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Topics. spatial encoding - part 1. K-space, the path to MRI. ENTER IF YOU DARE. What is k-space?. a mathematical device not a real “space” in the patient nor in the MR scanner key to understanding spatial encoding of MR images. y. x. f(x,y). k-space and the MR Image. k y. k x.

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  1. Topics • spatial encoding - part 1

  2. K-space, the path to MRI. ENTER IF YOU DARE

  3. What is k-space? • a mathematical device • not a real “space” in the patient nor in the MR scanner • key to understanding spatial encoding of MR images

  4. y x f(x,y) k-space and the MR Image ky kx F(kx,ky) Image-space K-space

  5. k-space and the MR Image • each individual point in the MR image is reconstructed from every point in the k-space representation of the image • like a card shuffling trick: you must have all of the cards (k-space) to pick the single correct card from the deck • all points of k-space must be collected for a faithful reconstruction of the image

  6. y x f(x,y) Discrete Fourier Transform F(kx,ky) is the 2D discrete Fourier transform of the image f(x,y) ky  kx F(kx,ky) image-space K-space

  7. k-space and the MR Image • If the image is a 256 x 256 matrix size, then k-space is also 256 x 256 points. • The individual points in k-space represent spatial frequencies in the image. • Contrast is represented by low spatial frequencies; detail is represented by high spatial frequencies.

  8. Low Spatial Frequency

  9. Higher Spatial Frequency

  10. low spatial frequencies high spatial frequencies all frequencies

  11. Spatial Frequencies • low frequency = contrast • high frequency = detail • The most abrupt change occurs at an edge. Images of edges contain the highest spatial frequencies.

  12. Waves and Frequencies • simplest wave is a cosine wave • properties • frequency (f) • phase () • amplitude (A)

  13. Cosine Waves ofdifferent frequencies

  14. Cosine Waves ofdifferent amplitudes

  15. Cosine Waves ofdifferent phases

  16. k-space Representation of Waves image space, f=4 k-space

  17. k-space Representation of Waves image space, f=16 k-space

  18. k-space Representation of Waves image space, f=64 k-space

  19. Complex Waveform Synthesis f4 + 1/2 f16 + 1/4 f32 Complex waveforms can be synthesized by adding simple waves together.

  20. k-space Representation of Complex Waves image space k-space f4 + 1/2 f16 + 1/4 f32

  21. k-space Representation of Complex Waves image space k-space “square” wave

  22. Reconstruction of square wave from truncated k-space image space k-space reconstructed waveform truncated space (16)

  23. Reconstruction of square wave from truncated k-space image space k-space reconstructed waveform truncated space (8)

  24. Reconstruction of square wave from truncated k-space image space k-space reconstructed waveform truncated space (240)

  25. Properties of k-space • k-space is symmetrical • all of the points in k-space must be known to reconstruct the waveform faithfully • truncation of k-space results in loss of detail, particularly for edges • most important information centered around the middle of k-space • k-space is the Fourier representation of the waveform

  26. MRI and k-space • The nuclei in an MR experiment produce a radio signal (wave) that depends on the strength of the main magnet and the specific nucleus being studied (usually H+). • To reconstruct an MR image we need to determine the k-space values from the MR signal.

  27. RF signal FT A/D conversion image space k-space

  28. MRI • Spatial encoding is accomplished by superimposing gradient fields. • There are three gradient fields in the x, y, and z directions. • Gradients alter the magnetic field resulting in a change in resonance frequency or a change in phase.

  29. MRI • For most clinical MR imagers using superconducting main magnets, the main magnetic field is oriented in the z direction. • Gradient fields are located in the x, y, and z directions.

  30. MRI • The three magnetic gradients work together to encode the NMR signal with spatial information. • Remember:the resonance frequency depends on the magnetic field strength. Small alterations in the magnetic field by the gradient coils will change the resonance frequency.

  31. Gradients • Consider the example of MR imaging in the transverse (axial) plane. • Z gradient: slice select • X gradient: frequency encode (readout) • Y gradient: phase encode

  32. Slice Selection • For axial imaging, slice selection occurs along the long axis of the magnet. • Superposition of the slice selection gradient causes non-resonance of tissues that are located above and below the plane of interest.

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