Assume object does not vary in y Signal equation for constant Gx gradient, no y gradient Assume object does not vary in y How does s(t) and F(m) relate? For F(m), look first at F(0) and then F(1)? How does the phase across f(n) applied by the kernel change between these 2 Fourier points? Between F(2) and F(1)?

Lecture 5 MR: Spatial Resolution, etc. Readout vs. Phase Encode Recall, Readout Gx t t Phase Encode Gy Gy t ty Each phase encode adds an incremental phase shift across the FOV. Phase shift over FOVy

Phase encode number 0, 1, 2, 3, 4, 5, 6 y=FOV/2 Turn off the readout gradient and look at the phase of individual signal spins at 5 locations along the y axis. We will first look at the ky=0 phase encode and then move up in k-space. The phase of the spins at each y level are written for each phase encode below. Phase encode number 0, 1, 2, 3, 4, 5, 6 6 5 4 3 2 1 Gy t y=FOV/2 0, p, 2p, 3p, 4p, 5p, 6p y=FOV/4 0, p/2, p, 3 p/2, 2p, 5p/2, 3p y=0 0, 0, 0 , 0, 0 , 0, 0 y=-FOV/4 0, -p/2, p,-3 p/2,-2p,-5p/2,-3p y=-FOV/2 0, -p, -2p, -3p, -4p, -5p, 6p

Readout Direction Note: Each excitation line across kx (fast) Next measure sample new ky (slow) For a given time sampling rate, use an anti-aliasing filter (LPF) I(t) sreceive LPF A/D cos(ot) Bandwidth is set so it’s width is

Readout Direction (2) We should never have aliasing in readout direction. FOV is free in the readout direction. Note: If we have hardware (receiver speed), we can increase the sampling rate, increase the low pass filter cutoff, to reduce t - If we reduce kx, what happens to FOV? - SNR effects? - Where should we place the long axis?

Aliasing in Phase Encode Direction Referenced to 2D FT Referred to as “phase wrap” Left) ky = 1/FOV Right) sampling ky = 2/FOV

Phase encode number 0, 1, 2, 3, 4, 5, 6 y=FOV/2 Turn off the readout gradient and look at the phase of individual signal spins at 5 locations along the y axis. We will first look at the ky=0 phase encode and then move up in k-space. Phase encode number 0, 1, 2, 3, 4, 5, 6 6 5 4 3 2 1 Gy t y=FOV/2 0, p, 2p, 3p, 4p, 5p, 6p y=FOV/4 0, p/2, p, 3 p/2, 2p, 5p/2, 3p y=0 0, 0, 0 , 0, 0 , 0, 0 y=-FOV/4 0, -p/2, p,-3 p/2,-2p,-5p/2,-3p y=-FOV/2 0, -p, -2p, -3p, -4p, -5p, 6p

Aliasing in Phase Encode Direction Ignore x encoding and look at phase of individual spins after each phase encoding step when ky = 2/FOV Phase encode number 0, 1, 2, 3, 4, 5, 6 y=FOV2 0, 2 p, 4 p, 6 p , 8 p , 10 p, 12 p y=FOV/4 0, p, 2p, 3 p, 4p, 5p, 6p y=0 0, 0, 0 , 0, 0 , 0, 0 y=-FOV/4 0, -p , -2p, -3 p, -4p, -5p, -6p y=-FOV/2 0, -2p, -4p, -6p, -8p, -10p, -12p

Why didn’t neck wrap around? Sagittal Phase Wrap Phase direction Frequency Direction Why didn’t neck wrap around?

Spatial Resolution Why? Main Lobe Width = in x, in y Definition: Spatial Resolution Element True for large kx # of samples · spatial resolution = FOV

k-Space Acquisition ky kx Phase Direction One line of k-space Encode Sampled Signal DAQ kx ky Phase Direction One line of k-space acquired per TR Frequency Direction

Fast Fourier Transform  FFT

Characteristics of Rectilinear Sampling Unaliased FOV kx Spatial Resolution ky

512 x 512 8 x 8

512 x 512 16 x 16

512 x 512 32 x 32

512 x 512 64 x 64

512 x 512 128 x 128

512 x 512 256 x 256

Resolution is lower, bigger pixels Avoiding Aliasing Phase Encode Direction need “small”, but 1) If we increase Npe to compensate for reducing ky scan time increases Resolution depends on FOV increases 2) Same Npe reducing ky reduce wky What happens? Resolution is lower, bigger pixels FOV increases same scan time

2D FT or Spin Warp Gx t Tread Gy max Gy t Why this 2? extent in kx about kx = 0 Gy max Gy t Why this 2?

Fundamental Limits – What are they? Limits to Resolution - Max gradient strength (<4 G/cm) (<1 G/cm) - Readout duration As Tread , T2 decay blurs - FOVy vs. For fixed scan time, improved resolution costs in FOV. - Sequence-dependent effects Gibbs Ringing Fundamental Limits – What are they?