^{1,a)}, Brian W. Pogue

^{1}, Hamid Dehghani

^{2}and Keith D. Paulsen

^{3}

### Abstract

Diffuse optical tomography (DOT) involves estimation of tissue optical properties using noninvasive boundary measurements. The image reconstruction procedure is a nonlinear, ill-posed, and ill-determined problem, so overcoming these difficulties requires regularization of the solution. While the methods developed for solving the DOT image reconstruction procedure have a long history, there is less direct evidence on the optimal regularization methods, or exploring a common theoretical framework for techniques which uses least-squares (LS) minimization. A generalized least-squares (GLS) method is discussed here, which takes into account the variances and covariances among the individual data points and optical properties in the image into a structured weight matrix. It is shown that most of the least-squares techniques applied in DOT can be considered as special cases of this more generalized LS approach. The performance of three minimization techniques using the same implementation scheme is compared using test problems with increasing noise level and increasing complexity within the imaging field. Techniques that use spatial-prior information as constraints can be also incorporated into the GLS formalism. It is also illustrated that inclusion of spatial priors reduces the image error by at least a factor of 2. The improvement of GLS minimization is even more apparent when the noise level in the data is high (as high as 10%), indicating that the benefits of this approach are important for reconstruction of data in a routine setting where the data variance can be known based upon the signal to noiseproperties of the instruments.

The authors are grateful to Professor Daniel R. Lynch for the useful discussions and valuable comments on this article. P.K.Y. acknowledges the DOD Breast Cancer predoctoral fellowship (BC050309). This work has been sponsored by the National Cancer Institute through Grant Nos. RO1CA78734, PO1CA80139, and DAMD17-03-1-0405.

I. INTRODUCTION

II. DOT FORWARD PROBLEM

III. LEAST-SQUARES MINIMIZATION TECHNIQUES

III.A. Without spatial priors

III.A.1. Levenberg–Marquardt minimization

III.A.2. Tikhonov minimization

III.A.3. GLS minimization

III.A.4. Choice of

III.A.5. Choices of

III.B. With spatial priors

III.B.1. Soft-priors

III.B.2. Hard-priors

III.B.3. Important notes about minimization schemes

III.B.4. Special cases of GLS minimization

III.B.5. Stopping criterion

IV. TEST PROBLEM

V. RESULTS AND DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Optical properties
- 35.0
- Image reconstruction
- 18.0
- Medical imaging
- 16.0
- Medical image noise
- 9.0
- Inverse problems
- 8.0

## Figures

An illustration of the forward and inverse problem in diffuse optical tomography is shown (see Ref. 64), where (a) the data is estimated given values of and and source/detector positions. In the inverse problem (b), the values of and must be obtained given a set of measurements .

An illustration of the forward and inverse problem in diffuse optical tomography is shown (see Ref. 64), where (a) the data is estimated given values of and and source/detector positions. In the inverse problem (b), the values of and must be obtained given a set of measurements .

The chosen optical property distribution/domain for the generation of synthetic data is shown. The diameter of the domain was .

The chosen optical property distribution/domain for the generation of synthetic data is shown. The diameter of the domain was .

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using noiseless data (bias calculations) (a) without spatial priors and (b) with spatial priors. The top row contains images of and bottom row shows images.

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using noiseless data (bias calculations) (a) without spatial priors and (b) with spatial priors. The top row contains images of and bottom row shows images.

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using 5% noisy data (a) without spatial priors and (b) with spatial priors. The top row gives images of and bottom row shows images.

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using 5% noisy data (a) without spatial priors and (b) with spatial priors. The top row gives images of and bottom row shows images.

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using 10% noisy data (a) without spatial priors and (b) with spatial priors. The top row gives images of and bottom row shows images.

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using 10% noisy data (a) without spatial priors and (b) with spatial priors. The top row gives images of and bottom row shows images.

A plot of the rms error in the estimated optical properties is shown as a function of increasing noise level for all reconstruction techniques.

A plot of the rms error in the estimated optical properties is shown as a function of increasing noise level for all reconstruction techniques.

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using 3% noisy data (a) without spatial priors and (b) with spatial priors for four targets in the tissue as shown. The top row gives images of and bottom row shows images. The actual and with target numbers are given in the first column of (a).

Reconstruction results (top of the first row, abbreviations are given in Appendix A) are shown using 3% noisy data (a) without spatial priors and (b) with spatial priors for four targets in the tissue as shown. The top row gives images of and bottom row shows images. The actual and with target numbers are given in the first column of (a).

Plot of the rms error in the estimated optical properties is shown for increasing number of targets with 3% noise in the data for all reconstruction techniques (legend of the figure). Abbreviations used for the techniques are given in Appendix A. The targets used are numbered in the images presented in Fig. 7(a).

Plot of the rms error in the estimated optical properties is shown for increasing number of targets with 3% noise in the data for all reconstruction techniques (legend of the figure). Abbreviations used for the techniques are given in Appendix A. The targets used are numbered in the images presented in Fig. 7(a).

## Tables

Mean and standard deviation of the reconstructed: (a) and (b) values (in ) for different regions [labeled in first column of Fig. 3(a)] recovered with data having 0%, 5%, 10% noise for images shown in Figs. 3–5.

Mean and standard deviation of the reconstructed: (a) and (b) values (in ) for different regions [labeled in first column of Fig. 3(a)] recovered with data having 0%, 5%, 10% noise for images shown in Figs. 3–5.

Mean and standard deviation of the reconstructed: (a) and (b) values (in ) for different regions [labeled in first column of Fig. 7(a)] recovered with data having 3% noise for images shown in Fig. 7.

Mean and standard deviation of the reconstructed: (a) and (b) values (in ) for different regions [labeled in first column of Fig. 7(a)] recovered with data having 3% noise for images shown in Fig. 7.

Comparison of computation time per iteration for different reconstruction techniques on Pentium IV (dual core) , RAM Linux work station. the abbreviations used for the reconstruction techniques are given in Appendix A.

Comparison of computation time per iteration for different reconstruction techniques on Pentium IV (dual core) , RAM Linux work station. the abbreviations used for the reconstruction techniques are given in Appendix A.

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