discussed in this paper. The process of discussion shows that the condition of this extension is strict, and attempts to analyze the reason why the theorem cannot be generalized from mathematical derivation. Combined with numerical results, the theoretical conclusions are further illustrated. It is worth noting that the strict generalization of the theorem does not mean that the fractional Fourier transform cannot be applied to three-dimensional reconstruction problems, only that some solutions are no longer used. To some extent, the new method based on fractional Fourier transform is proposed.