Resolution in the Fourier Domain

There exists a practical limit to the number of sine waves to be included in the Fourier representation (A1.6) of an image. Obviously, sine waves (or complex exponentials) with wavelengths smaller than the size of the smallest features contained in the image carry no information.

This information limit can be expressed in terms of the spatial frequency radius R = Vu2 + v2; beyond a certain radius R = R0 (i.e., outside of a roughly circular domain in the Fourier plane) no meaningful Fourier components are encountered. This boundary is called the bondlimit or resolution limit. [Note that usage of the term resolution is inconsistent in the literature; it is used to denote either the smallest distance resolved (in real space units, dimension length) or the resolution limit defined above (in spatial frequency units, dimension 1/length.]

The resolution limitation invariably can be traced back to a physical limitation of the imaging process: either there is an aperture in the optical system that limits the spatial frequency radius of the object's Fourier components (e.g., the objective lens aperture in the electron microscope), or the recording of the image itself may give rise to some blurring (as, for instance, the lateral spread of electrons in the photographic emulsion).

By virtue of the convolution theorem, any such spread or blurring in the image is expressed by a multiplication of the object's Fourier transform with a function that has a finite radius in the spatial frequency domain.

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