Fourier Transform Microscopic images

It is well known that under the weak-phase-object approximation (WPOA) [19], the image intensity function is linear to the convolution of the projected potential distribution function cpt (x, y) and the inverse Fourier transform (FT) of the contrast transfer function (CTF) r(u) of the electron microscope ... 

Equation (1) shows that HREM images contain information of the crystal projected potential as well as all microscopic aberrations, which is characterized by the CTF. According to Equation (2), under WPOA, Fourier transform of the HREM image yields the crystallographic structure factors that are multiplied by the CTF. Therefore, it is reasonable to... 

Ouyang, H, Sherman, P. J., Paschalis, E. P., Boskey, A. L. and Mendelsohn, R. (2004) Fourier transform infrared microscopic imaging effects of estrogen and estrogen deficiency on fracture healing in rat femurs. Appl. Spectrosc. 58, 1-9. 

Lattice images of cellulose can be obtained from cellulose samples in the electron microscope and subjected to the same Fourier transformations as AFM images. Both of these techniques confirm the idea that cellulose chains are very extended in crystalline microfibrils and emphatically do not undergo folding within linear microfibrilar structures, as had been proposed by some authors. 

Microscopically particle texture may be defined using fractals or Fourier transforms. The introduction of quantitative image microscopy has made such approaches to particle texture analysis practical. 

High-resolution cryo-EM data can be collected in two forms as electron images (69) or as electron diffraction patterns. Cryo-EM images contain information on both amplitude and phase, which can be analyzed after Fourier transformation. The quality of the amplitude data can be improved if combined with electron diffraction data, which contains only amplitude information. In this way, EM overcomes one of the main difficulties in XRC. In XRC, only diffraction patterns are obtained. X-rays caimot be used to form an image of the crystal therefore, the phase information is lost. In contrast, electron microscopes contain electron lenses that can capture phase information. 

The electron density equation very simple structures such as NaCl can be solved by comparison of the relative intensities of the diffraction spots. For more complicated structures, the power of Fourier transform methods was soon appreciated [27]. In order to produce an image of the structure, the diffracted rays must be combined. In the light microscope this is achieved by the focussing power of the objective lens (Fig. 3b). For X-rays the refractive index of almost all substances is close to 1 and it is not possible to construct a lens. The diffracted rays must be combined mathematically. This is achieved with the electron density equation. 

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