G16.4427 Practical MRI 1 The Fourier Transform It’s a different representation for signals and LTI systems

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.

Jean Baptiste Joseph Fourier 21st March 1768 - 16th May 1830 French mathematicians, physicists and politicians * In 1827 he described for the first time global warming due to greenhouse effect phenomenon

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

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

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

Fourier and MRI Analog-to-Digital Host Computer Converter (ADC) Sample K-space (Fourier series) Host Computer Discrete Fourier Transform

Spatial Encoding in MRI …

Standard MR Image Acquisition ... ...

SMASH: Combining Coil and Gradient Encoding y constant cosΔkyy sinΔkyy cos2Δkyy sin2Δkyy Sodickson DK, Manning WJ. Magn Reson Med 1997; 38: 591-603

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 data measured in the Fourier space (k-space), whereas ρ(x) is the desired image function (e.g. spin density)

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

Properties Uniqueness: Linearity: Shifting theorem: Conjugate symmetry: Scaling property: Derivative:

Convolution Theorem Substitute t = x – z, then dt = dx:

Any questions?

Discrete-Time Fourier Transform As is periodic of period 2π, the interval of integration can be taken as any interval of length 2π

Example Find the discrete-time Fourier transform of the following rectangular pulse: 1 … … k -N1 N1

Example (Continue) From the solution of the geometric series we saw last time: Substituting in the previous equation:

Example (Continue) 1 k … -2 2 For N1 = 2 5 … … ω -2π -π π 2π

Discrete Fourier Transform (DFT) For signals of finite duration: The DFT corresponds to samples of the discrete-time Fourier transform taken every 2π/N:

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

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.

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

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 3 4 ifftshift undoes the result of fftshift To reconstruct an image i from a 2D matrix of MR data d: i = ifftshift ( ifft2 ( fftshift ( d ) ) )

FFTSHIFT And IFFTSHIFT i = ifft2 ( d ) N x N k-space data (d) i = ifftshift ( ifft2 ( fftshift ( d ) ) )

Any questions?

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

Two-Dimensional Radon Transform The line (ray) integral path L is: = projection angle Another more convenient form of the 2D Radon transform is:

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

Example Sinogram (256 scans) Original Image 256 x 256 (45 angular views) (180 angular views)

See you on Monday!