An Lp (0 ≤ p ≤ 1)-norm regularized image reconstruction scheme for breast DOT with non-negative-constraint

Abstract

Background: In diffuse optical tomography (DOT), the image reconstruction is often an ill-posed inverse problem, which is even more severe for breast DOT since there are considerably increasing unknowns to reconstruct with regard to the achievable number of measurements. One common way to address this ill-posedness is to introduce various regularization methods. There has been extensive research regarding constructing and optimizing objective functions. However, although these algorithms dramatically improved reconstruction images, few of them have designed an essentially differentiable objective function whose full gradient is easy to obtain to accelerate the optimization process.

Methods: This paper introduces a new kind of non-negative prior information, designing differentiable objective functions for cases of L₁-norm, L_p (0 < p < 1)-norm and L₀-norm. Incorporating this non-negative prior information, it is easy to obtain the gradient of these differentiable objective functions, which is useful to guide the optimization process.

Results: Performance analyses are conducted using both numerical and phantom experiments. In terms of spatial resolution, quantitativeness, gray resolution and execution time, the proposed methods perform better than the conventional regularization methods without this non-negative prior information.

Conclusions: The proposed methods improves the reconstruction images using the introduced non-negative prior information. Furthermore, the non-negative constraint facilitates the gradient computation, accelerating the minimization of the objective functions.

Keywords: Diffuse optical tomography; Inverse problem; Non-negative; Sparsity regularization.

Fig. 1

Structural schematic diagram of the medium. In both the numerical and phantom experiments for the spatial resolution evaluation, the CCS is set to 22, 26, and 30 mm, respectively. For the quantitativeness and gray resolution analysis, the CCS is fixed at 40 mm

Fig. 2

Selection of the optimal regularization parameter: a the generalized L-curve and the red rough range of the optimal λ; b comparison of the normalized metrics including RMSE, AR, CNR and TE. The optimal regularization parameter is λ = 5e − 16, as highlighted by the dotted vertical line

Fig. 3

Numerical experiments for the spatial resolution analysis with absorption coefficients being 0.008 mm⁻¹: a the evaluation metrics of CCS = 22, 26, and 30 mm, respectively, at SNR _min = 20 dB (left) and SNR _min = 30 dB (right); b the reconstructed images of CCS = 22 mm, at SNR _min = 20 dB (left) and SNR _min = 30 dB (right)

Fig. 4

Numerical experiments for the quantitativeness analysis with CCS = 40 mm: a the evaluation metrics with absorption coefficients being 0.005, 0.006, and 0.008 mm⁻¹, respectively, at SNR _min = 20 dB (left) and SNR _min = 30 dB (right); b the reconstructed images with absorption coefficients being 0.005 mm⁻¹, at SNR _min = 20 dB (left) and SNR _min = 30 dB (right)

Fig. 5

Numerical experiments for the grayscale resolution analysis of CCS = 40 mm: a the metrics with absorption coefficients paired as (0.0072, 0.0088 mm⁻¹), (0.0064, 0.0096 mm⁻¹), and (0.0056, 0.0104 mm⁻¹), respectively, at SNR _min = 20 dB (left) and SNR _min = 30 dB (right); b The reconstructed images of absorption coefficients paired as (0.0072, 0.0088 mm⁻¹), at SNR _min = 20 dB (left) and SNR _min = 30 dB (right)

Fig. 6

Comparisons of the execution time required by different regularization methods for the spatial resolution analysis with CCS = 22 mm: a the numerical experiments with SNR _min = 20 dB; b the numerical experiments with SNR _min = 30 dB; c the phantom experiments

Fig. 7

The phantom and optode arrangement: (a)The sketch of the phantom; (b) The phantom photo and optode arrangement

Fig. 8

The phantom experiments for the spatial resolution analysis with absorption coefficients being 0.008 mm⁻¹: a their evaluation metrics with CCS = 22, 26, and 30 mm, respectively; b the reconstructed images with CCS = 22 mm

Fig. 9

The phantom experiments for the quantitativeness analysis with CCS = 40 mm: a the metrics with absorption coefficients being 0.005, 0.006, and 0.008 mm⁻¹, respectively; b The reconstructed images with absorption coefficients being 0.005 mm⁻¹

Fig. 10

Phantom experiments for the grayscale resolution analysis with CCS = 40 mm: a the metrics with absorption coefficients paired as (0.0072, 0.0088 mm⁻¹), (0.0064, 0.0096 mm⁻¹), and (0.0056, 0.0104 mm⁻¹), respectively; b the reconstructed images with absorption coefficients paired as (0.0072, 0.0088 mm⁻¹)