G16.4427 Practical MRI 1 – 3rd February 2015

G16.4427 Practical MRI 1

The Fourier Transform

G16.4427 Practical MRI 1 – 3rd February 2015

Development of Fourier Analysis• In 1748 Leonhard Euler used linear combinations of “normal

modes” to describe the motion of a vibrating string– If the configuration at some point in time is a linear combination

of normal modes, so is the configuration at any subsequent time– The coefficients of the linear combination at a the later time

could be calculated from the coefficients at the earlier time• For Bernoulli (1753) all physical motion of a string could be

described as linear combinations of normal modes, whereas Lagrange (1759) strongly criticized the use of trigonometric series because they could not represent string configurations with corners. Euler himself discarded them.

G16.4427 Practical MRI 1 – 3rd February 2015

Jean Baptiste Joseph Fourier

French mathematicians, physicists and politicians

21st March 1768 - 16th May 1830

* In 1827 he described for the first time global warming due to greenhouse effect phenomenon

G16.4427 Practical MRI 1 – 3rd February 2015

Jean Baptiste Joseph Fourier• While serving as a prefect for Napoleon he developed his

ideas on trigonometric series– By 1807 he had shown that series of harmonically related sinusoids

could describe the temperature distribution through a body (i.e. Fourier series)

– He claimed that any periodic signal could be represented by such a series (Question: is it true?)

His paper on trigonometric series was never published because one of the reviewers (Lagrange) rejected it

• He also obtained a representation of aperiodic signals as weighted sums of sinusoids that are not all harmonically related (i.e. Fourier transform)

• In 1965 Cooley and Tukey independently discovered the fast Fourier transform (FFT) algorithm (although it was found in Gauss’ notebooks), which made many applications practical

G16.4427 Practical MRI 1 – 3rd February 2015

Jean Baptiste Joseph Fourier• While serving as a prefect for Napoleon he developed his

ideas on trigonometric series– By 1807 he had shown that series of harmonically related sinusoids

could describe the temperature distribution through a body (i.e. Fourier series)

– He claimed that any periodic signal could be represented by such a series (Question: is it true?)

ANSWER: No, it’s true only if the convergence conditions (there are several theorems about that) are met

• He also obtained a representation of aperiodic signals as weighted sums of sinusoids that are not all harmonically related (i.e. Fourier transform)

• In 1965 Cooley and Tukey independently discovered the fast Fourier transform (FFT) algorithm (although it was found in Gauss’ notebooks), which made many applications practical

G16.4427 Practical MRI 1 – 3rd February 2015

Jean Baptiste Joseph Fourier• While serving as a prefect for Napoleon he developed his

ideas on trigonometric series– By 1807 he had shown that series of harmonically related sinusoids

could describe the temperature distribution through a body (i.e. Fourier series)

– He claimed that any periodic signal could be represented by such a series (False!)

– His paper on trigonometric series was never published because one of the reviewers (Lagrange) rejected it

• He also obtained a representation of aperiodic signals as weighted sums of sinusoids that are not all harmonically related (i.e. Fourier transform)

• In 1965 Cooley and Tukey independently discovered the fast Fourier transform (FFT) algorithm (although it was found in Gauss’ notebooks), which made many applications practical

G16.4427 Practical MRI 1 – 3rd February 2015

Fourier and MRI

Analog-to-DigitalConverter(ADC)

SampleK-space(Fourier series)

HostComputer

DiscreteFourier Transform

G16.4427 Practical MRI 1 – 3rd February 2015

Spatial Encoding in MRI

…

G16.4427 Practical MRI 1 – 3rd February 2015

Standard MR Image Acquisition

......

G16.4427 Practical MRI 1 – 3rd February 2015

SMASH: Combining Coil and Gradient Encoding

y

constant

cosΔkyy

sinΔkyy

cos2Δkyy

sin2Δkyy

Sodickson DK, Manning WJ. Magn Reson Med 1997; 38: 591-603

G16.4427 Practical MRI 1 – 3rd February 2015

The Fourier Transform

ρ(x) = spatial function S(k) = frequency spectrum of ρ(x)

k = spatial frequency (cycles per unit distance)

magnitude

phase In MRI, S(k) are the experimental datameasured in the Fourier space (k-space),whereas ρ(x) is the desired image function(e.g. spin density)

G16.4427 Practical MRI 1 – 3rd February 2015

Two-Dimensional Fourier Transform

Higher-dimensional Fourier transforms can be expressed as sequential one-dimensional transforms along each dimension:

G16.4427 Practical MRI 1 – 3rd February 2015

PropertiesUniqueness:

Linearity:

Shifting theorem:

Conjugate symmetry:

Scaling property:

Derivative:

G16.4427 Practical MRI 1 – 3rd February 2015

Convolution Theorem

Substitute t = x – z, then dt = dx:

G16.4427 Practical MRI 1 – 3rd February 2015

Any questions?

