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# 240 spatial encoding

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### 240 spatial encoding

1. 1. Topics • spatial encoding - part 1
2. 2. K-space, the path to MRI.K-space, the path to MRI. ENTER IF YOU DAREENTER IF YOU DARE
3. 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. 4. k-space and the MR Image x y f(x,y) kx ky K-spaceK-space F(kx,ky) Image-spaceImage-space
5. 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. 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. 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. 8. Low Spatial Frequency
9. 9. Higher Spatial Frequency
10. 10. low spatial frequencies high spatial frequencies all frequencies
11. 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. 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. 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. 14. Cosine Waves of different amplitudes -4 -3 -2 -1 0 1 2 3 4
15. 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. 16. k-space Representation of Waves image space, f=4 k-space -128 -96 -64 -32 0 32 64 96 128
17. 17. k-space Representation of Waves image space, f=16 k-space -128 -96 -64 -32 0 32 64 96 128
18. 18. k-space Representation of Waves image space, f=64 k-space -128 -96 -64 -32 0 32 64 96 128
19. 19. Complex Waveform Synthesis f4 + 1/2 f16 + 1/4 f32 Complex waveforms can be synthesized by adding simple waves together.
20. 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. 21. k-space Representation of Complex Waves “square” wave image space k-space -128 -96 -64 -32 0 32 64 96 128
22. 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. 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. 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. 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. 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. 27. RF signal A/D conversion image space FT k-space
28. 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. 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. 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. 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. 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.