EE16B M. Lustig, EECS UC Berkeley

EE16BDesigning Information

Devices and Systems IILecture 15A (!)

DFTThe end

MRI

EE16B M. Lustig, EECS UC Berkeley

Properties of the DFT

• Scaling and superposition:

EE16B M. Lustig, EECS UC Berkeley

Properties of the DFT

• Parseval’s relation (Energy conservation)

EE16B M. Lustig, EECS UC Berkeley

Conjugate Symmetry

• When the DFT coefficients satisfy:

0 1 2 3 40 1 2 3

Proof concept: Use properties of: WN(N-k) = (WN

k)*, DFT and the realness of x

EE16B M. Lustig, EECS UC Berkeley

Example

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Example

• FFTSHIFT

EE16B M. Lustig, EECS UC Berkeley

Example

• Often, display only half (if signal is real)• If the spectrum (DFT) is not conjugate symmetric, then the signal is complex!

EE16B M. Lustig, EECS UC Berkeley

Example

0 500 1000 1500 2000 2500 3000 3500 40000

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500 4000 45000

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000 6000 7000 80000

50

100

150

200

250

EE16B M. Lustig, EECS UC Berkeley

Modulation and Circular shift

Similarly, circular shift - modulation

Modulation – Circular shift

EE16B M. Lustig, EECS UC Berkeley

DFT Matrix and Circulant Matrices

• DFT diagonalizes Circulant matrices:

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DFT Matrix and Circulant Matrices

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• If, then,

Fast Circulant Matrix Vector Multiplication

• Given :

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Fast Circulant Matrix Vector Multiplication

• Why bother?• Option I, compute:

• Option II, compute:

Using the fast Fourier Transform (FFT) calculation of the DFT (and inverse) is O(N log N)

For N = 1024: N2 = 1,048,576 whereas, N log N = 10240

EE16B M. Lustig, EECS UC Berkeley

Fast Convolution Sum using the DFT

• We can write linear operators on finite sequences as matrix vector multiplication

• Recall… convolution sum.....

EE16B M. Lustig, EECS UC Berkeley

Graphical Example of Convolution

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

EE16B M. Lustig, EECS UC Berkeley

Graphical Example of Convolution

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

EE16B M. Lustig, EECS UC Berkeley

Graphical Example of Convolution

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

EE16B M. Lustig, EECS UC Berkeley

Graphical Example of Convolution

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

EE16B M. Lustig, EECS UC Berkeley

Graphical Example of Convolution

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

1 2 3 4 5 6 70 8 9 10-1

EE16B M. Lustig, EECS UC Berkeley

Example:If h[n] is length 2 and x[n] is length 5, what is thelength of their convolution sum?

1 2 3 401 2 3 40 1 2 3 4 50

EE16B M. Lustig, EECS UC Berkeley

Example

1 2 3 40

1 2 3 40

EE16B M. Lustig, EECS UC Berkeley

1 2 3 40

1 2 3 40

Example

EE16B M. Lustig, EECS UC Berkeley

1 2 3 40

1 2 3 40

Example

EE16B M. Lustig, EECS UC Berkeley

Example

• This matrix is called a Toeplitz matrix– But.. Not square… not circulant....

EE16B M. Lustig, EECS UC Berkeley

Example• Convert system to be square circulant by zero-padding

• Now can compute using the DFT!

EE16B M. Lustig, EECS UC Berkeley

General Case for Convolution Sum

• Given:

• Zeropad both to M+N-1

• Compute:

• Finally:

EE16B M. Lustig, EECS UC Berkeley

Spectrum of filtering?

• Example:

0 5 10 15 20 25 30 35-0.05

0

0.05

0.1

0.15

0.2

EE16B M. Lustig, EECS UC Berkeley

Where from here….

