Photon-measurement density functions. Part I: Analytical forms

Simon R. Arridge

doi:10.1364/AO.34.007395

Applied Optics
Vol. 34,
Issue 31,
pp. 7395-7409
(1995)
•https://doi.org/10.1364/AO.34.007395

Photon-measurement density functions. Part I: Analytical forms

Simon R. Arridge

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Simon R. Arridge, "Photon-measurement density functions. Part I: Analytical forms," Appl. Opt. 34, 7395-7409 (1995)

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Abstract

This paper addresses the problem of tomographic reconstruction of absorption and scattering parameters in the optical region from measurements of transilluminated light. Specifically, the question of the sensitivity of different measurement schemes on the boundary of an object to perturbations of the optical parameters within the object are addressed. The concept of a photon-sampling volume [Appl. Opt. 33, 448 (1994)] and a photon-hitting density [Appl. Opt. 32, 448 (1993)] is extended to a photon-measurement density function (PMDF). The PMDF is derived from the Green’s function of the diffusion equation and can be expressed for measurements such as the time-varying intensity, integrated intensity, temporal moments, and phase shift, as well as for both absorption and diffusion perturbations. Closed-form solutions are given for a number of these functions in infinite space, half-space, and slab geometries.

Example results are given in terms of three-dimensional images.

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Variable	Formula
σ	[(γ + iω)/κ]^1/2, ω > 0
σ₀	(γ/κ)^1/2
h ₂₃	[(x ₃ − x ₂)² + (y ₃ − y ₂)²]^1/2
h ₁₂	[(x ₂ − x ₁)² + (y ₂ − y ₁)²]^1/2
ρ₁₂₋	[h ₁₂ ² + (z ₂ − z ₁)²]^1/2
ρ₁₂₊	[h ₁₂ ² + (z ₂ + z ₁)²]^1/2
z ₂₃₊ _n	(2n + 1)d + z ₂
z ₂₃₋ _n	(2n + 1)d − z ₂
z ₁₂₊ _n _′	2nd + z ₂ + z ₁
z ₁₂₋ _n _′	2nd + z ₂ − z ₁
ρ₂₃₊ _n	(h ₂₃ ² + z ₂₃₊ _n ²)^1/2
ρ₂₃₋ _n	(h ₂₃ ² + z ₂₃₋ _n ²)^1/2
ρ₁₂₊ _n _′	(h ₁₂ ² + z ₁₂₊ _n _′ ²)^1/2
ρ₁₂₋ _n _′	(h ₁₂ ² + z ₁₂₋ _n _′ ²)^1/2

	Formula

Function	Time Domain	Frequency Domain
f(x)	${(2 κ t)}^{- 3 / 2} exp (- γ t - \frac{x^{2}}{4 κ t})$	$\frac{1}{2 κ} \frac{exp (- σ x)}{x}$
S ^(Γ)(x, y)	$\frac{1}{2 {(2 π)}^{3}} (\frac{1}{y^{2}} + \frac{x + y}{2 κ t} [\frac{1}{x} + \frac{1}{y}]) f (x + y, t)$	$\frac{1}{4 {(2 π)}^{3} κ} \frac{1 + σ y}{x y^{2}} exp [- σ (x + y)]$
T ^(Γ)(x, y)	$[{(\frac{1}{x} + \frac{1}{y})}^{3} (\frac{x y}{2 κ t^{2}}) + (\frac{1}{x^{3}} + \frac{1}{y^{3}}) \frac{1}{t}] \frac{1}{4 κ {(2 π)}^{3}} f (x + y, t)$	$\frac{1}{4 {(2 π)}^{3} κ} (\frac{1 + σ y}{y^{3}}) (\frac{1 + σ x}{x^{3}}) exp [- σ (x + y)]$
W ^(Γ)(x, y)	$[\frac{3 x}{y^{4} t} + \frac{{(x + y)}^{2} (3 x^{2} - 2 x y + y^{2})}{x^{2} y^{3}} (\frac{1}{2 κ t^{2}}) + \frac{{(x + y)}^{4}}{x y^{2}} \frac{1}{4 κ^{2} t^{3}}] \times \frac{1}{4 κ {(2 π)}^{3}} f (x + y, t)$	$\frac{1}{4 {(2 π)}^{3} κ} (\frac{3 + 3 σ y + σ^{2} y^{2}}{y^{4}}) (\frac{1 + σ x}{x^{2}}) exp [- σ (x + y)]$

Variable	Formula
σ	[(γ + iω)/κ]^1/2, ω > 0
σ₀	(γ/κ)^1/2
h ₂₃	[(x ₃ − x ₂)² + (y ₃ − y ₂)²]^1/2
h ₁₂	[(x ₂ − x ₁)² + (y ₂ − y ₁)²]^1/2
ρ₁₂₋	[h ₁₂ ² + (z ₂ − z ₁)²]^1/2
ρ₁₂₊	[h ₁₂ ² + (z ₂ + z ₁)²]^1/2
z ₂₃₊ _n	(2n + 1)d + z ₂
z ₂₃₋ _n	(2n + 1)d − z ₂
z ₁₂₊ _n _′	2nd + z ₂ + z ₁
z ₁₂₋ _n _′	2nd + z ₂ − z ₁
ρ₂₃₊ _n	(h ₂₃ ² + z ₂₃₊ _n ²)^1/2
ρ₂₃₋ _n	(h ₂₃ ² + z ₂₃₋ _n ²)^1/2
ρ₁₂₊ _n _′	(h ₁₂ ² + z ₁₂₊ _n _′ ²)^1/2
ρ₁₂₋ _n _′	(h ₁₂ ² + z ₁₂₋ _n _′ ²)^1/2

	Formula

Function	Time Domain	Frequency Domain
f(x)	${(2 κ t)}^{- 3 / 2} exp (- γ t - \frac{x^{2}}{4 κ t})$	$\frac{1}{2 κ} \frac{exp (- σ x)}{x}$
S ^(Γ)(x, y)	$\frac{1}{2 {(2 π)}^{3}} (\frac{1}{y^{2}} + \frac{x + y}{2 κ t} [\frac{1}{x} + \frac{1}{y}]) f (x + y, t)$	$\frac{1}{4 {(2 π)}^{3} κ} \frac{1 + σ y}{x y^{2}} exp [- σ (x + y)]$
T ^(Γ)(x, y)	$[{(\frac{1}{x} + \frac{1}{y})}^{3} (\frac{x y}{2 κ t^{2}}) + (\frac{1}{x^{3}} + \frac{1}{y^{3}}) \frac{1}{t}] \frac{1}{4 κ {(2 π)}^{3}} f (x + y, t)$	$\frac{1}{4 {(2 π)}^{3} κ} (\frac{1 + σ y}{y^{3}}) (\frac{1 + σ x}{x^{3}}) exp [- σ (x + y)]$
W ^(Γ)(x, y)	$[\frac{3 x}{y^{4} t} + \frac{{(x + y)}^{2} (3 x^{2} - 2 x y + y^{2})}{x^{2} y^{3}} (\frac{1}{2 κ t^{2}}) + \frac{{(x + y)}^{4}}{x y^{2}} \frac{1}{4 κ^{2} t^{3}}] \times \frac{1}{4 κ {(2 π)}^{3}} f (x + y, t)$	$\frac{1}{4 {(2 π)}^{3} κ} (\frac{3 + 3 σ y + σ^{2} y^{2}}{y^{4}}) (\frac{1 + σ x}{x^{2}}) exp [- σ (x + y)]$

Abstract

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Equations (65)

Applied Optics