Graph- and finite element-based total variation models for the inverse problem in diffuse optical tomography

Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to complex and unstructured geometries, non-linearity of the data fitting and regularization terms, and non-differentiability of the regularization term. We develop several approaches to overcome these difficulties by: i) defining discrete differential operators for unstructured geometries using both finite element and graph representations; ii) developing an optimization algorithm based on the alternating direction method of multipliers (ADMM) for the non-differentiable and non-linear minimization problem; iii) investigating isotropic and anisotropic variants of TV regularization, and comparing their finite element- and graph-based implementations. These approaches are evaluated on experiments on simulated data and real data acquired from a tissue phantom. Our results show that both FEM and graph-based TV regularization is able to accurately reconstruct both sparse and non-sparse distributions without the over-smoothing effect of Tikhonov regularization and the over-sparsifying effect of L$_1$ regularization. The graph representation was found to out-perform the FEM method for low-resolution meshes, and the FEM method was found to be more accurate for high-resolution meshes.Comment: 24 pages, 11 figures. Reviced version includes revised figures and improved clarit

Direct estimation of kinetic parametric images for dynamic PET.

Dynamic positron emission tomography (PET) can monitor spatiotemporal distribution of radiotracer in vivo. The spatiotemporal information can be used to estimate parametric images of radiotracer kinetics that are of physiological and biochemical interests. Direct estimation of parametric images from raw projection data allows accurate noise modeling and has been shown to offer better image quality than conventional indirect methods, which reconstruct a sequence of PET images first and then perform tracer kinetic modeling pixel-by-pixel. Direct reconstruction of parametric images has gained increasing interests with the advances in computing hardware. Many direct reconstruction algorithms have been developed for different kinetic models. In this paper we review the recent progress in the development of direct reconstruction algorithms for parametric image estimation. Algorithms for linear and nonlinear kinetic models are described and their properties are discussed

Two-dimensional Tissue Image Reconstruction Based on Magnetic Field Data

This paper introduces new possibilities within two-dimensional reconstruction of internal conductivity distribution. In addition to the electric field inside the given object, the injected current causes a magnetic field which can be measured either outside the object by means of a Hall probe or inside the object through magnetic resonance imaging. The Magnetic Resonance method, together with Electrical impedance tomography (MREIT), is well known as a bio-imaging modality providing cross-sectional conductivity images with a good spatial resolution from the measurements of internal magnetic flux density produced by externally injected currents. A new algorithm for the conductivity reconstruction, which utilizes the internal current information with respect to corresponding boundary conditions and the external magnetic field, was developed. A series of computer simulations has been conducted to assess the performance of the proposed algorithm within the process of estimating electrical conductivity changes in the lungs, heart, and brain tissues captured in two-dimensional piecewise homogeneous chest and head models. The reconstructed conductivity distribution using the proposed method is compared with that using a conventional method based on Electrical Impedance Tomography (EIT). The acquired experience is discussed and the direction of further research is proposed

Nonlinear tube-fitting for the analysis of anatomical and functional structures

We are concerned with the estimation of the exterior surface and interior summaries of tube-shaped anatomical structures. This interest is motivated by two distinct scientific goals, one dealing with the distribution of HIV microbicide in the colon and the other with measuring degradation in white-matter tracts in the brain. Our problem is posed as the estimation of the support of a distribution in three dimensions from a sample from that distribution, possibly measured with error. We propose a novel tube-fitting algorithm to construct such estimators. Further, we conduct a simulation study to aid in the choice of a key parameter of the algorithm, and we test our algorithm with validation study tailored to the motivating data sets. Finally, we apply the tube-fitting algorithm to a colon image produced by single photon emission computed tomography (SPECT) and to a white-matter tract image produced using diffusion tensor imaging (DTI).Comment: Published in at http://dx.doi.org/10.1214/10-AOAS384 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

WARP: Wavelets with adaptive recursive partitioning for multi-dimensional data

Effective identification of asymmetric and local features in images and other data observed on multi-dimensional grids plays a critical role in a wide range of applications including biomedical and natural image processing. Moreover, the ever increasing amount of image data, in terms of both the resolution per image and the number of images processed per application, requires algorithms and methods for such applications to be computationally efficient. We develop a new probabilistic framework for multi-dimensional data to overcome these challenges through incorporating data adaptivity into discrete wavelet transforms, thereby allowing them to adapt to the geometric structure of the data while maintaining the linear computational scalability. By exploiting a connection between the local directionality of wavelet transforms and recursive dyadic partitioning on the grid points of the observation, we obtain the desired adaptivity through adding to the traditional Bayesian wavelet regression framework an additional layer of Bayesian modeling on the space of recursive partitions over the grid points. We derive the corresponding inference recipe in the form of a recursive representation of the exact posterior, and develop a class of efficient recursive message passing algorithms for achieving exact Bayesian inference with a computational complexity linear in the resolution and sample size of the images. While our framework is applicable to a range of problems including multi-dimensional signal processing, compression, and structural learning, we illustrate its work and evaluate its performance in the context of 2D and 3D image reconstruction using real images from the ImageNet database. We also apply the framework to analyze a data set from retinal optical coherence tomography