Relationship between an object shape and its Fourier transform



    Due to the definition of the Fourier transform of an object, it exists close relationships between the shape of this object and the shape of its Fourier transform. This relationship is used in practice for example in order to observe and characterize very small objects such as atoms within a crystal (X-ray diffraction).

    The first example uses as objects two disks of different radii. One can observe that their FT have the same cylindrical symmetry than the initial objects, displayed as concentric and alternating bright and dark rings. However, the spatial extension in the Fourier space is inverted compared to the direct (object) space: the large disk leads to a FT having small rings, whereas the small disk results in an extended FT.

   

   

   
Top : initial objects: disks differing in radii. Middle: Fourier transforms of these disks. Bottom: profiles of the moduli of the FT


    The second example is composed of an white elongated segment on a black background. Its Fourier transform is then composed of white peaks, regularly spaced around the highest peak which is displayed at the origin of the Fourier space. As can be foreseen from the first example, oscillations in the horizontal direction are narrower than the one in the vertical direction since the spatial extension of the object is more important horizontally than vertically.

   

Top: initial object (white segment) and its Fourier transform. Middle: vertical profile of the moduli of the FT. Bottom: horizontal profile of the moduli of the FT

    Fom this stage, if one increases the length of the segment up to the full width of the image, its Fourier transform will then shrink in the horizontal direction:

   
White segment fully extended in the horizontal direction, and its Fourier transform

Indeed, the equidistance observed between the nodes (zeros) of the moduli of the FT being inversly proportional to the object extension in the considered direction, if the size of the object is infinitely increased in a given direction (the object occupies all the image width in this direction) then the nodes of the FT tend to the origin of the Fourier space i.e. the centre of the FT (cf. Dirac delta function).


    The foolowinf example underline the close relationship between an object and its Fourier transform: considering a cross as the object, its FT displays the same asymmetry (i.e. has also a cross shape) but it should be noted that there are non zero Fourier coefficients also in the diagonal direction of the Fourier space. The intensity of the moduli of the FT is maximal at the centre (low resolution) and decrease with oscillations toward the edges of the Fourier map (high resolution).



   
Top: a cross that can be seen as two white segments. Bottom: moduli of the FT of the cross using two different gray scales in order to show low and high value moduli


    Starting from this unique cross one can form a second object by translations in the vertical and horizontal directions. This new object is what one calls a lattice, periodic recurrence of a motif (the cross). The FT of such an object is then also periodic as can be seen from the small white spots (because this particular object is not perfectly periodic with respect to the image width, its FT consists of spots not infinitely sharp but having a given extension).
One can see that the variation of the intensity accros the FT of this new object is the same as the one of the FT of the unique cross, being maximal at the centre and then decreasing with oscillations in the vertical and horizontal directions (cf. Convolution theorem).



   
Top: lattice formed by the recurrence of the unique cross in vertical and horizontal directions. Bottom: moduli of the FT of this lattice using to different gray scales in order to show low and high value moduli



    The shape of an object and of its Fourier transform are then closely related, and the study of an FT can then give information about the periodicity, the spatial extension of the object.
    This is the underlying principle used in crysallography. Atoms in crystal can not be directly (easily) observed; however they scatter X-rays and it can be shown that the pattern formed during X-ray diffraction is linked to the crystal atomic structure by a Fourier transformation. The analysis of this diffraction pattern (Fourier space) allows to obtain the atomic positions in the crystal (direct space), as the inverse Fourier transform do on one of the preceding examples.