Teaching set #1 - Image form factor
When a image consisting of a repeating object undergoes a Fourier transformation, the shape and size of the object influences the intensity of the spots originating from the periodicity of the image.
While the spots' position depends on the lattice describing how the object repeats, the spots' intensity (that is, their brightness) depends on the form factor of the object (that is, its Fourier transform). The scope of the following series of figures is to highlight this relationship.
Fourier transforms of repeating objects (here called "crystals") and those of single objects are shown using a different intensity scale for the sake of clarity. All Fourier transforms of crystals are reported with the same exact intensity scale, and all form factors also have the same intensity scale.
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01.a Crystal of large circles (40 circles radius) and its Fourier transform.
01.b Large circle as in 01.a, and its Fourier transform.
Note: the intensity of the spots in 01.a is proportional to the intensity of the corresponding positions in this curve. Where the curve is dark, spots are not visible, when it's at its brightest, spots have the highest intensity.
02.a Crystal of medium circles (40 circles radius) and its Fourier transform.
02.b Medium circle as in 02.a, and its Fourier transform.
Note: the same behaviour as in 01.a-b can be seen here, but the form factor is very different. As an object becomes smaller, its form factor undergoes the opposite effect.
03.a Crystal of small circles (40 circles radius) and its Fourier transform.
03.b Small circle as in 03.a, and its Fourier transform.
04.a Crystal of rhombuses (40 rhombuses radius) and its Fourier transform.
04.b Rhombus as in 04.a, and its Fourier transform.
05.a Crystal of triangles (40 triangles radius) and its Fourier transform.
05.b Triangle as in 05.a, and its Fourier transform.
Note: even though a triangle is not a centrosymmetric object, its form factor is. The reason is rather simple: the Fourier transformation is a function of every pixel-pixel spatial relationship, which is naturally centrosymmetric. Indeed, given two points A and B, we have two spatial relationships involved: A to B and B to A. These relationships are described by vectors and called "spatial correlations". In this case these vectors lie on the same line, have same lengths, and go in opposite directions. This is what, in simple terms, makes every Fourier transform centrosymmetric.
06.a Crystal of dashes (40 dashes radius) and its Fourier transform.
06.b Dash as in 06.a, and its Fourier transform.
07.a Crystal of ducks (40 ducks radius) and its Fourier transform.
Ducks are considerably more complex in shape than a dash or a triangle. This allows to gain a feeling of what the Fourier space of more "realistic" objects and their ordered repetition look like. Why do we care? Because in science the behaviour of the Fourier transform is closely linked to the behaviour of a crystal (an ordered repetition of atoms) in a diffraction experiment. In these experiments, a source of light is diffracted in patterns that are analysed by crystallographers to decipher the atomic structure of materials, pharmaceuticals, minerals, etc. How do they to? Using the notion that diffraction patterns originate from the Fourier transform of an atomic structure.
07.b Duck as in 07.a, and its Fourier transform.
