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A diffraction tomographic model of the forward problem using diffuse photon density waves

Charles L. Matson

Open Access Open Access

Abstract

We present a model of the forward problem for diffuse photon density waves in turbid medium using a diffraction tomographic problem formulation. We consider a spatially-varying inhomogeneous structure whose absorption properties satisfy the Born approximation and whose scattering properties are identical to the homogeneous turbid media in which it is imbedded. The two-dimensional Fourier transform of the scattered field, measured in a plane, is shown to be related to the three-dimensional Fourier transform of the object evaluated on a surface which in many cases is approximately a plane.

©1997 Optical Society of America

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Figures (1)
Figure 1
Figure 1 Plots of the two-dimensional projection of the surface on which the Fourier transform of the convolved object function is obtained with diffraction tomography. The dotted line is for real values of k, the dashed line is for Re{k2}<<Im{k2}, and the solid line is for Re{k2}>>Im{k2}. The horizontal axis is ωx and the vertical axis is ωy.

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Equations (22)

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( ∇ 2 + k 2 ) u B ( r → ) = − o ( r → ) u o ( r → )
u B ( r → ) = ∫ o ( r → ′ ) u o ( r → ′ ) g ( r → − r → ′ ) d r′ →
g ( r → ) = exp [ ik ∣ r → ∣ ] 4 π ∣ r → ∣
= 1 8 π 2 ∫ ∫ 1 α x 2 + α y 2 − k 2 exp { − ∣ z ∣ α x 2 + α y 2 − k 2 + ix α x + iy α y } dα x dα y
u B ( r → ) = 1 8 π 2 ∫ o ( r → ′ ) u o ( r → ′ ) ∫ ∫ 1 α x 2 + α y 2 − k 2 exp { − ∣ z − z′ ∣
× α x 2 + α y 2 − k 2 + i ( x − x′ ) α x + i ( y − y′ ) α y } dα x dα y d r′ →
= 1 8 π 2 ∫ o ( r ′ → ) u o ( r → ′ ) ∫ ∫ 1 γ exp { − ∣ z − z′ ∣ ( γ r + i γ i )
+ i ( x − x′ ) α x + i ( y − y′ ) α y } dα x dα y d r′ →
γ ≡ γ r + i γ i
= Re { α x 2 + α y 2 − k 2 } + i Im { α x 2 + α y 2 − k 2 }
u B ( x , y , z o ) = 1 8 π 2 ∫ ∫ exp { ix α x + iy α y − iz o γ i } γ
× ∫ ∫ ∫ o ( x′ , y′ , z′ ) u o ( x′ , y′ , z′ ) exp { − ∣ z o − z′ ∣ γ r }
× exp { − ix′ α x − iy ′ α y − iz′ ( − γ i ) } dx ′ dy ′ dz′ dα x dα y
u B ( x , y , z o ) = 1 8 π 2 ∫ ∫ exp { ix α x + iy α y − i z o γ i } γ O u γ ( α x , α y , − γ i ) dα x dα y
u B ( ω x , ω y , z o ) = 1 8 π 2 ∫ ∫ exp { − iz o γ i } γ O u γ ( α x , α y , − γ i )
× ∫ ∫ exp { i ( x α x + y α y ) } exp { − i ( xω x + yω y ) } dxdyd α x d α y
U B ( ω x , ω y , z o ) = 1 2 exp { − iz o γ i } γ O u γ ( ω x , ω y , − γ i )
γ i = Im { ω x 2 + ω y 2 − k 2 }
k 2 = ( − v μ a + i 2 π f t ) 3 μ′ s v
k 2 = − 15 + i 9 f t
G ( ω x , ω y , ω z ) = 4 π 2 δ ( ω x ) δ ( ω y ) 2 γ r γ r 2 + ω z 2
O u γ ( ω x , ω y , − γ i ) ≈ O u γ ( ω x , ω y , 0 )
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