2D FT Review MP/BME 574

1D to 2D Sampling Signal under analysis is periodic Signal is ‘essentially bandlimited’ Sampling rate is high enough to satisfy Nyquist criterion Other assumptions (for convenience) Signal is sampled with uniformly spaced intervals

2D Sampling/Discrete-Space Signal 1 n n2

2D Functions Impulses Step Sequences Separable Sequences Periodic Sequences

Line Impulse 1 n n2

2D step n2 1 n

2D step 1 n n2

2D Step n2 1 n

Separable Sequences n2 1 n

Periodic Sequences 1 n n2

2D Convolution h(k1,k2) x(k1,k2) k1 k2 (3) (4) (1) (2) k2 k1

2D Convolution h(-k1,-k2) k1 k2 x(k1,k2) k2 (2) (1) (4) (3) k1

2D Convolution h(4-k1,3-k2) x(k1,k2) k1 k2 k2 (2) (1) (4) (3) 3-k2 k1

2D Convolution h(n1-k1,n2-k2) x(k1,k2) k1 k2 k2 (2) (1) (4) (3) (2)

2D Convolution h(n1-k1,n2-k2) y(n1,n2) k2 n2 (2) (1) (4) (3) (7) (7) (6) (4) (6) n1 (2) k1 (1) (3) (3) n1-1

2D Convolution h(n1-k1,n2-k2) y(n1,n2) k2 n2 (2) (1) (4) (3) (7) (7) (6) (10) (4) (6) n1 (2) k1 (1) (3) (3) n1-1

2DFT Imaging in MRI MP/BME 574

Abbe’s Theory of Image Formation From Meyer-Arendt

No Magnetic Field = No Net Magnetization Random Orientation

Dipole Moments from Entire Sample Magnetic Field (B0) Magnetic Field (B0) m m Positive Orientation Negative Orientation

Precession

Precession and Electromotive Force (emf) or Voltage emf derives from Faraday’s law Time-dependent magnetic flux through a coil of wire Induces current flow Proportional to the magnetic field strength and the frequency of the field oscillation

Example x y z B1(t)

Example x y z B1(t)

Complex Voltage/Signal: General Case

rf-excitation By reciprocity, Lab Frame Rotating Frame After Haacke, 1999

Quadrature Conversion in MRI (and Ultrasound) Signal Processing Received Radio Frequency Echo Signal x(t) (fc = 10MHz; 40MS/s) X LPF xc(t) I—Channel 2 cos wct -p/2 Phase Shift -2 sin wct Q—Channel xs(t) X LPF In a high-end ultrasound/MR imaging system this conversion is done in the digital domain. In a lower-end system the conversion is done in the analog domain. Why?

Spatial Encoding

Slice Selection Ideal, non-selective rf: S(t) =rect(t/Dt) B1ideal(t)

Non-selective rf-pulse Entire Volume Excited

FTdemo: Rect modulated Cosine

FTdemo: Rect modulated Cosine

Spatial Encoding Gradients z B(r) r y x

Slice Selection Selective rf: Ssel(t) = sinc(t/t) rect(t/Dt) B1ideal(t) Apply spatial gradient simultaneous to rf-pulse.

Slice Selective rf-pulse Slice of width Dz Excited

FTdemo: Cosine modulated Sinc

Summary Spin ½ nuclei will precess in a magnetic field Bo Excite and receive signal with coils (antennae) by Faraday’s Law Complex representation of real signals Quadrature detection Reciprocity Spatial magnetic field gradients Bandwidth of precessing “spins” Non-Selective rf pulses using Fourier transform principles Shift theorem etc… applies

Spatial Encoding Gradients z B(r) r y x

Frequency Encoding f, B Df B=Bo xmin xmax FOVx

Frequency Encoding … … Recall Lab 2, Problem 4: Piano Keyboard E, 660 Hz A, 220 Hz Middle C

