Spin-Warp Pulse Sequence

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1 Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 25 Linear Systems Lecture Spin-Warp Pulse Sequence RF G x (t G y (t k y k x K-space trajectories EPI Spiral k y k y k x k x Credit: Larry Frank

2 K-space At each point in time, the received signal is the Fourier transform of the object s(t = M( k x (t (t = F[ m(x, y ] kx (t (t evaluated at the spatial frequencies: k x (t = γ 2π k y (t = γ 2π t t G x (τdτ G y (τdτ Thus, the gradients control our position in k-space. The design of an MRI pulse sequence requires us to efficiently cover enough of k-space to form our image. Rectangle Function Π(x = x >/2 x /2 -/2 /2 x Also called rect(x Π(x, y = Π(xΠ(y y /2 -/2 /2 -/2 x Kronecker Delta Function δ[n] = for n = otherwise δ[n] δ[n-2] n n 2

3 Kronecker Delta Function for m =,n = δ[m,n] = otherwise δ[m,n] δ[m-2,n] δ[m,n-2] δ[m-2,n-2] Discrete Signal Expansion g[n] = g[k]δ[n k] g[m,n] = k= k= l= g[k,l]δ[m k,n l] δ[n] g[n] n -δ[n-] n n.5δ[n-2] n n Dirac Delta Function Notation : δ(x - D Dirac Delta Function δ(x, y or 2 δ(x, y - 2D Dirac Delta Function δ(x, y,z or 3 δ(x, y,z - 3D Dirac Delta Function δ( r - N Dimensional Dirac Delta Function 3

4 D Dirac Delta Function δ(x = when x and δ(xdx = Can interpret the integral as a limit of the integral of an ordinary function that is shrinking in width and growing in height, while maintaining a constant area. For example, we can use a shrinking rectangle function such that δ(xdx = lim τ Π(x /τdx. τ τ -/2 /2 x 2D Dirac Delta Function δ(x, y = when x 2 + y 2 and δ(x, ydxdy = where we can consider the limit of the integral of an ordinary 2D function that is shrinking in width but increasing in height while maintaining constant area. δ(x,ydxdy = lim τ 2 Π x /τ, y /τ τ ( dxdy. Useful fact : δ(x,y = δ(xδ(y τ Generalized Functions Dirac delta functions are not ordinary functions that are defined by their value at each point. Instead, they are generalized functions that are defined by what they do underneath an integral. The most important property of the Dirac delta is the sifting property δ(x x g(xdx = g(x where g(x is a smooth function. This sifting property can be understood by considering the limiting case lim τ Π x /τ τ ( g(xdx = g(x x g(x Area = (height(width= (g(x / τ τ = g(x 4

5 Working with Dirac Delta Functions What is δ(ax - b? What is dδ(x/dx? How do we define generalized functions? There are two main approaches : Look at the limit of an integral with sequences. 2 Consider the behavior of the function when integrated with a nice test function. Two generalized functions δ (t and δ 2 (t are equivalent in the distributional sense when δ (tφ(tdt = δ 2 (tφ(tdt - - Example : δ(ax =?? Representation of D Function From the sifting property, we can write a D function as g(x = g(ξδ(x ξdξ. To gain intuition, consider the approximation g(x n= g(nδx Δx Π x nδx Δx. Δx g(x Representation of 2D Function Similarly, we can write a 2D function as g(x,y = g(ξ,ηδ(x ξ,y ηdξdη. To gain intuition, consider the approximation g(x, y g(nδx,mδy Δx Π x nδx n= Δx Δy Π y mδy ΔxΔy. m= Δy 5

6 Impulse Response Intuition: the impulse response is the response of a system to an input of infinitesimal width and unit area. Original Image Blurred Image Since any input can be thought of as the weighted sum of impulses, a linear system is characterized by its impulse response(s. The Fourier Transform The Fourier Transform (FT is simply given by the basis coefficients G( f = e j 2πft,g(t = g(te j 2πft dt = F{ g(t } The Inverse Fourier Transform is the continuous- time integral expansion for g(t : g(t = G( f b f (t df = G( f e j 2πft df = F { G( f } This can also be written as an inner product in Fourier Space g(t = e j 2πft,G( f Units Temporal Coordinates, e.g. t in seconds, f in cycles/second G( f = e j 2πft,g(t = g(te j 2πft dt Fourier Transform g(t = e j 2πft,G( f = G( f e j 2πft df Inverse Fourier Transform Spatial Coordinates, e.g. x in cm, k x is spatial frequency in cycles/cm = e j 2πkxx,g(x = g(xe j 2πk xx dx Fourier Transform g(x = e j 2πk x x, = e j 2πk x x dk x Inverse Fourier Transform 6

