Object localization in the presence of a strong heterogeneous background in fluorescent tomography

Pouyan Mohajerani; Ali A. Eftekhar; Ali Adibi

doi:10.1364/JOSAA.25.001467

Object localization in the presence of a strong heterogeneous background in fluorescent tomography

Pouyan Mohajerani, Ali A. Eftekhar, and Ali Adibi

Journal of the Optical Society of America A
Vol. 25,
Issue 6,
pp. 1467-1479
(2008)
•https://doi.org/10.1364/JOSAA.25.001467

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Pouyan Mohajerani, Ali A. Eftekhar, and Ali Adibi, "Object localization in the presence of a strong heterogeneous background in fluorescent tomography," J. Opt. Soc. Am. A 25, 1467-1479 (2008)

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Related Topics
Table of Contents Category
- Medical Optics and Biotechnology
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Medical optics and biotechnology (170.0170)
- Image reconstruction techniques (170.3010)
- Medical and biological imaging (170.3880)
- Turbid media (170.7050)

About this Article
History
- Original Manuscript: October 19, 2007
- Revised Manuscript: March 11, 2008
- Manuscript Accepted: April 1, 2008
- Published: May 30, 2008
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Abstract

We propose a method for object localization in fluorescent tomography (FT) in the presence of a highly heterogeneous background. Existing approaches typically assume a homogeneous background distribution; thus, they are incapable of accurately accounting for the more general case of an unconstrained, possibly heterogeneous, background. The proposed method iteratively solves the inverse problem over a solution space partitioned into a background subspace and an object subspace to simultaneously estimate the background and localize the target fluorescent objects. Simulation results of this algorithm applied to continuous-wave FT demonstrate effective localization of target objects in the presence of highly heterogeneous background distributions.

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(a) Initialization; $i = 0$ , $X_{b}^{(0)} = 0_{K}$ , and $X_{o}^{(0)} = 0_{K}$ . Start in the OMP mode.
(b) Estimate the background distribution given the current estimation of the object distribution;
$X_{b}^{(i + 1)} = \underset{x ∊ {\bar{X}}_{b}}{\arg \min} ‖ Z x - (M - M_{o}^{(i)}) ‖ = \sum_{j = 1}^{P} a_{j} B_{j}^{ω},$ where
$(a_{1}, \dots, a_{P}) = \underset{a_{1}, \dots, a_{P} ⩾ 0}{\arg \min} ‖ Z \sum_{j = 1}^{P} a_{j} B_{j}^{ω} - (M - M_{o}^{(i)}) ‖ .$ ^a
(c) Estimate the object distribution given the current estimation of the background distribution;
If in the OMP mode, then
$X_{o}^{(i + 1)} = \underset{x ∊ {\bar{X}}_{o}, ‖ x ‖_{0} ⩽ N_{o}}{\arg \min} ‖ Z x - (M - M_{b}^{(i + 1)}) ‖ = \sum_{j ∊ I} b_{j} δ_{j},$ where
$((b_{j})_{j ∊ I}, I) = \underset{b_{j} ⩾ 0, ∣ I ∣ ⩽ N_{o}}{\arg \min} ‖ Z \sum_{j ∊ I} b_{j} δ_{j} - (M - M_{b}^{(i + 1)}) ‖,$
else
$X_{o}^{(i + 1)} = \underset{x ∊ R^{+ K}}{\arg \min} ‖ Z x - (M - M_{b}^{(i + 1)}) ‖ + λ ‖ x ‖_{1} .$ ^b
(d) Calculate the estimation error; $E_{i + 1} = ‖ M - M_{b}^{(i + 1)} - M_{o}^{(i + 1)} ‖$ .
(e) If the estimation $E_{i}$ is not stable,^cthen $i = i + 1$ and go to (b).
(f) If in the OMP mode, then switch to the BP mode and go to (b), else finished.


