2D-FT MR- Imaging and related Nobel awards
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
, [1])
of solutions for molecular weights (
) of the dissolved polymers up to about 50,000
. 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
, 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.
Definition 0.1
A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables,

, carried first in the first variable

, followed by the Fourier transform
in the second variable

of the resulting function

. (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
yielding a real 2D-FT NMR `spectrum' (collection of 1D FT-NMR
spectra) represented by a matrix
whose elements are
where
and
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:
with
taking all possible single-quantum frequency
values and with the summation carried out over all discrete, double quantum
frequencies
.
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 .
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.
-
- 1
-
Kurt Wütrich: 1986, NMR of Proteins and Nucleic Acids., J. Wiley and Sons: New York, Chichester, Brisbane, Toronto, Singapore.
(Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules); 2D-FT NMR Instrument Image Example: a JPG color image of a 2D-FT NMR Imaging `monster' Instrument
- 2
-
Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. Nobel Lecture, on December 9, 1992.
- 3
-
Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.
- 4
-
D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. (lots of 2D-FT images of mathematical, brain scans.)
PDF of 2D-FT Imaging Applications to MRI in Medical Research.
- 5
-
Paul Lauterbur. 2003.
Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.
- 6
-
Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampere International Summer School, Basko Polje, unpublished. A verbatim quote follows from Richard R. Ernst's Nobel Laureate Lecture
delivered on December 2nd, 1992, ``A new approach to measure two-dimensional (2D) spectra has been
proposed by Jean Jeener at an Ampère Summer School in Basko Polje, Yugoslavia, 1971 ([6]). He suggested a 2D Fourier transform experiment consisting of two
pulses with a variable time
between the pulses and the time variable
measuring the time elapsed after the second pulse as shown in Fig. 6 that expands the principles of Fig. 1. Measuring the response
of the two-pulse sequence and Fourier-transformation with
respect to both time variables produces a two-dimensional spectrum
of the desired form. This two-pulse experiment by Jean Jeener is the forefather of a whole class of
experiments that can also easily be expanded to multidimensional spectroscopy.''
- 7
-
A 2D-FT NMRI article
and Spectroscopy.
- 8
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Cardiac infarct movies by 2D-FT NMR Imaging
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As of this snapshot date, this entry was owned by bci1.