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Gibbs Phenomenon
 
In mathematics, the Gibbs phenomenon (also known as ringing artifacts, named after the American physicist J. Willard Gibbs) is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function f behaves at a jump discontinuity: the nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit.
In MRI, artifactual ripples parallel to abrupt and intense changes are caused by the Fourier transformation.

See Gibbs Artifact, Truncation Artifact, Ringing Artifact Reduction.
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Further Reading:
  Basics:
Gibbs phenomenon
   by en.wikipedia.org    
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Integral
 
An integral is a mathematical object that can be interpreted as an area or a generalization of area. A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed (summation in the limit). In MRI for example this mathematical function is used in the Fourier transformation.
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Integral
   by en.wikipedia.org    
Integral
   by mathworld.wolfram.com    
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Lorentzian Line
 
Usual shape of the lines in a NMR spectrum, characterized by a central peak with long tails; proportional to 1/[(1/T2)2 + (f - fo)2], where f is frequency and fo is the frequency of the peak (i.e., central resonance frequency). A Lorentzian function is the Fourier transformation of a decaying exponential.
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Nuclear Magnetic ResonanceMRI Resource Directory:
 - NMR -
 
(NMR) Nuclear Magnetic Resonance is a physical phenomenon of the magnetic property of nuclei, which have a positive nuclear spin quantum number.
Under the influence of an external static magnetic field this nuclei will precess about the direction of the magnetic field with an angular frequency (Larmor frequency). Through absorption and emission of RF energy (gradients, RF coils) at the resonance frequency (Larmor equation) and the processing of this raw data by the Fourier transformation - physical, chemical, electronic, and structural information about molecules can be obtained (NMR Magnetic Resonance Spectroscopy, Magnetic Resonance Imaging).
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Further Reading:
  Basics:
MRI's inside story
Thursday, 4 December 2003   by www.economist.com    
Nuclear magnetic resonance with no magnets
Wednesday, 18 May 2011   by www.physorg.com    
  News & More:
Neuromelanin-Sensitive MRI Identified as a Potential Biomarker for Psychosis
Sunday, 10 February 2019   by www.nimh.nih.gov    
A powder to enhance NMR signals
Thursday, 12 December 2013   by phys.org    
New Paradigm for Nanoscale Resolution MRI Experimentally Achieved
Friday, 27 September 2013   by www.sciencedaily.com    
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Parallel Imaging TechniqueForum -
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In parallel MR imaging, a reduced data set in the phase encoding direction(s) of k-space is acquired to shorten acquisition time, combining the signal of several coil arrays. The spatial information related to the phased array coil elements is utilized for reducing the amount of conventional Fourier encoding.
First, low-resolution, fully Fourier-encoded reference images are required for sensitivity assessment. Parallel imaging reconstruction in the Cartesian case is efficiently performed by creating one aliased image for each array element using discrete Fourier transformation. The next step then is to create an full FOV image from the set of intermediate images. Parallel reconstruction techniques can be used to improve the image quality with increased signal to noise ratio, spatial resolution, reduced artifacts, and the temporal resolution in dynamic MRI scans.
Parallel imaging algorithms can be divided into 2 main groups:
Image reconstruction produced by each coil (reconstruction in the image domain, after Fourier transform): SENSE (Sensitivity Encoding), PILS (Partially Parallel Imaging with Localized Sensitivity), ASSET.
Reconstruction of the Fourier plane of images from the frequency signals of each coil (reconstruction in the frequency domain, before Fourier transform): GRAPPA.
Additional techniques include SMASH, SPEEDER™, IPAT (Integrated Parallel Acquisition Techniques - derived of GRAPPA a k-space based technique) and mSENSE (an image based enhanced version of SENSE).
 
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Further Reading:
  Basics:
Parallel MRI Using Multiple Receiver Coils
   by www-math.mit.edu    
Coil Arrays for Parallel MRI: Introduction and Overview.
   by www.mr.ethz.ch    
  News & More:
Cardiac MRI Becoming More Widely Available Thanks to AI and Reduced Exam Times
Wednesday, 19 February 2020   by www.dicardiology.com    
The Effects of Breathing Motion on DCE-MRI Images: Phantom Studies Simulating Respiratory Motion to Compare CAIPIRINHA-VIBE, Radial-VIBE, and Conventional VIBE
Tuesday, 7 February 2017   by www.kjronline.org    
Implementation of Dual-Source RF Excitation in 3 T MR-Scanners Allows for Nearly Identical ADC Values Compared to 1.5 T MR Scanners in the Abdomen
Wednesday, 29 February 2012   by www.plosone.org    
Clinical evaluation of a speed optimized T2 weighted fast spin echo sequence at 3.0 T using variable flip angle refocusing, half-Fourier acquisition and parallel imaging
Wednesday, 25 October 2006
MRI Resources 
Pathology - Guidance - Jobs pool - Diffusion Weighted Imaging - Portals - Online Books
 
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