Diffuse optical tomography (DOT) reconstructs the distributions of the optical properties, such as the scattering and absorption coefficients in the biological media. Near infrared (NIR) light incident on the surface of the biological medium propagates diffusively inside the biological medium, and the fractions of the NIR light are reemitted from and detected at the surface of the medium. By solving the inverse problem based on the photon diffusion equation with the measurement data of the detected light, DOT images are obtained noninvasively [1
The reconstructed optical properties contain the information about not only the structure but also the constituents in the measured object. By using the spectroscopic technique, the concentrations of hemoglobin, lipid etc. can be calculated from the optical properties. The abnormal absorption coefficients which can be caused by the tumor with angiogenesis are detected by DOT [3
]. Therefore DOT attracts attentions as a new imaging modality for breast cancer diagnoses. DOT also can monitor the brain activities. The large change in the regional blood flow due to the higher brain activities is detectable by DOT [8
The advantage of the near infrared optical imaging modalities is that the measuring instruments are smaller in size and easier to use than the other modalities. Therefore, it is desirable that the number of the detectors is small. However, when the number of the detectors is limited, the inverse problem of DOT is highly ill-posed. The reconstructed optical properties usually have a low spatial resolution, and measurement noises significantly affect the reconstructed images. One of the solutions of these problems is to use a regularization method in the image reconstruction. So, recently the regularization techniques are actively studied for improvement of the DOT image quality.
The important approach is to incorporate the structural prior information. Pogue et al [10
] uses a spatially variant Tikhonov regularization to reduce high frequency noises in the reconstructed images. Boverman et al employed prior segmentation of breast into glandular and adipose tissues [11
]. Yalavarthy et al uses MRI-derived breast geometry to regularize the inverse solution [12
]. Douiri et al applied anisotropic diffusion regularization with a priori edge information of the object to preserve the edge of the inner structure [13
]. An algorithm assuming that the targets have blocky structures for regularization is introduced by Hiltunen et al [14
]. Mutual information and joint entropy are used by Panagiotou et al to reflect the structure obtained from an alternative high resolution modality in the DOT images [15
Another possible approach to improve the spatial resolution of the DOT image is to use a sparsity constraint. Cao et al implements L1
norm minimization by use of an expectation maximization algorithm for a linearized DOT inverse problem, and show that the reconstructed region with abnormal optical properties are localized more than the other methods they use [16
Sparsity regularization with L1
norm and other methods to obtain sparse solution are used to several applications of inverse problems. Image restoration with sparsity constrained regularization is proposed by Shankar et al [17
]. The L1
sparsity constraint is applied to fluorescence/bioluminescence diffuse optical tomography (F/BDOT) [18
]. Okawa et al used recursive spatial filtering for F/BDOT [20
]. And other applications are found in the functional brain imaging such as electroencephalography [21
When the changes in the optical properties, which can be caused by the regional blood flow changes or the breast cancer in early stage, are expected to be localized, the sparsity regularization will provide a good reconstruction in DOT inverse problem.
In this paper, we apply the lp
(0 < p
≤ 1) sparsity regularization by partial use of the focal underdetermined system solver (FOCUSS) algorithm, which was introduced by He et al. [23
] to solve linear underdetermined inverse problems, and investigate the effects of the regularization on the non-linear time-domain DOT image reconstruction.
The changes in the absorption coefficients are described by a parameter to solve the difficulty in lp minimization. And the DOT images are reconstructed by minimizing the residual error between the measurement and predicted data sets and the lp norm of the changes in the absorption coefficients simultaneously. Numerical simulations and a phantom experiment demonstrate that the regularization improves the localization of the changes in the distribution of the absorption coefficient.