EE16B M. Lustig, EECS UC Berkeley

Readers: Sherwin Afshar, Ryan Tjitro, Sicen Luan, Mohammed Shaikh, Khanh Dang Lab Assistants: Tiffany Cappellari, Jenna Wen, Alyssa Huang, Bikramjit Kukreja, Justin Lu, Nirmaan Shanker, Nathan Mar, Rahul Tewari, Daniel Zu, Raymond Gu, Ilya Chugunov, Chufan Liang, Dre Mahaarachchi, Benjamin Carlson, Victor Lee, Rafael Calleja, Jackson Paddock, David Allan Vakshlyak, Christian Castaneda, Ashley Lin, Jason Wang, Jove Yuan, Ryan Purpura, Merryle Wang, Avanthika Ramesh, Adilet Pazylkarim, Mariyam Jivani, Angela WangAcademic Interns: David Dongwon Kim, Vin Ramamurti, Tanner Yamada, Carolyn Schwendeman, Akash Velu, Mikaela Frichtel, Steven Lu, Grace Park Sun

M. Lustig, EECS UC Berkeley

MRI vs CT• MRI is VERY VERY different from CT

MRICTBased on MagnetismNo moving partsNo ionizing radiationSensitive to soft tissueComplicated to operate

Based on X-rayRapidly moving partsUses ionizing radiationLess sensitive to soft tissueEasy to operate

M. Lustig, EECS UC Berkeley

How Does MRI Work?

•Magnetic Polarization-- Very strong uniform magnet• Excitation

-- Very powerful RF transmitter• Acquisition

-- Location is encoded by gradient magnetic fields-- Very powerful audio amps

M. Lustig, EECS UC Berkeley

Polarization

•Protons have a magnetic moment•Protons have spins•Like rotating magnets

1H

M. Lustig, EECS UC Berkeley

Polarization

• Body has a lot of protons• In a strong magnetic field B0, spins align with B0 giving a net

magnetization

B0

*Graphic rendering Bill Overall

• 0.1 to 12 Tesla• 0.5 to 3 T common• 1 T is 10,000 Gauss• Earth’s field is 0.5G• Typically a superconducting

magnet

M. Lustig, EECS UC Berkeley

Polarizing Magnet

B0

Typical MRI Scanner

M. Lustig, EECS UC Berkeley

Polarizaion

M. Lustig, EECS UC Berkeley

Free Precession•Much like a spinning top•Frequency proportional to the field• f = 127MhZ @ 3T

MIT physics demos

M. Lustig, EECS UC Berkeley

Free Precession•Precession induces magnetic flux•Flux induces voltage in a coil

Signal

M. Lustig, EECS UC Berkeley

Frequency Demo

M. Lustig, EECS UC Berkeley

Intro to MRI - The NMR signalB0

γB0

time frequency

• Signal from 1H (mostly water)

• Magnetic field ⇒ Magnetization

• Radio frequency ⇒ Excitation

• Frequency ∝ Magnetic field

M. Lustig, EECS UC Berkeley

Intro to MRI - The NMR signal• Signal from 1H (mostly water)

• Magnetic field ⇒ Magnetization

• Radio frequency ⇒ Excitation

• Frequency ∝ Magnetic field

γB0

B0

time frequency

γB0

M. Lustig, EECS UC Berkeley

Intro to MRI - Imaging

γB0

center

B0

time frequency

• B0 Missing spatial information

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Phone Imaging I

M. Lustig, EECS UC Berkeley

Intro to MRI - ImagingB0

center

γB0

G

Stronger field

Weaker field

unchanged

time frequency

• B0 Missing spatial information

• Add gradient field, G

M. Lustig, EECS UC Berkeley

Intro to MRI - ImagingB0

Reference (γB0)

center

time

• B0 Missing spatial information

• Add gradient field, G • Mapping:

spatial position ⇒ frequency

G

0

frequency

M. Lustig, EECS UC Berkeley

Phone Imaging II

M. Lustig, EECS UC Berkeley

MR Imagingmagnitude k-space (Raw Data) Image

Fourier transform

Video courtesy Brian Hargreaves

Fourier