G16.4427 Practical MRI 1 – 29 th January 2015 G16.4427 Practical MRI 1 Introduction to the course...

transcript

G16.4427 Practical MRI 1 – 29th January 2015

G16.4427 Practical MRI 1

Introduction to the courseMathematical fundamentals

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

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

Instructors• Prof. Riccardo Lattanzi (course director)– All lectures and first lab session– Riccardo.Lattanzi@nyumc.org (212-263-4860)– Office hours: after class or by appointment

• Prof. Kaveh Vahedipour– Pulse sequence programming lab sessions– Kaveh.Vahedipour@nyumc.org

• Dr. Ryan Brown– RF coil lab sessions– Ryan.Brown@nyumc.org

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

Any questions?

Vectors• Cartesian representation

Magnitude:

Direction:

Question:Can you provide examples of

vectors quantity in MRI?

Complex Notation• In MR a complex notation is often used for 2D

vectors:

with:vector of length A0 rotating counterclockwise at an angularspeed equal to ω0

Commonly Used Functions• Unit Step Function

• Rectangular Window Function

• Kronecker Delta Function

Sinc Function

• It is an even function• Zero crossings at x = ± nπ• Sinusoidal oscillation of period 2π with

amplitude decreasing continuously as 1/x

sinc(x)

xπ 2π-2π -π

Any questions?

Matlab Demonstration

Convolution• A concept central to Fourier theory and the

analysis of linear systems

• Symbolically often written as:

Properties of Convolution• Commutativity

• Associativity

• Distributivity

• Differentiation:

Example• Calculate

• Differentiation property:

• Fundamental theorem of calculus:

• Then:

x1/2-1/2

Graphical Method• Flip (or reverse) one function in time:

• Slide the flipped function over the other from –∞ to + ∞:

x1/2-1/2

1/2-1/2

Graphical Method• Integrate where both functions overlap:

1/2-1/2 Integral equal to zero (no overlap)

1/2-1/2

Integral equal to:

1/2-1/2 Integral equal to zero (no overlap)

Graphical Method• Putting everything together

Problem:

Given:

Calculate:

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– –

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.

Impulse Response• Impulse response h(t) is the response to δ(t):

• LTI system response:

Commutative and Associative Properties

• Commutative property:

• Associative property:

Distributive Property

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.

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

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

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

Problem:

What is ?

Any questions?

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 ?

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!

Question:Can you provide examples of

sampling in MRI?

Impulse-Train Sampling

T = sampling period

ωs= 2π/T = sampling frequency

x(nT) = samplest

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

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

Example: Question• Find y[n] given:

with 0 < α < 1

……

Example: Solution

……n

for n < 0 and for

k > n as the signals do not overlap

elsewhere

Therefore, for 0 ≤ k ≤ n:

……

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 = 4)

……

T-T-2T

See you on Tuesday!

G16.4427 Practical MRI 1 – 29 th January 2015 G16.4427 Practical MRI 1 Introduction to the course...

Documents