2DFTImaging 2008-12-24 18:31:57 2008-12-24 18:31:57 Topic two-dimensional Fourier transform imaging % 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}} \textbf{Preliminary Data} 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 \emph{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-NMR, \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 molecular structures, 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 NIR hyperspectral imaging}, 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 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 $$S(\nu_1,\nu_2) = \textbf{Re} \int \int cos(\nu_1 t_1)exp^{(-i\nu_2 t_2)} s(t_1, t_2)dt_1 dt_2$$ where $\nu_1$ and $\nu_2$ denote the discrete indirect double-quantum and single-quantum(detection) axes, respectively, in the 2D NMR experiments. Next, the \emph{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 $\nu_1$.\\ \textbf{Example 0.2} \PMlinkexternal{2D-FT STEM Images (obtained at Cornell University) of electron distributions in a high-temperature cuprate superconductor `paracrystal'}{http://www.physorg.com/multimedia/pix1815/} reveal both the domains (or `location') and the local symmetry of the ``pseudo-gap'' in the electron-pair correlation band responsible for the high--temperature superconductivity effect (a definite possibility for the next Nobel (?) iff the mathematical physics treatment is also developed to include also such results). \\ \textbf{Remark:} 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 \PMlinkexternal{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. \PMlinkexternal{(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}; \PMlinkexternal{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. \PMlinkexternal{Nobel Lecture}{http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf}, on December 9, 1992. \bibitem{PM2k3} Peter Mansfield. 2003. \PMlinkexternal{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}.) \PMlinkexternal{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. \PMlinkexternal{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 Ampere 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 (62,63). This two-pulse experiment by Jean Jeener is the forefather of a whole class of $2D$ experiments (8,63) that can also easily be expanded to multidimensional spectroscopy.'' \end{thebibliography}