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- 1. Topics • spatial encoding - part 1
- 2. K-space, the path to MRI.K-space, the path to MRI. ENTER IF YOU DAREENTER 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. k-space and the MR Image x y f(x,y) kx ky K-spaceK-space F(kx,ky) Image-spaceImage-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. Discrete Fourier Transform F(kx,ky) is the 2D discrete Fourier transform of the image f(x,y) x y f(x,y) kx ky ℑ K-space F(kx,ky) f x y N F k k e xk yk kk x y j N x j N yNN yx ( , ) ( , )= + = − = − ∑∑ 1 2 2 2 0 1 0 1 π π image-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 (φ) –amplitude (A) f x A f x( ) cos ( )= +2π φ
- 13. Cosine Waves of different frequencies -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
- 14. Cosine Waves of different amplitudes -4 -3 -2 -1 0 1 2 3 4
- 15. Cosine Waves of different phases -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
- 16. k-space Representation of Waves image space, f=4 k-space -128 -96 -64 -32 0 32 64 96 128
- 17. k-space Representation of Waves image space, f=16 k-space -128 -96 -64 -32 0 32 64 96 128
- 18. k-space Representation of Waves image space, f=64 k-space -128 -96 -64 -32 0 32 64 96 128
- 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 f4 + 1/2 f16 + 1/4 f32 -128 -96 -64 -32 0 32 64 96 128 image space k-space
- 21. k-space Representation of Complex Waves “square” wave image space k-space -128 -96 -64 -32 0 32 64 96 128
- 22. Reconstruction of square wave from truncated k-space truncated space (16) image space k-space -128 -96 -64 -32 0 32 64 96 128 reconstructed waveform
- 23. Reconstruction of square wave from truncated k-space truncated space (8) image space k-space -128 -96 -64 -32 0 32 64 96 128 reconstructed waveform
- 24. Reconstruction of square wave from truncated k-space truncated space (240) image space k-space -128 -96 -64 -32 0 32 64 96 128 reconstructed waveform
- 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 A/D conversion image space FT 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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