G Practical MRI 1 – 29 th January 2015 G Practical MRI 1 Introduction to the course Mathematical fundamentals

G Practical MRI 1 – 29 th 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

G Practical MRI 1 – 29 th January 2015 Course Information Website: – Format – Twice per week from the 29 th January to 7 th May – mins lectures, 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

G Practical MRI 1 – 29 th January 2015 Instructors Prof. Riccardo Lattanzi (course director) – All lectures and first lab session – ( ) – Office hours: after class or by appointment Prof. Kaveh Vahedipour – Pulse sequence programming lab sessions – Dr. Ryan Brown – RF coil lab sessions –

G Practical MRI 1 – 29 th 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

G Practical MRI 1 – 29 th January 2015 Any questions?

G Practical MRI 1 – 29 th January 2015 Vectors Cartesian representation Magnitude: Direction:

G Practical MRI 1 – 29 th January 2015 Question: Can you provide examples of vectors quantity in MRI?

G Practical MRI 1 – 29 th January 2015 Complex Notation In MR a complex notation is often used for 2D vectors: with: vector of length A 0 rotating counterclockwise at an angular speed equal to ω 0

G Practical MRI 1 – 29 th January 2015 Commonly Used Functions Unit Step Function Rectangular Window Function Kronecker Delta Function

G Practical MRI 1 – 29 th 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) x π 2π -2π -π

G Practical MRI 1 – 29 th January 2015 Any questions?

G Practical MRI 1 – 29 th January 2015 Matlab Demonstration

G Practical MRI 1 – 29 th January 2015 Convolution A concept central to Fourier theory and the analysis of linear systems Symbolically often written as:

G Practical MRI 1 – 29 th January 2015 Properties of Convolution Commutativity Associativity Distributivity Differentiation:

G Practical MRI 1 – 29 th January 2015 Example Calculate Differentiation property: Fundamental theorem of calculus: Then: 1 x 1/2 -1/2

G Practical MRI 1 – 29 th January 2015 Graphical Method Flip (or reverse) one function in time: Slide the flipped function over the other from –∞ to + ∞: 1 x 1/2 -1/2 1 x 1/2 -1/2 1 1/2 -1/2

G Practical MRI 1 – 29 th 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)

G Practical MRI 1 – 29 th January 2015 Graphical Method Putting everything together 1 x 1

G Practical MRI 1 – 29 th January 2015 Problem: Given: Calculate:

G Practical MRI 1 – 29 th January 2015 Matlab Demonstration

G Practical MRI 1 – 29 th 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 – –

G Practical MRI 1 – 29 th 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.

G Practical MRI 1 – 29 th January 2015 Impulse Response Impulse response h(t) is the response to δ(t): LTI system response:

G Practical MRI 1 – 29 th January 2015 Commutative and Associative Properties Commutative property: Associative property:

G Practical MRI 1 – 29 th January 2015 Distributive Property

G Practical MRI 1 – 29 th 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.

G Practical MRI 1 – 29 th 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

G Practical MRI 1 – 29 th 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

G Practical MRI 1 – 29 th 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

G Practical MRI 1 – 29 th January 2015 Problem: What is ?

G Practical MRI 1 – 29 th January 2015 Any questions?

G Practical MRI 1 – 29 th 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 ?

G Practical MRI 1 – 29 th 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!

G Practical MRI 1 – 29 th January 2015 Question: Can you provide examples of sampling in MRI?

G Practical MRI 1 – 29 th January 2015 Impulse-Train Sampling t 0 t 0 T 1 T = sampling period ω s = 2π/T = sampling frequency x(nT) = samples t 0 T

G Practical MRI 1 – 29 th 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 impulse train to discrete-time sequence C/D conversion

G Practical MRI 1 – 29 th 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

G Practical MRI 1 – 29 th January 2015 Example: Question Find y[n] given: with 0 < α < 1 1 n 0 … … 1 n 0 … …

G Practical MRI 1 – 29 th January 2015 Example: Solution 1 k 0 … … n  for n < 0 and for k > n as the signals do not overlap  elsewhere Therefore, for 0 ≤ k ≤ n: n 0 … …

G Practical MRI 1 – 29 th 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] n 0 … … (N 0 = 4) t 0 … … T -T -2T

G Practical MRI 1 – 29 th January 2015 Matlab Demonstration

G Practical MRI 1 – 29 th January 2015 See you on Tuesday!