(a) Initialization; set $n = 1$ and $\tilde{M} = M$ .
(b) Choose the next best index $i_{n} = \underset{i \neq i_{1}, \dots, i_{n - 1}}{\arg \max} ⟨ \tilde{M}, V_{i} ⟩ ∕ (‖ \tilde{M} ‖ ‖ V_{i} ‖)$ .
(c) Form the set of indices $I = (i_{1}, \dots, i_{n})$ .
(d) Estimate the optimal coefficients given the indices $(i_{1}, \dots, i_{n});$
$(a_{1}, \dots, a_{n}) = \underset{a_{1}, \dots, a_{n} ⩾ 0}{\arg \min} ‖ \sum_{j = 1}^{n} a_{j} V_{i_{j}} - M ‖ .$
(e) Calculate the residual signal $\tilde{M} = M - \sum_{j = 1}^{n} a_{j} V_{i_{j}} .$
(f) If $n < N_{o}$ ,then $n = n + 1$ and go to (b).


(a) Initialization; set $i = 0$ , $a_{0} = 0_{N}$ , and $b_{0} = 0_{N}$ .
(b) Find the best vector in $\bar{A}$ given $b_{i}$ ;
$a_{i + 1} = \underset{x ∊ \bar{A}}{\arg \min} ‖ m - b_{i} - x ‖$ .
(c) Find the best vector in $\bar{B}$ given $a_{i + 1}$ ;
$b_{i + 1} = \underset{x ∊ \bar{B}}{\arg \min} ‖ m - a_{i + 1} - x ‖$ .
(d) Calculate the estimation error $E_{i + 1} = ‖ m - a_{i + 1} - b_{i + 1} ‖$ .
(e) If the estimation error $E_{i}$ has converged,^a then $i = i + 1$ and go to (b).
(f) Set $a_{o} = a_{i + 1}$ and $b_{o} = b_{i + 1}$ .


(a) Initialization; $i = 0$ , $X_{b}^{(0)} = 0_{K}$ , and $X_{o}^{(0)} = 0_{K}$ . Start in the OMP mode.
(b) Estimate the background distribution given the current estimation of the object distribution;
$X_{b}^{(i + 1)} = \underset{x ∊ {\bar{X}}_{b}}{\arg \min} ‖ Z x - (M - M_{o}^{(i)}) ‖ = \sum_{j = 1}^{P} a_{j} B_{j}^{ω},$ where
$(a_{1}, \dots, a_{P}) = \underset{a_{1}, \dots, a_{P} ⩾ 0}{\arg \min} ‖ Z \sum_{j = 1}^{P} a_{j} B_{j}^{ω} - (M - M_{o}^{(i)}) ‖ .$ ^a
(c) Estimate the object distribution given the current estimation of the background distribution;
If in the OMP mode, then
$X_{o}^{(i + 1)} = \underset{x ∊ {\bar{X}}_{o}, ‖ x ‖_{0} ⩽ N_{o}}{\arg \min} ‖ Z x - (M - M_{b}^{(i + 1)}) ‖ = \sum_{j ∊ I} b_{j} δ_{j},$ where
$((b_{j})_{j ∊ I}, I) = \underset{b_{j} ⩾ 0, ∣ I ∣ ⩽ N_{o}}{\arg \min} ‖ Z \sum_{j ∊ I} b_{j} δ_{j} - (M - M_{b}^{(i + 1)}) ‖,$
else
$X_{o}^{(i + 1)} = \underset{x ∊ R^{+ K}}{\arg \min} ‖ Z x - (M - M_{b}^{(i + 1)}) ‖ + λ ‖ x ‖_{1} .$ ^b
(d) Calculate the estimation error; $E_{i + 1} = ‖ M - M_{b}^{(i + 1)} - M_{o}^{(i + 1)} ‖$ .
(e) If the estimation $E_{i}$ is not stable,^cthen $i = i + 1$ and go to (b).
(f) If in the OMP mode, then switch to the BP mode and go to (b), else finished.

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