7 2D Fourier Transform Fourier Transform = F[ g( x, y ] = e j 2π ( k x x +k y y,g = g(x,y Inverse Fourier Transform g(x, y = ( dkx dk y e j 2π k xx +k yy ( dxdy e j 2π k x x +k y y D Fourier Transform KPBS KIFM Fourier Transform KIOZ Plane Waves e j 2π (k xx +k y y = cos( 2π(k x x + k y y + j sin( 2π(k x x + k y y k x 2 + k y 2 /k y /k x cos(2πk x x cos(2πk y y cos(2πk x x +2πk y y 7

8 Figure 2.5 from Prince and Link Plane Waves ΔABC ~ ΔBDC AC AB = BC BD A /ky θ D C θ B BC BD = AB = AC k x ky + k x2 k y2 = k x2 + k y2 k θ = arctan y kx /kx Linearity Basic Properties F [ ag(x, y + bh(x, y] = a + bh(kx Scaling F [ g(ax,by] = kx kx G, ab a b Shift F [ g(x a, y b] = e j 2 π (kx a +ky b Modulation F [ g(x, ye j 2 π (xa +yb ] = G(kx a b 8

9 Computing Transforms F(δ(x = δ(xe j 2πk xx dx = F(δ(x x = δ(x x e j 2πk x x dx = e j 2πk x x / 2 F( Π( x = e j 2πk xx / 2 dx = e jπk x e jπk x j2πk x = sin(πk x πk x = sinc(k x Scaling Theorem F{ g(ax } = a G k x a Separable Functions g( x, y is said to be a separable function if it can be written as g( x, y = g X ( xg Y ( y The Fourier Transform is then separable as well. = g(x, y e j 2π ( k x x +k y y dxdy = g X ( xe j 2πk x x dx g Y ( y e j 2πkyy dy = G X (k x G Y (k y Example g(x, y = Π(xΠ(y = sinc(k x sinc(k y 9

?? Computing Transforms Define h( k x = e j 2πk x x dx and see what it does under an

10 Example g(x, y = Π(xΠ(y = sinc(k x sinc(k y Example (sinc/rect y /2 -/2 /2 x -/2 Example (sinc/rect F( = e j 2πk x x dx =??? Computing Transforms Define h( k x = e j 2πk x x dx and see what it does under an integral. G( k x h( k x dk x = G( k x e j 2πk xx dxdk x = G( k x e j 2πk x x dk x dx = g( xdx = G( Therefore, F( = e j 2πk xx dx = δ(k x

11 Linearity The Fourier Transform is linear. F{ ag(x + bh(x } = a + bh(k x F[ ag(x,y + bh(x,y ] = a + bh(k x Similarly, Computing Transforms F e j 2πk x { } = δ(k x k F{ cos2πk x} = ( 2 δ(k x k + δ(k x + k F{ sin2πk x} = ( 2 j δ(k x k δ(k x + k Examples g(x, y = δ(x, y = δ(xδ(y = g(x, y = δ(x = δ(k y!!!

12 Examples g(x, y =+ e j 2πax = δ k x ( + δ(k x + aδ(k y g(x, y =+ e j 2πay = δ(k x + δ(k x δ(k y a Examples g(x, y = cos(2π(ax + by = 2 δ(k x aδ(k y b + 2 δ(k x + aδ(k y + b Examples = δ(k x + g(x, y =??? δ(k x + cδ(k y + δ(k x δ(k y d + 2 δ(k x aδ(k y b + 2 δ(k x + aδ(k y + b 2

13 Duality Note the similarity between these two transforms { } = δ(k x a F e j 2πax F{ δ(x a } = e j 2πk x a These are specific cases of duality F{ G(x } = g( k x Application of Duality F{ sinc(x } = sinπx πx e j 2πk xx dx =?? Recall that F{ Π(x } = sinc( k x. Therefore from duality, F{ sinc( x } = Π( k x = Π(k x Shift Theorem F{ g(x a } = G( k x e j 2πak x Shifting the function doesn't change its spectral content, so the magnitude of the transform is unchanged. Each frequency component is shifted by a. This corresponds to a relative phase shift of - 2πa /(spatial period = - 2πak x For example, consider exp( j2πk x x. Shifting this by a yields exp( j2πk x (x a = exp( j2πk x xexp( j2πak x 3

14 Modulation F g(xe j 2πk [ x ] = δ(k x k = G( k x k F[ g(xcos( 2πk x ] = 2 G ( k k x + 2 G ( k + k x F[ g(xsin( 2πk x ] = 2 j G ( k k x 2 j G ( k + k x Example Amplitude Modulation (e.g. AM Radio g(t 2g(t cos(2πf t 2cos(2πf t G(f -f f G(f-f + G(f+f Modulation Example x = * = 4

Topics. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2004 Lecture 4 2D Fourier Transforms

Topics. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2004 Lecture 4 2D Fourier Transforms Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2004 Lecture 4 2D Fourier Transforms Topics 1. 2D Signal Representations 2. 2D Fourier Transform 3. Transform Pairs 4. FT Properties 1