Frequency Encoding Time (t) FT Temporal Frequency (f) Position (x) Proportionality Temporal Frequency (f) Position (x)

Frequency Encoding

Frequency Encoding Spatial Frequency (k) Time (t) FT Temporal Frequency (f) FT Proportionality Position (x) Spatial Frequency (k)

Phase Encoding y f, B Df B=Bo xmin xmax FOVx

Phase Encoding y f, B Df B=Bo xmin xmax FOVx

Phase Encoding y f, B Df B=Bo xmin xmax FOVx

Zero gradient for time, T Phase Encoding y B ¶ Zero gradient for time, T y

Positive gradient for time, T Phase Encoding y B ¶ Positive gradient for time, T y

Positive gradient for time, T Phase Encoding y B ¶ Positive gradient for time, T y

Frequency Encoding Spatial Frequency (k-ko) Time (t) FT FT Proportionality FT FT Proportionality Temporal Frequency (f) Position (x) e-igGyT

2D Fast GRE Imaging Gx Gy Gz RF TR = 6.6 msec TE Phase Encode Dephasing/ Rewinder Asymmetric Readout Gy Dephasing/ Rewinder Gz Shinnar- LaRoux RF RF TR = 6.6 msec

2D FT y x k n Start Finish The elliptical centric view order is centric in both phase encode directions. Views are sampled based on their distance from the center of k-space. This leads to a radially symmetric distribution of contrast enhancement about the center of k-space.

3D Fast GRE Imaging Gx Gy Gz RF TR = 6.6 msec TE Phase Encode Dephasing/ Rewinder Asymmetric Readout Gy Dephasing/ Rewinder Phase Encode Gz Shinnar- LaRoux RF RF TR = 6.6 msec

3D FT k z y x Tscan =Ny Nz TR NEX i.e. Time consuming! n The elliptical centric view order is centric in both phase encode directions. Views are sampled based on their distance from the center of k-space. This leads to a radially symmetric distribution of contrast enhancement about the center of k-space.

Summary Frequency encoding Phase Entirely separable Bandwidth of precessing frequencies Phase Incremental phase in image space Implies shift in k-space Entirely separable 1D column-wise FFT 1D row-wise FFT

Navigating in 2D k-Space Goals Improve your intuition Specific examples Effects of: Apodization windowing “Zero-Padding” or Sinc interpolation Vendors refer to this as “ZIP” Sampling the corners of k-Space

Elliptical Centric View Order k z High Detail Information k y Overall Image Contrast The elliptical centric view order is centric in both phase encode directions. Views are sampled based on their distance from the center of k-space. This leads to a radially symmetric distribution of contrast enhancement about the center of k-space. Sampled Points

MRI: Image Acquisition FT FT K-space Image space

Case I Case II Case III ky kz

Case I k-space: Image Space: kz ky DFT Bernstein MA, Fain SB, and Riederer SJ, JMRI 14: 270-280 (2001)

Case II k-space: Image Space: kz ky FT

Case III k-space: Image Space: kz ky FT

a b

Zero-padding/Sinc Interpolation Recall that the sampling theorem Restoration of a compactly supported (band-limited) function Equivalent to convolution of the sampled points with a sinc function

Recovering or “Restoring” f(x) from f(n):

Recovering or “restoring” f(x) from f(n): Dx

Recovering or “Restoring” f(x) from f(n):

Recovering or “restoring” f(x) from f(n): Dx

Recovering or “restoring” f(x) from f(n): f(n’) where Dx

Case I k-space: Image Space: kz ky DFT Bernstein MA, Fain SB, and Riederer SJ, JMRI 14: 270-280 (2001)

Methods: Sampling Case I: Zero-filled k-space: Image Space: kz ky FT

Case II k-space: Image Space: kz ky FT

Case II k-space: Image Space: kz ky FT

Methods: Sampling Case III k-space: Image Space: kz ky FT

Methods: Sampling Case III: Zero-Filled k-space: Image Space: kz ky FT