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 $$900,000 M_w$$, 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.