Image Reconstruction from Projection

Transcription

1 Image Reconstruction from Projection Reconstruct an image from a series of projections X-ray computed tomography (CT) Computed tomography is a medical imaging method employing tomography where digital geometry processing is used to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around a single axis of rotation. 3/7/18 1

2 Backprojection In computed tomography or other imaging techniques requiring reconstruction from multiple projections, an algorithm for calculating the contribution of each voxel of the structure to the measured ray data, to generate an image; the oldest and simplest method of image reconstruction. 3/7/18 2

3 Medical Imaging (Before 1895) Only way to see is to cut!

4 Engineering Advances 1 st x-ray (1895) x-ray today

5 Projection X-Ray Projection Format Small Dose Fast Inexpensive Pneumonia Projection

6 Computed Tomography (CT) Head CT abdominal CT Many Projections

7 Soft, uniform tissue Image Reconstruction: Introduction Intensity is proportional to absorption Uniform with higher absorption Tumor 3/7/18 7

8 Image Reconstruction: Introduction 3/7/18 8

9 3/7/18 9

10 3/7/18 10

11 Other CTs Electron beam CT (Fifth-generation CT) Electron beam tomography (EBCT) was introduced in the early 1980s, by medical physicist Andrew Castagnini, as a method of improving the temporal resolution of CT scanners. High cost of EBCT equipment, and poor flexibility Helical (or spiral) cone beam computed tomography (sixth-generation) A type of three dimensional computed tomography (CT) in which the source (usually of x-rays) describes a helical trajectory relative to the object while a two dimensional array of detectors measures the transmitted radiation on part of a cone of rays emitting from the source 3/7/18 11

12 Other CTs Multislice CT (seventh-generation) The major benefit of multi-slice CT Significant increase in detail Utilizes X-ray tubes more economically Reducing cost and potentially reducing dosage 3/7/18 12

13 Impulse lines and line-integrals In 1D Z 1 In 2D Z f(x) (x)dx = f(0) f(x, y) (x)dx = f(0,y) A 1D Line Integral Z 1 1 Z 1 1 f(x, y) (x)dy dx= Z 1 f(0,y)dy 1 Integral of the 1D

14 Impulse lines and line-integrals y f(x,y) 1 2 x 1 dy Z f(x, y) (x)dx = f(0,y)=u(y) Z 1 dy =1 y 1 dy Z f(x, y) (x 1)dx = f(1,y)=u(2y) what about line integral with δ(x-u)? Z 1 dy y =0.5

15 Line Integral and Projection y f(x,y) 1 2 x p(u) 1 2 u p(u) = Z 1 1 Z 1 1 f(x, y) (x u)dxdy = Z 1 1 f(u,y)dy

16 Projections and the Radon Transform 3/7/18 16

17 Projections and the Radon Transform g( ρ, θ ) = f ( x, y) δ ( x cosθ + y sin θ ρ ) dxdy j k k k j 3/7/18 17

18 General Projections p(, ) = Z 1 1 Z 1 1 f(x, y) ( x cos y sin )dxdy Line integral projection x-ray f(x, y) p ( )

19 Many Projections - Tomography

20 Radon Transform p(, ) = Z 1 1 Z 1 1 f(x, y) ( x cos y sin )dxdy 0.5 y f(x,y) 0.5 y f(x,y) x x p(ρ,0) 1 p(ρ,0.4π) ρ 2 ρ

21 Radon Transform: Sinogram Also called Sinogram Impulse Sinusoid y x p(, ) 2

22 g( ρ) = 0 otherwise A r ρ ρ r 3/7/18 22

23 Computed Tomography Sinogram cross-section FBP x-ray source

24 Image Reconstruction f ( x, y) = g( x cosθ + y sin θ, θ ) θ π f ( x, y) = f ( x, y) dθ θ π f ( x, y) = f ( x, y) θ 0 θ = 0 A back-projected image formed is referred to as a laminogram 3/7/18 24

25 Examples: Laminogram 3/7/18 25

26 Projection Slice Theorem (Bracewell) f(x, y) F 1D {p(, )} = F ( cos, sine cos ) y F (f x,f y ) x p(, 0) = p(x) 1D FT F (f x, 0)

27 Projection Slice Theorem (Bracewell) f(x, y) F 1D {p(, )} = F ( cos, cos ) y F (f x,f y ) sine x p(, ) 1D FT F ( cos, cos )

28 Projection Slice Theorem (Bracewell) f(x, y) F 1D {p(, )} = F ( cos, cos ) y F (f x,f y ) sine x p(, ) 1D FT F ( cos, cos )

