ee16b designing information devices and systems iiee16b/fa18/lectures/lastlecture.pdf · ee16b m....
TRANSCRIPT
![Page 1: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/1.jpg)
EE16B M. Lustig, EECS UC Berkeley
EE16BDesigning Information
Devices and Systems IILecture 15A (!)
DFTThe end
MRI
![Page 2: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/2.jpg)
EE16B M. Lustig, EECS UC Berkeley
Properties of the DFT
• Scaling and superposition:
![Page 3: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/3.jpg)
EE16B M. Lustig, EECS UC Berkeley
Properties of the DFT
• Parseval’s relation (Energy conservation)
![Page 4: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/4.jpg)
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
![Page 5: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/5.jpg)
EE16B M. Lustig, EECS UC Berkeley
Example
![Page 6: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/6.jpg)
EE16B M. Lustig, EECS UC Berkeley
Example
• FFTSHIFT
![Page 7: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/7.jpg)
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!
![Page 8: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/8.jpg)
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
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250
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EE16B M. Lustig, EECS UC Berkeley
Modulation and Circular shift
Similarly, circular shift - modulation
Modulation – Circular shift
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EE16B M. Lustig, EECS UC Berkeley
DFT Matrix and Circulant Matrices
• DFT diagonalizes Circulant matrices:
![Page 11: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/11.jpg)
EE16B M. Lustig, EECS UC Berkeley
DFT Matrix and Circulant Matrices
![Page 12: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/12.jpg)
EE16B M. Lustig, EECS UC Berkeley
• If, then,
Fast Circulant Matrix Vector Multiplication
• Given :
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EE16B M. Lustig, EECS UC Berkeley
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
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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.....
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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
![Page 16: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/16.jpg)
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
![Page 17: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/17.jpg)
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
![Page 18: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/18.jpg)
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
![Page 19: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/19.jpg)
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
![Page 20: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/20.jpg)
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
![Page 21: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/21.jpg)
EE16B M. Lustig, EECS UC Berkeley
Example
1 2 3 40
1 2 3 40
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EE16B M. Lustig, EECS UC Berkeley
1 2 3 40
1 2 3 40
Example
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EE16B M. Lustig, EECS UC Berkeley
1 2 3 40
1 2 3 40
Example
![Page 24: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/24.jpg)
EE16B M. Lustig, EECS UC Berkeley
Example
• This matrix is called a Toeplitz matrix– But.. Not square… not circulant....
![Page 25: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/25.jpg)
EE16B M. Lustig, EECS UC Berkeley
Example• Convert system to be square circulant by zero-padding
• Now can compute using the DFT!
![Page 26: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/26.jpg)
EE16B M. Lustig, EECS UC Berkeley
General Case for Convolution Sum
• Given:
• Zeropad both to M+N-1
• Compute:
• Finally:
![Page 27: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/27.jpg)
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
![Page 28: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/28.jpg)
EE16B M. Lustig, EECS UC Berkeley
Where from here….
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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
![Page 30: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/30.jpg)
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
![Page 31: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/31.jpg)
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
![Page 32: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/32.jpg)
M. Lustig, EECS UC Berkeley
Polarization
•Protons have a magnetic moment•Protons have spins•Like rotating magnets
1H
![Page 33: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/33.jpg)
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
![Page 34: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/34.jpg)
• 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
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Typical MRI Scanner
![Page 36: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/36.jpg)
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M. Lustig, EECS UC Berkeley
Polarizaion
![Page 38: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/38.jpg)
M. Lustig, EECS UC Berkeley
Free Precession•Much like a spinning top•Frequency proportional to the field• f = 127MhZ @ 3T
MIT physics demos
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M. Lustig, EECS UC Berkeley
Free Precession•Precession induces magnetic flux•Flux induces voltage in a coil
Signal
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M. Lustig, EECS UC Berkeley
Frequency Demo
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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
![Page 43: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/43.jpg)
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
![Page 44: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/44.jpg)
M. Lustig, EECS UC Berkeley
Intro to MRI - Imaging
γB0
center
B0
time frequency
• B0 Missing spatial information
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M. Lustig, EECS UC Berkeley
Phone Imaging I
![Page 46: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/46.jpg)
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
![Page 47: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/47.jpg)
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
![Page 48: EE16B Designing Information Devices and Systems IIee16b/fa18/lectures/LastLecture.pdf · EE16B M. Lustig, EECS UC Berkeley EE16B Designing Information Devices and Systems II Lecture](https://reader033.vdocuments.us/reader033/viewer/2022050918/600b2f6ddc155d57173aee7d/html5/thumbnails/48.jpg)
M. Lustig, EECS UC Berkeley
Phone Imaging II
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M. Lustig, EECS UC Berkeley
MR Imagingmagnitude k-space (Raw Data) Image
Fourier transform
Video courtesy Brian Hargreaves
Fourier