# PlanetPhysics/2DFT Imaging
### Two-dimensional Fourier transform imaging

A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, both involving `standard', one-dimensional Fourier transforms. However, the second stage Fourier transform is not the inverse Fourier transform (which would result in the original function that was transformed at the first stage), but a Fourier transform in a second variable-- which is `shifted' in value-- relative to that involved in the result of the first Fourier transform. Such 2D-FT analysis is a very powerful method for three-dimensional reconstruction of polymer and biopolymer structures by two-dimensional Nuclear Magnetic resonance (2D-FT NMR ,) of solutions for molecular weights ($M_{w}$) of the dissolved polymers up to about 50,000 $M_{w}$. For larger biopolymers or polymers, more complex methods have been developed to obtain the desired resolution needed for the 3D-reconstruction of higher molecular structures, e.g. for $900,000M_{w}$, methods that can also be utilized in vivo . The 2D-FT method is also widely utilized in optical spectroscopy, such as 2D-FT NIR Hyperspectral Imaging, or in MRI imaging for research and clinical, diagnostic applications in Medicine.
A more precise mathematical definition of the `double' Fourier transform involved is specified next, and a precise example follows the definition.
A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables, $f(x_{1},x_{2})$, carried first in the first variable $x_{1}$, followed by the Fourier transform in the second variable $x_{2}$ of the resulting function $F(s_{1},x_{2})$. (For further specific details and example for 2D-FT Imaging v. URLs provided in the following recent Bibliography).
Example 0.1 A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals $s(t_{1},t_{2})$ yielding a real 2D-FT NMR `spectrum' (collection of 1D FT-NMR spectra) represented by a matrix $S$ whose elements are $S(\nu _{1},\nu _{2})='''Re'''\int \int cos(\nu _{1}t_{1})exp^{(-i\nu _{2}t_{2})}s(t_{1},t_{2})dt_{1}dt_{2},$ where $\nu _{1}$ and $\nu _{2}$ denote the discrete indirect double-quantum and single-quantum(detection) axes, respectively, in the 2D NMR experiments. Next, the covariance matrix is calculated in the frequency domain according to the following equation: $C(\nu _{2}',\nu _{2})=S^{T}S=\sum _{\nu ^{1}}[S(\nu _{1},\nu _{2}')S(\nu _{1},\nu _{2})],$
with $\nu _{2},\nu _{2}'$ taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies $\nu _{1}$.\\

### Example 0.2

Atomic structure reconstruction by 2D-FT of STEM Images(obtained at Cornell University) reveals the electron distributions in a high-temperature cuprate superconductor `paracrystal'; both the domains (or `location') and the local symmetry of the "pseudo-gap" are seen in the electron-pair correlation band responsible for the high--temperature superconductivity effect .

### Remarks

So far there have been three Nobel prizes awarded for 2D-FT NMR/MRI during 1992-2003, and an additional, earlier Nobel prize for 2D-FT of X-ray data (`CAT scans'); recently the advanced possibilities of 2D-FT techniques in Chemistry, Physiology and Medicine received very significant recognition.