Introduction to MRI 1Psy 8960, Fall ‘06

Fourier transforms

• 1D: square wave

• 2D: kx and ky

• 2D: FOV and resolution• 2D: spike artifacts• 3D

Introduction to MRI 2Psy 8960, Fall ‘06

Fourier (de)composition of a square wave

Fundamental frequency:

Fundamental + 1st harmonic:

Fundamental + 2 harmonics:

Fundamental + 3 harmonics:

Introduction to MRI 3Psy 8960, Fall ‘06

Fourier (de)composition of a square wave

16s

Introduction to MRI 4Psy 8960, Fall ‘06

The 0th Fourier component is the mean (DC)

Introduction to MRI 5Psy 8960, Fall ‘06

Even symmetry = lack of imaginary component in transform

Introduction to MRI 6Psy 8960, Fall ‘06

A real image should have symmetric k-space

Introduction to MRI 7Psy 8960, Fall ‘06

seconds cycles per second

Discrete Fourier transform: the effect of sampling rate

Introduction to MRI 8Psy 8960, Fall ‘06

Discrete Fourier transform: the effect of sampling window

seconds cycles per second

Introduction to MRI 9Psy 8960, Fall ‘06

Fourier relationships

• Big step size in one domain = small FOV in the other• Large extent (FOV) in one domain = small step size in the

other• Multiplication in one domain = convolution in the other• Symmetry in one domain = no imaginary part in the other

Introduction to MRI 10Psy 8960, Fall ‘06

Time domain Frequency domain

seconds cycles per second

realimag

Introduction to MRI 11Psy 8960, Fall ‘06

Time domain Frequency domain

seconds cycles per second

realimag

Introduction to MRI 12Psy 8960, Fall ‘06

Time domain Frequency domain

seconds cycles per second

Introduction to MRI 13Psy 8960, Fall ‘06

Time domain Frequency domain

seconds cycles per second

Introduction to MRI 14Psy 8960, Fall ‘06

Time domain Frequency domain

seconds cycles per second

Introduction to MRI 15Psy 8960, Fall ‘06

original image filtered with gaussian filter filtered with hard filter