EE16B Designing Information Devices and Systems II Lecture 15A (!) DFT The end MRI
Properties of the DFT Scaling and superposition:
Properties of the DFT Parseval s relation (Energy conservation)
Conjugate Symmetry When the DFT coefficients satisfy: 0 1 2 3 0 1 2 3 4 Proof concept: Use properties of: W N (N-k) = (W N k) *, DFT and the realness of x
Example
Example FFTSHIFT
Example Often, display only half (if signal is real) If the spectrum (DFT) is not conjugate symmetric, then the signal is complex!
Example 120 100 80 60 40 20 0 0 500 1000 1500 2000 2500 3000 3500 4000 80 70 60 50 40 30 20 10 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 250 200 150 100 50 0 0 1000 2000 3000 4000 5000 6000 7000 8000
Modulation and Circular shift Modulation Circular shift Similarly, circular shift - modulation
DFT Matrix and Circulant Matrices DFT diagonalizes Circulant matrices:
DFT Matrix and Circulant Matrices
Fast Circulant Matrix Vector Multiplication Given : If, then,
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
Fast Convolution Sum using the DFT We can write linear operators on finite sequences as matrix vector multiplication Recall convolution sum...
Graphical Example of Convolution -1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10
Graphical Example of Convolution -1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10
Graphical Example of Convolution -1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10
Graphical Example of Convolution -1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10
Graphical Example of Convolution -1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10-1 0 1 2 3 4 5 6 7 8 9 10
Example: If h[n] is length 2 and x[n] is length 5, what is the length of their convolution sum? 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 5
Example 0 1 2 3 4 0 1 2 3 4
Example 0 1 2 3 4 0 1 2 3 4
Example 0 1 2 3 4 0 1 2 3 4
Example This matrix is called a Toeplitz matrix But.. Not square not circulant...
Example Convert system to be square circulant by zero-padding Now can compute using the DFT!
General Case for Convolution Sum Given: Zeropad both to M+N-1 Compute: Finally:
Spectrum of filtering? 0.2 Example: 0.15 0.1 0.05 0-0.05 0 5 10 15 20 25 30 35
Where from here.
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
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
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 B 0 M. Lustig, EECS UC Berkeley *Graphic rendering Bill Overall
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
Typical MRI Scanner
Polarizaion M. Lustig, EECS UC Berkeley
Free Precession Much like a spinning top Frequency proportional to the field f = 127MhZ @ 3T M. Lustig, EECS UC Berkeley MIT physics demos
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 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
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
Intro to MRI - Imaging B 0 Missing spatial information B 0 center time frequency γb 0 M. Lustig, EECS UC Berkeley
Phone Imaging I M. Lustig, EECS UC Berkeley
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
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
Phone Imaging II M. Lustig, EECS UC Berkeley
MR Imaging Fourier magnitude k-space (Raw Data) Image Fourier transform M. Lustig, EECS UC Berkeley Video courtesy Brian Hargreaves