EE16B Designing Information Devices and Systems II

Transcription

1 EE16B Designing Information Devices and Systems II Lecture 15A (!) DFT The end MRI

2 Properties of the DFT Scaling and superposition:

3 Properties of the DFT Parseval s relation (Energy conservation)

4 Conjugate Symmetry When the DFT coefficients satisfy: Proof concept: Use properties of: W N (N-k) = (W N k) *, DFT and the realness of x

5 Example

6 Example FFTSHIFT

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

8 Example

9 Modulation and Circular shift Modulation Circular shift Similarly, circular shift - modulation

10 DFT Matrix and Circulant Matrices DFT diagonalizes Circulant matrices:

11 DFT Matrix and Circulant Matrices

12 Fast Circulant Matrix Vector Multiplication Given : If, then,

13 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: N 2 = 1,048,576 whereas, N log N = 10240

14 Fast Convolution Sum using the DFT We can write linear operators on finite sequences as matrix vector multiplication Recall convolution sum...

15 Graphical Example of Convolution

16 Graphical Example of Convolution

17 Graphical Example of Convolution

18 Graphical Example of Convolution

19 Graphical Example of Convolution

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

21 Example

22 Example

23 Example

24 Example This matrix is called a Toeplitz matrix But.. Not square not circulant...

25 Example Convert system to be square circulant by zero-padding Now can compute using the DFT!

26 General Case for Convolution Sum Given: Zeropad both to M+N-1 Compute: Finally:

27 Spectrum of filtering? 0.2 Example:

28 Where from here.

29 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 Wang Academic Interns: David Dongwon Kim, Vin Ramamurti, Tanner Yamada, Carolyn Schwendeman, Akash Velu, Mikaela Frichtel, Steven Lu, Grace Park Sun

30 MRI vs CT MRI is VERY VERY different from CT CT MRI Based on Magnetism No moving parts No ionizing radiation Sensitive to soft tissue Complicated to operate Based on X-ray Rapidly moving parts Uses ionizing radiation Less sensitive to soft tissue Easy to operate M. Lustig, EECS UC Berkeley

31 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

32 Polarization Protons have a magnetic moment Protons have spins Like rotating magnets 1H M. Lustig, EECS UC Berkeley

33 Polarization Body has a lot of protons In a strong magnetic field B0, spins align with B0 giving a net magnetization B 0 M. Lustig, EECS UC Berkeley *Graphic rendering Bill Overall

34 Polarizing Magnet 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 B 0 M. Lustig, EECS UC Berkeley

35 Typical MRI Scanner

36

37 Polarizaion M. Lustig, EECS UC Berkeley

38 Free Precession Much like a spinning top Frequency proportional to the field f = 3T M. Lustig, EECS UC Berkeley MIT physics demos

39 Free Precession Precession induces magnetic flux Flux induces voltage in a coil Signal M. Lustig, EECS UC Berkeley

40

41 Frequency Demo M. Lustig, EECS UC Berkeley

42 Intro to MRI - The NMR signal Signal from 1 H (mostly water) Magnetic field Magnetization B 0 Radio frequency Excitation Frequency Magnetic field time frequency γb 0 M. Lustig, EECS UC Berkeley

43 Intro to MRI - The NMR signal Signal from 1 H (mostly water) Magnetic field Magnetization B 0 Radio frequency Excitation Frequency Magnetic field time frequency γb 0 M. Lustig, EECS UC Berkeley

44 Intro to MRI - Imaging B 0 Missing spatial information B 0 center time frequency γb 0 M. Lustig, EECS UC Berkeley

45 Phone Imaging I M. Lustig, EECS UC Berkeley

46 Intro to MRI - Imaging B 0 Missing spatial information Add gradient field, G B 0 G time Weaker field center unchanged frequency Stronger field γb 0 M. Lustig, EECS UC Berkeley

47 Intro to MRI - Imaging B 0 Missing spatial information Add gradient field, G Mapping: spatial position frequency B 0 G center time frequency 0 Reference (γb 0 ) M. Lustig, EECS UC Berkeley

48 Phone Imaging II M. Lustig, EECS UC Berkeley

49 MR Imaging Fourier magnitude k-space (Raw Data) Image Fourier transform M. Lustig, EECS UC Berkeley Video courtesy Brian Hargreaves