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G16.4427 Practical MRI 1 – 29th January 2015
G16.4427 Practical MRI 1
Introduction to the courseMathematical fundamentals
G16.4427 Practical MRI 1 – 29th January 2015
Course Summary• Practical introduction to the basic components of
signal detection and excitation in magnetic resonance imaging (MRI)
• Organized in 3 modules (lectures + labs):– Part 1 Fundamental mathematical tools needed to
describe an MRI experiment and their implementation in Matlab
– Part 2 Basic concepts of MR pulse sequences– Part 3 Principles of RF coil design and development
G16.4427 Practical MRI 1 – 29th January 2015
Course Information• Website:
– https://alex.med.nyu.edu/portal/site/practicalmri1 • Format
– Twice per week from the 29th January to 7th May– 15 120-mins lectures, 10 240-mins labs, 2 exams– All sessions at the Center for Biomedical Imaging
• Grading policy:– Course participation (10%), midterm exam (25%), lab
projects (40%), final exam (25%)• Reference textbooks
– J. T. Vaughan and J. R. Griffiths RF coils for MRI, Wiley-Liss, 2012
– M. A. Bernstein, K. F. King and X. J. Zhou, Handbook of MRI Pulse Sequences, Academic Press, 2004
G16.4427 Practical MRI 1 – 29th January 2015
Instructors• Prof. Riccardo Lattanzi (course director)– All lectures and first lab session– [email protected] (212-263-4860)– Office hours: after class or by appointment
• Prof. Kaveh Vahedipour– Pulse sequence programming lab sessions– [email protected]
• Dr. Ryan Brown– RF coil lab sessions– [email protected]
G16.4427 Practical MRI 1 – 29th January 2015
Matlab • NYU has an institutional license• Installation instructions with license info are
also posted on the course website on ALEX• If you have one, bring your laptop with Matlab
for next class
G16.4427 Practical MRI 1 – 29th January 2015
Any questions?
G16.4427 Practical MRI 1 – 29th January 2015
Vectors• Cartesian representation
Magnitude:
Direction:
G16.4427 Practical MRI 1 – 29th January 2015
Question:Can you provide examples of
vectors quantity in MRI?
G16.4427 Practical MRI 1 – 29th January 2015
Complex Notation• In MR a complex notation is often used for 2D
vectors:
with:
with:vector of length A0 rotating counterclockwise at an angularspeed equal to ω0
G16.4427 Practical MRI 1 – 29th January 2015
Commonly Used Functions• Unit Step Function
• Rectangular Window Function
• Kronecker Delta Function
G16.4427 Practical MRI 1 – 29th January 2015
Sinc Function
• It is an even function• Zero crossings at x = ± nπ• Sinusoidal oscillation of period 2π with
amplitude decreasing continuously as 1/x
1
sinc(x)
xπ 2π-2π -π
G16.4427 Practical MRI 1 – 29th January 2015
Any questions?
G16.4427 Practical MRI 1 – 29th January 2015
Matlab Demonstration
G16.4427 Practical MRI 1 – 29th January 2015
Convolution• A concept central to Fourier theory and the
analysis of linear systems
• Symbolically often written as:
G16.4427 Practical MRI 1 – 29th January 2015
Properties of Convolution• Commutativity
• Associativity
• Distributivity
• Differentiation:
G16.4427 Practical MRI 1 – 29th January 2015
Example• Calculate
• Differentiation property:
• Fundamental theorem of calculus:
• Then:
1
x1/2-1/2
G16.4427 Practical MRI 1 – 29th January 2015
Graphical Method• Flip (or reverse) one function in time:
• Slide the flipped function over the other from –∞ to + ∞:
1
x1/2-1/2
1
x1/2-1/2
1
1/2-1/2
G16.4427 Practical MRI 1 – 29th January 2015
Graphical Method• Integrate where both functions overlap:
1
1/2-1/2 Integral equal to zero (no overlap)
1
1/2-1/2
Integral equal to:
1
1/2-1/2 Integral equal to zero (no overlap)
G16.4427 Practical MRI 1 – 29th January 2015
Graphical Method• Putting everything together
1
x1-1
G16.4427 Practical MRI 1 – 29th January 2015
Problem:
Given:
Calculate:
G16.4427 Practical MRI 1 – 29th January 2015
Matlab Demonstration
G16.4427 Practical MRI 1 – 29th January 2015
Linear System• A linear system is a system that possesses the
important property of superposition: if an input consists of the weighted sum of several signals, then the output is the superposition (i.e. the weighted sum) of the responses of the system to each of those signals– –
G16.4427 Practical MRI 1 – 29th January 2015
Linear Time-Invariant (LTI) System
• A linear system for which whether we apply an input to the system now or T seconds from now, the output will be identical except for a time delay if T seconds–
• Any LTI system can be characterized entirely by a single function called the system’s impulse response– The output of the system is simply the convolution
of the input with the impulse response.
