title: Derivative encoding for parallel magnetic resonance imaging.
abstract: OBJECTIVE
  To introduce a linear shift-invariant relationship between the partial derivatives of k space signals acquired using multichannel receive coils and to demonstrate that k space derivatives can be used for image unwrapping.
  
  
  METHODS
  Fourier transform of k space derivatives contains information on the spatial origins of aliased pixels; therefore, images can be reconstructed by k space derivatives. Fully sampled phantom and brain images acquired at 3 T using a standard eight channel receive coil were used to validate the k space derivatives theorem by unwrapping aliased images.
  
  
  RESULTS
  Derivative encoding leads to new methods for parallel imaging reconstruction in both k space and image domains. Noise amplification in sensitivity encoding image reconstruction, which is considered to produce the optimal SNR, can be further reduced using k space derivative encoding without making any assumptions on the characteristics of the images to be reconstructed.
  
  
  CONCLUSIONS
  This work demonstrated that the partial derivative of the k space signal acquired from one coil with respect to one direction can be expressed as a sum of partial derivatives of signals from multiple coils with respect to the perpendicular k space direction(s). This relationship between the partial derivatives of k space signals is linear and shift-invariant in the Cartesian coordinate system.
authors: National Institute of Mental Health Intramural Research Program, NIH, Bethesda, MD 20892-1527, USA. shenj@intra.nimh.nih.gov
authors: Jun
authors: J
authors: Shen
mesh_terms: D000465:Algorithms; D001921:Brain / Q000473:pathology; D002138:Calibration; D003198:Computer Simulation; D005583:Fourier Analysis; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D008279:Magnetic Resonance Imaging / Q000379:methods*; D015233:Models, Statistical; D008962:Models, Theoretical; D009010:Monte Carlo Method; D019047:Phantoms, Imaging; D012815:Signal Processing, Computer-Assisted