%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: 2D-FT MR- Imaging and related Nobel awards
%%% Primary Category Code: 83.85.Fg
%%% Filename: 2DFTImaging.tex
%%% Version: 35
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
% this is the default PlanetPhysics preamble. as your
% define commands here
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym}
\usepackage{xypic}
\usepackage[mathscr]{eucal}
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}
\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@
}}}
\numberwithin{equation}{section}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
\newcommand{\G}{\mathcal G}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathbb G}}
\newcommand{\dgrp}{{\mathbb D}}
\newcommand{\desp}{{\mathbb D^{\rm{es}}}}
\newcommand{\Geod}{{\rm Geod}}
\newcommand{\geod}{{\rm geod}}
\newcommand{\hgr}{{\mathbb H}}
\newcommand{\mgr}{{\mathbb M}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathbb G)}}
\newcommand{\obgp}{{\rm Ob(\mathbb G')}}
\newcommand{\obh}{{\rm Ob(\mathbb H)}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\ghomotop}{{\rho_2^{\square}}}
\newcommand{\gcalp}{{\mathbb G(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\glob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\med}{\medbreak}
\newcommand{\medn}{\medbreak \noindent}
\newcommand{\bign}{\bigbreak \noindent}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\oset}[1]{\overset {#1}{\ra}}
\newcommand{\osetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}
\begin{document}
\subsection{Two-dimensional Fourier transform imaging}
A \htmladdnormallink{two-dimensional Fourier transform}{http://planetphysics.us/encyclopedia/2DFFT.html} (2D-FT) is computed numerically or carried out in two stages, both involving `standard', one-dimensional \htmladdnormallink{Fourier transforms}{http://planetphysics.us/encyclopedia/FourierTransforms.html}. However, the second stage Fourier transform is \emph{not the inverse} Fourier transform (which would result in the original
\htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} Nuclear Magnetic \htmladdnormallink{resonance}{http://planetphysics.us/encyclopedia/QualityFactorOfAResonantCircuit.html} (\htmladdnormallink{2D-FT NMR}{http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance#Nuclear_spin_and_magnets} , \cite{KurtWutrich86})
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 \htmladdnormallink{molecular structures}{http://planetphysics.us/encyclopedia/FCS3.html}, e.g. for $900,000 M_w$, methods that can also be utilized \emph{in vivo}. The 2D-FT method is also widely utilized in optical spectroscopy, such as \emph{2D-FT \htmladdnormallink{NIR}{http://planetphysics.us/encyclopedia/SpectralImaging.html} \htmladdnormallink{Hyperspectral Imaging}{http://planetphysics.us/encyclopedia/SpectralImaging.html}}, or in \emph{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.
\begin{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).
\end{definition}
\textbf{Example 0.1}
A 2D Fourier transformation and phase correction is applied to a set of 2D \htmladdnormallink{NMR}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} (FID) signals $s(t_1, t_2)$ yielding a real 2D-FT \htmladdnormallink{NMR `spectrum}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html}' (collection of 1D \htmladdnormallink{FT-NMR}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} spectra) represented by a \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} $S$ whose elements are
$$S(\nu_1,\nu_2) = \textbf{Re} \int \int cos(\nu_1 t_1)exp^{(-i\nu_2 t_2)} s(t_1, t_2)dt_1 dt_2,$$ where $\nu_1$ and $\nu_2$ denote the discrete indirect double-quantum and single-quantum(\htmladdnormallink{detection}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html}) axes, respectively, in the 2D NMR experiments. Next, the \emph{\htmladdnormallink{covariance}{http://planetphysics.us/encyclopedia/Covariance.html} matrix} is calculated in the frequency \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} 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 $\nu_1$.\\
\subsection{Example 0.2}
\htmladdnormallink{Atomic structure reconstruction by 2D-FT of STEM Images(obtained at Cornell University)}{http://www.physorg.com/news129395045.html } 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 \htmladdnormallink{superconductivity}{http://planetphysics.us/encyclopedia/QuantumStatisticalTheories.html} effect .
\subsection{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 \htmladdnormallink{X-ray}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} data (`CAT scans'); recently the advanced possibilities of 2D-FT techniques in
\htmladdnormallink{Chemistry}{http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf}, Physiology and Medicine received very significant recognition.
\begin{thebibliography}{9}
\bibitem{KurtWutrich86}
Kurt W\"{u}trich: 1986, \emph{NMR of Proteins and Nucleic Acids.}, J. Wiley and Sons: New York, Chichester, Brisbane, Toronto, Singapore.
\htmladdnormallink{(Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules)}{http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/wutrich-lecture.pdf}; \htmladdnormallink{2D-FT NMR Instrument Image Example: a JPG color image of a 2D-FT NMR Imaging `monster' Instrument}{http://upload.wikimedia.org/wikipedia/en/b/bf/HWB-NMRv900.jpg}
\bibitem{RICHARDRERNST1992}
Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. \htmladdnormallink{Nobel Lecture}{http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf}, on December 9, 1992.
\bibitem{PM2k3}
Peter Mansfield. 2003. \htmladdnormallink{Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.}{http://www.parteqinnovations.com/pdf-doc/fandr-Gaz1006.pdf}
\bibitem{MRI-2DFT}
D. Benett. 2007. \emph{PhD Thesis}. Worcester Polytechnic Institute. ({\em lots of 2D-FT images of mathematical, brain scans}.)
\htmladdnormallink{PDF of 2D-FT Imaging Applications to MRI in Medical Research}{http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdf}.
\bibitem{PL2k3}
Paul Lauterbur. 2003.
\htmladdnormallink{Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.}{http://nobelprize.org/nobel_prizes/medicine/laureates/2003/}
\bibitem{JeanJeneer1971}
Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampere International Summer School, Basko Polje, \emph{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 Amp\`ere Summer School in Basko Polje, Yugoslavia, 1971 (\cite{JeanJeneer1971}). He suggested a 2D Fourier transform experiment consisting of two $\pi/2$ pulses with a variable time $t_1$ between the pulses and the time variable $t_2$ measuring the time elapsed after the second pulse as shown in Fig. 6 that expands the principles of Fig. 1. Measuring the response $s(t_1,t_2)$ of the two-pulse sequence and Fourier-transformation with
respect to both time variables produces a two-dimensional spectrum $S(O_1,O_2)$ of the desired form. This two-pulse experiment by Jean Jeener is the forefather of a whole class of $2D$ experiments that can also easily be expanded to multidimensional spectroscopy.''
\bibitem{bci2k9}
A \htmladdnormallink{2D-FT NMRI article}{http://en.wikipedia.org/wiki/2D-FT_NMRI_and_Spectroscopy} and Spectroscopy.
\bibitem{infarct}
Cardiac infarct movies by \htmladdnormallink{2D-FT NMR Imaging}{http://www.mr-tip.com/exam_gifs/cardiac_infarct_short_axis_cine_6.gif}
\end{thebibliography}
\end{document}