G16.4427 Practical MRI 1 – 3rd February 2015

Discrete-Time Fourier Transform

As is periodic of period 2π, the interval of integrationcan be taken as any interval of length 2π

G16.4427 Practical MRI 1 – 3rd February 2015

ExampleFind the discrete-time Fourier transform of the following rectangular pulse:

1

k0

……-N1 N1

G16.4427 Practical MRI 1 – 3rd February 2015

Example (Continue)From the solution of the geometric series we saw last time:

Substituting in the previous equation:

G16.4427 Practical MRI 1 – 3rd February 2015

Example (Continue)

1

k0

……-2 2

ω0

……

π-π-2π 2π

5

For N1 = 2

G16.4427 Practical MRI 1 – 3rd February 2015

Discrete Fourier Transform (DFT)For signals of finite duration:

The DFT corresponds to samples of the discrete-time Fouriertransform taken every 2π/N:

G16.4427 Practical MRI 1 – 3rd February 2015

Two-Dimensional DFT

For example, in MRI, we use the 2D DFT to reconstruct an image (i.e. a slice) from k-space samples

G16.4427 Practical MRI 1 – 3rd February 2015

Fast Fourier Transform (FFT)• Direct calculation of the 1D DFT requires O(N2)

operations, as there are N output X[k] and each one requires the sum of N terms

• FFT is a method to compute the same results in O(NlogN) operations– There are several algorithms to perform the FFT– The basic idea is to decompose the problem in

smaller subproblems. E.g. For an N-point DTF, when = N1N2, the Cooley-Tukey algorithm first computes N1 transforms of size N2, then N2 transform of size N1.

G16.4427 Practical MRI 1 – 3rd February 2015

MR Image Reconstruction With 2D FFT • Since the MR data is the Fourier transform of the

image, we can reconstruct the image with an inverse Fourier transform

• To reconstruct a slice, we need the Matlab function fft2, which returns the 2D DTF

• As FFT assumes that the origin of both image and k-space is at sample (1,1), we need to use the fftshift Matlab function in order to move the origin in the center of the matrix

G16.4427 Practical MRI 1 – 3rd February 2015

FFTSHIFT And IFFTSHIFT

fftshift rearranges the ouput of fft2 by moving the zero-frequency component from the upper-left corner to the center of the matrix

1 2

34ifftshift undoes the result of fftshift

To reconstruct an image i from a 2D matrix of MR data d:

i = ifftshift ( ifft2 ( fftshift ( d ) ) )

G16.4427 Practical MRI 1 – 3rd February 2015

FFTSHIFT And IFFTSHIFT

i = ifftshift ( ifft2 ( fftshift ( d ) ) )

N x N k-space data (d)

i = ifft2 ( d )

G16.4427 Practical MRI 1 – 3rd February 2015

Any questions?

G16.4427 Practical MRI 1 – 3rd February 2015

The Radon Transform• It is named after the Austrian mathematician

Johann Karl August Radon (1887-1956)• It computes the projection (i.e. line integrals)

of a function (e.g. an image) along a set of angles

• The 2D Radon transform is a mapping from the Cartesian rectangular coordinates (x,y) to a distance and an angle (ρ,ϕ), also known as polar coordinates

G16.4427 Practical MRI 1 – 3rd February 2015

Two-Dimensional Radon Transform

The line (ray) integral path L is:

= projection angle

Another more convenient form of the 2D Radon transform is:

G16.4427 Practical MRI 1 – 3rd February 2015

The Central Section Theorem• The Radon transform is closely related to the

Fourier transform by the central section (or projection-slice) theorem:

• The theorem relates a one-dimensional projection to line of data in k-space

For a function ρ(x,y), the one-dimensional Fourier transform of the projection P(p,ϕ) along the p-axis for a fixed projection angle ϕ is identical to the 2D Fourier transform of ρ(x,y) evaluated along a line passing through the origin with the same orientation angle in the Fourier space

G16.4427 Practical MRI 1 – 3rd February 2015

Example

Original Image256 x 256

Sinogram (256 scans)

(45 angular views)

(180 angular views)

G16.4427 Practical MRI 1 – 3rd February 2015

See you on Monday!