"]] 

 == NMRI Principle== 
 Certain nuclei such as 1H nuclei, or `fermions' have spin-1/2, because there are two spin states, referred to as ""up"" and ""down"" states. The nuclear magnetic resonance absorption phenomenon occurs when samples containing such nuclear spins are placed in a static magnetic field and a very short radiofrequency pulse is applied with a center, or carrier, frequency matching that of the transition between the up and down states of the spin-1/2 1H nuclei that were polarized by the static magnetic field. Antoine Abragam. 1968. Principles of Nuclear Magnetic Resonance., 895 pp., Cambridge University Press: Cambridge, UK. 
 Very low field schemes have also been recently reported. 

 ==Chemical Shifts== 
 NMR is a very useful family of techniques for chemical and biochemical research because of the chemical shift; this effect consists in a frequency shift of the nuclear magnetic resonance for specific chemical groups or atoms as a result of the partial shielding of the corresponding nuclei from the applied, static external magnetic field by the electron orbitals (or molecular orbitals) surrounding such nuclei present in the chemical groups. Thus, the higher the electron density surounding a specific nucleus the larger the chemical shift will be. The resulting magnetic field at the nucleus is thus lower than the applied external magnetic field and the resonance frequencies observed as a result of such shielding are lower than the value that would be observed in the absence of any electronic orbital shielding. Furthermore, in order to obtain a chemical shift value independent of the strength of the applied magnetic field and allow for the direct comparison of spectra obtained at different magnetic field values, the chemical shift is defined by the ratio of the strength of the local magnetic field value at the observed (electron orbital-shielded) nucleus by the external magnetic field strength, Hloc/ H0. 
 The first NMR observations of the chemical shift, with the correct physical chemistry interpretation, were reported for 19F containing compounds in the early 1950's by Herbert S. Gutowsky and Charles P. Slichter from the University of Illinois at Urbana (USA). 

 ==NMR Imaging Principles== 
 A number of methods have been devised for combining magnetic field gradients and radiofrequency pulsed excitation to obtain an image. Two major maethods involve either 2D -FT or 3D-FT reconstruction from projections, somewhat similar to Computed Tomography, with the exception of the image interpretation that in the former case must include dynamic and relaxation/contrast enhancement information as well. Other schemes involve building the NMR image either point-by-point or line-by-line. Some schemes use instead gradients in the rf field rather than in the static magnetic field. The majority of NMR images routinely obtained are either by the Two-Dimensional Fourier Transform (2D-FT) technique (with slice selection), or by the Three-Dimensional Fourier Transform (3DFT) techniques that are however much more time consuming at present. 2D-FT NMRI is sometime called in common parlance a ""spin-warp"".  An NMR image corresponds to a spectrum consisting of a number of `spatial frequencies' at different locations in the sample investigated, or in a patient.* 
 
 A two–dimensional Fourier transformation of such a ""real"" image may be considered as a representation of such ""real waves"" by a matrix of spatial frequencies known as the k–space.  
 We shall see next in some mathematical detail how the 2D-FT  
 computation works to obtain 2D-FT NMR images. 

 == Two-dimensional Fourier transform imaging and spectroscopy== 

 A two-dimensional Fourier transform (2D-FT) is computed nume