29 Projection Slice Theorem (Bracewell)

30 Reconstruction Using Parallel-Beam Filtered Backprojections j2 ( ux vy) f ( x, y) = π + F( u, v) e dudv Let u = wcos θ, v = w sin θ, then dudv = wdwdθ, 2π j2 π w( xcosθ + ysin θ ) f ( x, y) = F( wcos, wsin ) e wdwd 0 0 = θ θ θ 2π j2 π w( xcosθ + ysin θ ) G( w, θ ) e wdwd 0 0! G( w, θ ) = G( w, θ ) π 2 ( cos sin ) (, ) (, ) j π f x y w G w e w x θ + y θ = θ dwd 0 θ θ 3/7/18 30

31 Reconstruction Using Parallel-Beam Filtered Backprojections π j2 π w( xcosθ + ysin θ ) f ( x, y) = w G( w, θ ) e dwd 0 It s not integrable = π 0 (, ) j2π wρ w G w θ e dw d ρ = xcosθ + ysinθ θ θ Approach: Window the ramp so it becomes zero outside of a defined frequency interval. That is, a window band-limits the ramp filter. 3/7/18 31

32 Hamming / Hann Widow 2π w c + ( c 1) cos 0 w ( M 1) h( w) = M 1 0 otherwise c c = = 0.54, the function is called the Hamming window 0.5, the function is called the Han window 3/7/18 32

33 The Plot of Hamming Widow 3/7/18 33

34 Filtered Backprojection The complete, filtered backprojection (to obtain the reconstructed image f(x,y) ) is described as follows: 1. Compute the 1-D Fourier transform of each projection 2. Multiply each Fourier transform by the filter function w which has been multiplied by a suitable (e.g., Hamming) window 3. Obtain the inverse 1-D Fourier transform of each resulting filtered transform 4. Integrate (sum) all the 1-D inverse transforms from step 3 3/7/18 34

35 Implementation of Filtered Backprojection in Spatial Domain Fourier transform of the product of two frequency domain functions is equal to the convolution of the spatial representation Let s(p) denote the inverse Fourier transform of w π j2π wρ f ( x, y) = w G( w, θ ) e dw d 0 ρ = xcosθ + ysinθ = π [ ( ) (, )] s ρ g ρ θ dθ 0 ρ = xcosθ + ysinθ π = g( ρ, θ ) s( x cosθ y sin θ ρ) d ρ + dθ 0 θ 3/7/18 35

36 Partly Discrete Reconstruction Let s assume continuous angle ϴ, discrete ρ f(x, y) y F (apple x, apple y ) apple =! 2 x DTFT p [n] ) P [apple]

37 Partly Discrete Reconstruction Let s assume continuous angle ϴ, discrete ρ f(x, y) y F (apple x, apple y ) apple =! 2 x DTFT p [n] ) P [apple]

38 Reconstruction From Polar Coordinates f[n, m] = = Z 0.5 Z Z Z F (apple x, apple y )e 2 j(apple xn+apple y m) dapple x dapple y F (, )e 2 j( cos( )n+ sin( )m) d d Polar frequency data must be multiplied by ρ Also called a rho filter

39 Discrete Reconstruction Let s assume discrete angle ϴm, discrete ρ f(x, y) y F (apple apple =! x, apple y ) 2 x DFT p m [n] ) P m [l]

40 Discrete Reconstruction Let s assume discrete angle ϴm, discrete ρ f(x, y) y F (apple apple =! x, apple y ) 2 x DFT p m [n] ) P m [l]

41 Filtered Back Projection Replace integrals with sums. Sum over radius and angle Define a (filtered) backprojection: C m [n x,n y ]= So, (N/2) 1 X l= N/2 F [l, m ]e 2 j(l/n cos( m)n x +l/n sin( m )n y ) l/n f[n x,n y ]= X m C m [n x,n y ] =

42 Example

43 Example Convolution Back Projection For ϴ=0 C 0 [n x,n y ]= (N/2) 1 X l= N/2 F [l, 0] l/n e 2 j(l/nn x) FFT Rho back proj. F [l, 0] l/n IFFT

44 Example Convolution Back Projection For ϴ=π/2 C /2 [n x,n y ]= (N/2) 1 X l= N/2 F [l, /2] l/n e 2 j(l/nn y) FFT Rho back proj. F [l, /2] l/n IFFT

45 Convolution Back Projection

46 Filtered Back Projection Back projection Filtered Back projection

47 How Many Projections? FOV F (apple x, apple y ) 1/FOV

48 How Many Projections? 256 Proj. 128 Proj 64 Proj

49 Fan Beam CT Single Source Many detectors How to reconstruct?

50 Fan Beam CT Single Source Many detectors How to reconstruct? Re-binning!

51 Fan Beam CT Single Source Many detectors How to reconstruct? Re-binning!