Convergence for nonconvex ADMM, with applications to CT imaging.

The alternating direction method of multipliers (ADMM) algorithm is a powerful and flexible tool for complex optimization problems of the form . ADMM exhibits robust empirical performance across a range of challenging settings including nonsmoothness and nonconvexity of the objective functions and , and provides a simple and natural approach to the inverse problem of image reconstruction for computed tomography (CT) imaging. From the theoretical point of view, existing results for convergence in the nonconvex setting generally assume smoothness in at least one of the component functions in the objective.

Simultaneous Activity and Attenuation Estimation in TOF-PET With TV-Constrained Nonconvex Optimization.

An alternating direction method of multipliers (ADMM) framework is developed for nonsmooth biconvex optimization for inverse problems in imaging. In particular, the simultaneous estimation of activity and attenuation (SAA) problem in time-of-flight positron emission tomography (TOF-PET) has such a structure when maximum likelihood estimation (MLE) is employed. The ADMM framework is applied to MLE for SAA in TOF-PET, resulting in the ADMM-SAA algorithm.

Black-box tests for algorithmic stability.

Algorithmic stability is a concept from learning theory that expresses the degree to which changes to the input data (e.g. removal of a single data point) may affect the outputs of a regression algorithm.

Constrained one-step material decomposition reconstruction of head CT data from a silicon photon-counting prototype.

Background: Spectral CT material decomposition provides quantitative information but is challenged by the instability of the inversion into basis materials. We have previously proposed the constrained One-Step Spectral CT Image Reconstruction (cOSSCIR) algorithm to stabilize the material decomposition inversion by directly estimating basis material images from spectral CT data. cOSSCIR was previously investigated on phantom data.

Pharmacokinetic Analysis of Enhancement-Constrained Acceleration (ECA) reconstruction-based high temporal resolution breast DCE-MRI.

The high spatial and temporal resolution of dynamic contrast-enhanced MRI (DCE-MRI) can improve the diagnostic accuracy of breast cancer screening in patients who have dense breasts or are at high risk of breast cancer. However, the spatiotemporal resolution of DCE-MRI is limited by technical issues in clinical practice. Our earlier work demonstrated the use of image reconstruction with enhancement-constrained acceleration (ECA) to increase temporal resolution.

Simultaneous activity and attenuation estimation in TOF-PET with TV-constrained nonconvex optimization.

An alternating direction method of multipliers (ADMM) framework is developed for nonsmooth biconvex optimization for inverse problems in imaging. In particular, the simultaneous estimation of activity and attenuation (SAA) problem in time-of-flight positron emission tomography (TOF-PET) has such a structure when maximum likelihood estimation (MLE) is employed. The ADMM framework is applied to MLE for SAA in TOF-PET, resulting in the ADMM-SAA algorithm.

Addressing CT metal artifacts using photon-counting detectors and one-step spectral CT image reconstruction.

Purpose: The constrained one-step spectral CT image reconstruction (cOSSCIR) algorithm with a nonconvex alternating direction method of multipliers optimizer is proposed for addressing computed tomography (CT) metal artifacts caused by beam hardening, noise, and photon starvation. The quantitative performance of cOSSCIR is investigated through a series of photon-counting CT simulations.

Methods: cOSSCIR directly estimates basis material maps from photon-counting data using a physics-based forward model that accounts for beam hardening.

Enhancement-constrained acceleration: A robust reconstruction framework in breast DCE-MRI.

In patients with dense breasts or at high risk of breast cancer, dynamic contrast enhanced MRI (DCE-MRI) is a highly sensitive diagnostic tool. However, its specificity is highly variable and sometimes low; quantitative measurements of contrast uptake parameters may improve specificity and mitigate this issue. To improve diagnostic accuracy, data need to be captured at high spatial and temporal resolution.

Fast and flexible estimation of effective migration surfaces.

Spatial population genetic data often exhibits 'isolation-by-distance,' where genetic similarity tends to decrease as individuals become more geographically distant. The rate at which genetic similarity decays with distance is often spatially heterogeneous due to variable population processes like genetic drift, gene flow, and natural selection. Petkova et al.

The bias of isotonic regression.

We study the bias of the isotonic regression estimator. While there is extensive work characterizing the mean squared error of the isotonic regression estimator, relatively little is known about the bias. In this paper, we provide a sharp characterization, proving that the bias scales as ( ) up to log factors, where 1 ≤ ≤ 2 is the exponent corresponding to Hölder smoothness of the underlying mean.

Estimating the spectrum in computed tomography via Kullback-Leibler divergence constrained optimization.

Purpose: We study the problem of spectrum estimation from transmission data of a known phantom. The goal is to reconstruct an x-ray spectrum that can accurately model the x-ray transmission curves and reflects a realistic shape of the typical energy spectra of the CT system.

Methods: Spectrum estimation is posed as an optimization problem with x-ray spectrum as unknown variables, and a Kullback-Leibler (KL)-divergence constraint is employed to incorporate prior knowledge of the spectrum and enhance numerical stability of the estimation process.

EigenPrism: inference for high dimensional signal-to-noise ratios.

Consider the following three important problems in statistical inference, namely, constructing confidence intervals for (1) the error of a high-dimensional ( > ) regression estimator, (2) the linear regression noise level, and (3) the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the [Formula: see text]-norm of the signal in high-dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well.

A Spectral CT Method to Directly Estimate Basis Material Maps From Experimental Photon-Counting Data.

The proposed spectral CT method solves the constrained one-step spectral CT reconstruction (cOSSCIR) optimization problem to estimate basis material maps while modeling the nonlinear X-ray detection process and enforcing convex constraints on the basis map images. In order to apply the optimization-based reconstruction approach to experimental data, the presented method empirically estimates the effective energy-window spectra using a calibration procedure. The amplitudes of the estimated spectra were further optimized as part of the reconstruction process to reduce ring artifacts.

MOCCA: Mirrored Convex/Concave Optimization for Nonconvex Composite Functions.

Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative algorithms. In this paper, we are interested in optimization problems of the form F() + G(), where is a fixed linear transformation, while F and G are functions that may be nonconvex and/or nondifferentiable. In particular, if either of the terms are nonconvex, existing alternating minimization techniques may fail to converge; other types of existing approaches may instead be unable to handle nondifferentiability.