Backprojection. Projections. Projections " $ & cosθ & Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2014 CT/Fourier Lecture 2

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1 Backprojection Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2014 CT/Fourier Lecture Projections Projections " r% " ' = cosθ # s& # sinθ sinθ %" cosθ& ' x % # y ' & " x% # y& ' = " cosθ # sinθ sinθ %" cosθ & ' r % # s ' & % I(r,θ) = I 0 exp' ( µ(x, y)ds* & Lr,θ ) % = I 0 exp' ( µ(rcosθ ssinθ,rsinθ + scosθ)ds* & Lr,θ ) 1

2 Projections % I(r,θ) = I 0 exp' & Lr,θ ( µ(rcosθ ssinθ,rsinθ + scosθ)ds* ) p(r,θ) = ln I θ (r) I 0 = µ(rcosθ ssinθ,rsinθ + scosθ)ds Lr,θ g(r,θ) = Radon Transform µ(x(s), y(s))ds = µ(rcosθ ssinθ,rsinθ + scosθ)ds ( ) = µ(x,y)δ x cosθ + y sinθ r dxdy " r% # s& ' = " cosθ # sinθ sinθ %" cosθ& ' x % # y ' & r = x cosθ + ysinθ " x% # y& ' = " cosθ # sinθ sinθ %" cosθ & ' r % # s ' & # f (x, y) = 1 x 2 + y 2 1 % 0 otherwise Example Example f (x, y) = rect(x, y) g(l,θ = 0) = 1 l 2 = dy 1 l 2 f (l, y)dy ') = 2 1 l2 l 1 ( *) 0 otherwise Calculate the projections at angles of 0, 45, and 90 degrees 2

3 Example f (x, y) = δ(x 10, y 10) In-class Exercise µ(x, y) = rect(x, y / 2) What is the projection of this object? Sketch this object. What are the projections at theta = 0 and 90 degrees? For what angle is the projection maximized? PollEv.com/be280a Sinogram y Backprojection x 0 x b(x 0, y) = p( l,θ = 0)Δθ = p(x 0,0)Δθ l b θ (x, y) = g(x cosθ + y sinθ,θ)δθ b(x, y) = B{ g( l,θ) } π = g(x cosθ + y sinθ,θ)dθ 0 3

4 Backprojection b(x, y) = B{ p( l,θ) } π = p(x cosθ + y sinθ,θ)dθ 0 Backprojection π b(x, y) = B{ p( l,θ) } = p(x cosθ + y sinθ,θ)dθ

5 Example Projection Slice Theorem j 2 πρl g(l,θ )e dl = f (x, y)δ (x cos θ + y sin θ l)e ( ) = f (x, y)e dx dy G( ρ,θ ) = j 2 πρl dx dy dl j 2 πρ x cosθ +y sin θ = F2D [ f (x, y)] u= ρ cosθ,v= ρ sin θ Prince & Links 2006 Prince&Links 2006 Signals and Images Discrete-time/space signal /image: continuous valued function with a discrete time/space index, denoted as s[n] for 1D, s[m,n] for 2D, etc. n n m Continuous-time/space signal / image: continuous valued function with a continuous time/space index, denoted as s(t) or s(x) for 1D, s(x,y) for 2D, etc. t y x x Figure 2.5 from Prince and Link 5

6 k-space Image space k-space y k y x Fourier Transform k x Hanson

7 Examples Examples Examples Examples 7

8 Examples Exercise PollEv.com/be280a Exercise PollEv.com/be280a Exercise PollEv.com/be280a 8

9 The Fourier Transform Fourier Transform (FT) G( f ) = g(t)e j 2πft Inverse Fourier Transform dt = F{ g(t) } g(t) = G( f )e j 2πft df = F 1 { G( f )} Units Temporal Coordinates, e.g. t in seconds, f in cycles/second G( f ) = g(t)e j 2πft dt Fourier Transform g(t) = G( f )e j 2πft df Inverse Fourier Transform Spatial Coordinates, e.g. x in cm, k x is spatial frequency in cycles/cm G(k x ) = g(x)e j 2πk x x dx Fourier Transform g(x) = G(k x )e j 2πk x x dk x Inverse Fourier Transform j = 1 Complex Numbers Complex Numbers z = 2 +1j j 2 =? (3+ 2 j)(3 2 j) =? j 2 = 1 (3+ 2 j)(3 2 j) = 9 4 j 2 =13 z = = 5 " θ = tan 1 1 % ' = 26.6 degrees # 2 & 9

10 e jθ Euler s Formula = cosθ + j sinθ z = x + jy = z e jθ G(k x ) = 1D Fourier Transform ( ) g(x)exp j2πk x x dx = g(x)cos(2πk x x)dx j g(x) sin(2πk x x) dx The part of g(x) that "looks" like cos(2πk x x) The part of g(x) that "looks" like sin(2πk x x) 1 k x 1 k x Computing Transforms F(δ(x)) = δ(x)e j 2πk x x dx =1 F(δ(x x 0 )) = δ(x x 0 )e j 2πk x x dx = e j 2πk x x 0 1/ 2 F( Π( x) ) = e j 2πk xx 1/ 2 dx = e jπk x e jπk x j2πk x = sin(πk x) πk x = sinc(k x ) 2D Fourier Transform Fourier Transform G(k x,k y ) = F[ g( x,y) ] = g(x, y) Inverse Fourier Transform g(x, y) = G(k x,k y ) ( ) dkx dk y e j 2π k xx +k y y ( ) dxdy e j 2π k x x +k y y 10

11 Plane Waves 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) ) 1/k y 1 k x 2 + k y 2 A 1/k y θ θ B D 1/k x C ΔABC ~ ΔBDC AC BC = AB BD BD = AB BC AC = θ = arctan k ' y & ) % ( k x 1 1 k x k y 1 k + 1 = 2 2 x k y 1 k x 2 + k y 2 1/k x cos(2πk x x) cos(2πk y y) cos(2πk x x +2πk y y) 11