G16.4427 Practical MRI 1 – 29th January 2015
Impulse Response• Impulse response h(t) is the response to δ(t):
• LTI system response:
G16.4427 Practical MRI 1 – 29th January 2015
Commutative and Associative Properties
• Commutative property:
• Associative property:
G16.4427 Practical MRI 1 – 29th January 2015
Distributive Property
G16.4427 Practical MRI 1 – 29th January 2015
Other Definitions• An LTI system is without memory if its output
at any time depends only on the value of the input at the same time.
G16.4427 Practical MRI 1 – 29th January 2015
Other Definitions• An LTI system is without memory if its output
at any time depends only on the value of the input at the same time
• An LTI system is causal if its output depends only on the present and past values of the input
G16.4427 Practical MRI 1 – 29th January 2015
Other Definitions• An LTI system is without memory if its output
at any time depends only on the value of the input at the same time
• An LTI system is causal if its output depends only on the present and past values of the input
• An LTI system is stable if every bounded input produces a bounded output
G16.4427 Practical MRI 1 – 29th January 2015
Unit Step Response• The unit step response s(t) correspond to the
output when the input is u(t)
• The unit step response can be used to characterize the system since we can calculate the impulse response from it
G16.4427 Practical MRI 1 – 29th January 2015
Problem:
What is ?
G16.4427 Practical MRI 1 – 29th January 2015
Any questions?
G16.4427 Practical MRI 1 – 29th January 2015
Example• Linear constant-coefficient differential
equations can be used to describe causal LTI systems:– Provide an implicit specification of the system– Must be solved in order to find an explicit
expression for the system output as a function of the input
What is ?
G16.4427 Practical MRI 1 – 29th January 2015
Sampling• Under certain conditions, a continuous-time
signal can be completely represented by and recoverable from knowledge of its samples at points equally spaced in times
• A convenient way to sample a continuous-time signal x(t) is to multiply it by a periodic impulse train p(t) (i.e. the sampling function)
• In MRI sampling is very important!
G16.4427 Practical MRI 1 – 29th January 2015
Question:Can you provide examples of
sampling in MRI?
G16.4427 Practical MRI 1 – 29th January 2015
Impulse-Train Sampling
t0
t0
T1
T = sampling period
ωs= 2π/T = sampling frequency
x(nT) = samplest
0
T
G16.4427 Practical MRI 1 – 29th January 2015
C/D Conversion• In many application there is a significant
advantage in processing a continuous-time signal by first converting it into a discrete-time signal
Conversion of impulsetrain to discrete-time
sequence
C/D conversion
G16.4427 Practical MRI 1 – 29th January 2015
Discrete-Time Convolution• The input x[n] and the output y[n] of a
discrete-time LTI system are related by the convolution sum:
• The same properties of the continuous case apply to the discrete case
G16.4427 Practical MRI 1 – 29th January 2015
Example: Question• Find y[n] given:
with 0 < α < 1
1
n0
……
1
n0
……
G16.4427 Practical MRI 1 – 29th January 2015
Example: Solution
1
k0
……n
for n < 0 and for
k > n as the signals do not overlap
elsewhere
Therefore, for 0 ≤ k ≤ n:
n0
……
G16.4427 Practical MRI 1 – 29th January 2015
Periodic Signals• A periodic continuous-time signal x(t) has the
property that there is a positive value of T for which x(t) = x(t +T)
• A discrete-time periodic signal x[n] is periodic with period N (integer) if it is unchanged by a time shift of N: x[n] = x[n + N]
n0
……
(N0 = 4)
t0
……
T-T-2T
G16.4427 Practical MRI 1 – 29th January 2015
Matlab Demonstration
G16.4427 Practical MRI 1 – 29th January 2015
See you on Tuesday!