Quantification and Reconstruction in Photoacoustic Tomography

Optical absorption is closely associated with many physiological important parameters, such as the concentration and oxygen saturation of hemoglobin. Conventionally, accurate quantification in PAT requires knowledge of the optical fluence attenuation, acoustic pressure attenuation, and detection bandwidth. We circumvent this requirement by quantifying the optical absorption coefficients from the acoustic spectra of PA signals acquired at multiple optical wavelengths. We demonstrate the method using the optical-resolution photoacoustic microscopy: OR-PAM) and the acoustical-resolution photoacoustic microscopy: AR-PAM) in the optical ballistic regime and in the optical diffusive regime, respectively. The data acquisition speed in photoacoustic computed tomography: PACT) is limited by the laser repetition rate and the number of parallel ultrasound detecting channels. Reconstructing an image with fewer measurements can effectively accelerate the data acquisition and reduce the system cost. We adapted Compressed Sensing: CS) for the reconstruction in PACT. CS-based PACT was implemented as a non-linear conjugate gradient descent algorithm and tested with both phantom and in vivo experiments. Speckles have been considered ubiquitous in all scattering-based coherent imaging technologies. As a coherent imaging modality based on optical absorption, photoacoustic: PA) tomography: PAT) is generally devoid of speckles. PAT suppresses speckles by building up prominent boundary signals, via a mechanism similar to that of specular reflection. When imaging smooth boundary absorbing targets, the speckle visibility in PAT, which is defined as the ratio of the square root of the average power of speckles to that of boundaries, is inversely proportional to the square root of the absorber density. If the surfaces of the absorbing targets have uncorrelated height fluctuations, however, the boundary features may become fully developed speckles. The findings were validated by simulations and experiments. The first- and second-order statistics of PAT speckles were also studied experimentally. While the amplitude of the speckles follows a Gaussian distribution, the autocorrelation of the speckle patterns tracks that of the system point spread function

Statistical Inference For High-Dimensional Linear Models

High-dimensional linear models play an important role in the analysis of modern data sets. Although the estimation problem has been well understood, there is still a paucity of methods and theories on the inference problem for high-dimensional linear models. This thesis focuses on statistical inference for high-dimensional linear models and consists of the following three parts. 1. The first part of the thesis considers confidence intervals for linear functionals in high-dimensional linear regression. We first establish the convergence rates of the minimax expected length for confidence intervals. Furthermore, we investigate the problem of adaptation to sparsity for the construction of confidence intervals and identify the regimes in which it is possible to construct adaptive confidence intervals. 2. In the second part of the thesis, we consider point and interval estimation of the $\ell_q$ loss of a given estimator in high-dimensional linear regression. For the class of rate-optimal estimators, we establish the minimax rates for estimating their $\ell_{q}$ losses, the minimax expected length of confidence intervals for their $\ell_{q}$ losses and the possibility of adaptivity of confidence intervals for their $\ell_q$ losses. 3. In the third part of the thesis, we consider the problem in the framework of high-dimensional instrumental variable regression and construct confidence intervals for the treatment effect in the presence of possibly invalid instrumental variables. We develop a novel selection procedure, Two-Stage Hard Thresholding (TSHT) to select valid instrumental variables and construct honest confidence intervals for the treatment effect using the selected instrumental variables

Inference for High-dimensional Maximin Effects in Heterogeneous Regression Models Using a Sampling Approach

Heterogeneity is an important feature of modern data sets and a central task is to extract information from large-scale and heterogeneous data. In this paper, we consider multiple high-dimensional linear models and adopt the definition of maximin effect (Meinshausen, B{\"u}hlmann, AoS, 43(4), 1801--1830) to summarize the information contained in this heterogeneous model. We define the maximin effect for a targeted population whose covariate distribution is possibly different from that of the observed data. We further introduce a ridge-type maximin effect to simultaneously account for reward optimality and statistical stability. To identify the high-dimensional maximin effect, we estimate the regression covariance matrix by a debiased estimator and use it to construct the aggregation weights for the maximin effect. A main challenge for statistical inference is that the estimated weights might have a mixture distribution and the resulted maximin effect estimator is not necessarily asymptotic normal. To address this, we devise a novel sampling approach to construct the confidence interval for any linear contrast of high-dimensional maximin effects. The coverage and precision properties of the proposed confidence interval are studied. The proposed method is demonstrated over simulations and a genetic data set on yeast colony growth under different environments

Confidence Intervals for High-Dimensional Linear Regression: Minimax Rates and Adaptivity

Confidence sets play a fundamental role in statistical inference. In this paper, we consider confidence intervals for high-dimensional linear regression with random design. We first establish the convergence rates of the minimax expected length for confidence intervals in the oracle setting where the sparsity parameter is given. The focus is then on the problem of adaptation to sparsity for the construction of confidence intervals. Ideally, an adaptive confidence interval should have its length automatically adjusted to the sparsity of the unknown regression vector, while maintaining a pre-specified coverage probability. It is shown that such a goal is in general not attainable, except when the sparsity parameter is restricted to a small region over which the confidence intervals have the optimal length of the usual parametric rate. It is further demonstrated that the lack of adaptivity is not due to the conservativeness of the minimax framework, but is fundamentally caused by the difficulty of learning the bias accurately