NRMS (Normalized root mean square) error between the reconstructed and original phase shift vs the number of iterations of 3 techniques in the case of:.
Iterative method using random signed feedback (RSF) accelerator for X‐ray phase contrast imaging
Nghia Vo Diamond Light Source May 2015 1
Formulation Define phase shift Absorption function
y x
ϕ (x, y ) = −(2π λ )∫ δ (x, y , z )dz A( x, y ) = (2π λ )∫ β (x, y, z )dz
y x
z
z
The absorption function Combine to form the transmission function
The phase shift
T ( x, y ) = exp[− A( x, y ) + iϕ ( x, y )] 3
Formulation The interaction between the material and the wave‐field is described as
U ( x, y ) = T ( x, y )U i ( x, y ) The wave‐field at distance D after free‐propagation
U D ( x, y ) ≡ Fr [U (x, y )] = F −1 [F [U ( x, y )]× K (u , v )] where
K (u , v ) = e i 2πD / λ exp[−iπλD(u 2 + v 2 )]
Fresnel propagator
The intensity that can be measured
I D (x D , y D ) = U D (x D , y D )
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4
Formulation
Phase retrieval
T ( x, y ) = exp[− A( x, y ) + iϕ ( x, y )]
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Methods for phase retrieval in the near field or Fresnel region Direct methods
λDw ⎞ * ⎛ λDw ⎞ −i 2πrw ⎛ F [I D (r )] ≡ IˆD (w ) = ∫∫ T ⎜ r − dr ⎟T ⎜ r + ⎟e 2 ⎠ ⎝ 2 ⎠ ⎝
Contrast Transfer Function (CTF)
T ( r ) ≈ 1 − A ( r ) + iϕ ( r )
Transport of Intensity Equation (TIE)
λ Dw ⎞ 1 ⎛ T ⎜r ± ⎟ ≈ T ( r ) ± λ Dw • ∇ T ( r ) 2 ⎠ 2 ⎝
Mix of TIE and CTF
T ( r ) = exp ⎡⎣ − A ( r ) ⎤⎦ ⎡⎣1 + iφ ( r ) ⎤⎦
Iterative methods Gerchberg‐Saxton (GS) algorithm ‐ Fourier transform input ‐ Replace the calculated modulus by measured modulus ‐ Inverse Fourier transform ‐ Apply constraints. Fourier transform Ù Fresnel propagator 6
Critical limitation of direct methods Contrast transfer function (CTF)
⎧⎪ 1 ⎡ ⎤ ⎫⎪ 2 2 2 2 ˆ ˆ ϕ ( x, y ) = F ⎨ ⎢C ∑ I Dn (u, v ) sin πλDn u + v − A∑ I Dn (u , v ) cos πλDn u + v ⎥ ⎬ ⎪⎩ 2Δ ⎣ Dn Dn ⎦ ⎪⎭
[
−1
(
)]
[
)]
(
Transport of intensity equation (TIE)
⎛ ⎧ 1 ⎡ −2 ⎛ I D (r ) − I 0 (r ) ⎞⎤ ⎫ ⎞⎟ ⎜ ϕ (r ) = − ∇ ∇ • ⎨ ∇ ⎢∇ ⎜ ⎟⎥ ⎬ ⎟ ⎜ ( ) r λ I D ⎝ ⎠⎦ ⎭ ⎠ ⎣ ⎩ 0 ⎝ 2π
Mixed TIE‐CTF
F [I 0 (r )ϕ k +1 (r )] =
∇ −2 = −
−2
1 4π 2
F −1
1 F 2 2 u +v
∑ 2 sin (πλD w ){Iˆ (w ) − Iˆ (w ) − (iλD )cos(πλD w )w • F [ϕ (r )∇I (r )]} 2
n
Dn
2
Dn
n
0
n
k
0
∑ 4 sin (πλD w ) 2
2
n
Dn
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Accelerator for iterative phase retrieval named Random Signed Feedback (RSF) constant λ, D1, D2, N, dr, θ; T=initial_ estimate_complex _matrix; For n=1 to Iterative_number Step1: T1=GS(T,D1,ID1); Step2: T2= GS(T1,D2,ID2); Step3: T3=(T1+ T2)/2; Step4: M1=Abs(FrD2 (T3)); Step5: M2=Sqrt(ID2) – M1; Step6: M3=Sgn(M2); Step7: M4=θ×Random([0,1],{N,N}); Step8: T=SmoothFilter(Re(T3))+i×(Im(T3)+M4×M3); End Fresnel transform
TD = Fr ⎡⎣T0 ( x, y ) ⎤⎦ = F −1 ⎡⎣ F ⎡⎣T0 ( x, y ) ⎤⎦ × K D ( u , v ) ⎤⎦
⎡ F ⎡⎣TD ( x, y ) ⎤⎦ ⎤ Inverse Fresnel transform T0 = Fr ⎡⎣TD ( x, y ) ⎤⎦ = F ⎢ K D ( u, v )⎥ ⎣ ⎦ −1
−1
GS() function returns the next estimate of the transmission after 1 round of forward and back propagation
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Simulation experiments
The simulated phantom for testing algorithms: (a) Absorption function, (b) phase shift
The calculated intensities at: (a) D1=50cm, (b) D2=80cm with dr=1μm, E=12 keV
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Optimum performance of the RSF accelerator
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Optimum performance of the RSF accelerator Which distance we need to use?
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Optimum performance of the RSF accelerator How to choose θ?
The NRMS error versus iteration of different values of : θ1 = 0.5%, θ2 = 1%, θ3 = 2%, θ4 = 3% and θ5 = 0:1% 12
Optimum performance of the RSF accelerator Change of the strength of smoothing filter
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Optimum performance of the RSF accelerator
The NRMS error versus iteration of different : (σ1 = 2; σ2 = 4; σ3 = 6) of the Gaussian filter
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Optimum performance of the RSF accelerator If there is the best choice of distances for measuring intensities?
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Comparison of the RSF accelerator with other techniques
NRMS (Normalized root mean square) error between the reconstructed and original phase shift vs the number of iterations of 3 techniques in the case of: (a) noise‐free data, (b) SNR=10 white noise data.
HIO: Hybrid input output CGS: Conjugate gradient search
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Comparison of the RSF accelerator with other techniques
Reconstructed phase shift from: (a,d) RSF technique, (b,e) CGS technique, (c,f) HIO technique. 17
Comparison of the RSF accelerator with other techniques
Reconstructed phase shift from: (a,d) RSF technique, (b,e) CGS technique, (c,f) HIO technique. 18
Comparison of the RSF accelerator with other techniques Reconstruction process (noisy data)
https://www.youtube.com/watch?v=lIW0RExGa8M 19
Apply on Diamond data Tomographic experiments
Projection of polymer spheres [polypropylene (PP), cellulose acetate (CA) and polyethylene (PE)] (a) The original image at 2.22 m (b) The reconstructed phase shift at zero plane 20
Tomographic experiments
https://www.youtube.com/watch?v=r1rHZqBwjyY
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Tomographic experiments
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Apply on ESRF data
Absorption function Tomographic reconstruction:
Phase shift
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