A Variational Reconstruction Method for Undersampled Dynamic X-ray Tomography based on Physical Motion Models
Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计学系）
Martin Burger, Hendrik Dirks, Lena Frerking, Andreas Hauptmann, Tapio Helin, Samuli Siltanen
发表信息：
Inverse Problems, Volume 33, Number 12, 124008 (2017)
In this paper we study the reconstruction of moving object densities from undersampled dynamic X-ray tomography in two dimensions. A particular motivation of this study is to use realistic measurement protocols for practical applications, i.e. we do not assume to have a full Radon transform in each time step, but only projections in few angular directions. This restriction enforces a space-time reconstruction, which we perform by incorporating physical motion models and regularization of motion vectors in a variational framework. The methodology of optical flow, which is one of the most common methods to estimate motion between two images, is utilized to formulate a joint variational model for reconstruction and motion estimation.
We provide a basic mathematical analysis of the forward model and the variational model for the image reconstruction. Moreover, we discuss the efficient numerical minimization based on alternating minimizations between images and motion vectors. A variety of results are presented for simulated and real measurement data with different sampling strategy. A key observation is that random sampling combined with our model allows reconstructions of similar amount of measurements and quality as a single static reconstruction.
Photoacoustic tomography (PAT) is an emerging imaging modality that aims at measuring the high-contrast optical properties of tissues by means of high-resolution ultrasonic measurements. The interaction between these two types of waves is based on the thermoacoustic effect. In recent years, many works have investigated the applicability of compressed sensing to PAT, in order to reduce measuring times while maintaining a high reconstruction quality. However, in most cases, theoretical guarantees are missing. In this work, we show that in many measurement setups of practical interest, compressed…
We present two methods that combine image reconstruction and edge detection in computed tomography (CT) scans. Our first method is as an extension of the prominent filtered backprojection algorithm. In our second method we employ $\ell^{1}$-regularization for stable calculation of the gradient. As opposed to the first method, we show that this approach is able to compensate for undersampled CT data.
Motion degradation is a central problem in Magnetic Resonance Imaging (MRI). This work addresses the problem of how to obtain higher quality, super-resolved motion-free, reconstructions from highly undersampled MRI data. In this work, we present for the first time a variational multi-task framework that allows joining three relevant tasks in MRI: reconstruction, registration and super-resolution. Our framework takes a set of multiple undersampled MR acquisitions corrupted by motion into a novel multi-task optimisation model, which is composed of an $L^2$ fidelity term that allows sharing repre…
In this paper we investigate the non-linear and ill-posed inverse problem of simultaneously identifying the conductivity and the reaction in diffuse optical tomography with noisy measurement data available on an accessible part of the boundary. We propose an energy functional method and the total variational regularization combining with the quadratic stabilizing term to formulate the identification problem to a PDEs constrained optimization problem. We show the stability of the proposed regularization method and the convergence of the finite element regularized solutions to the identification…
Diffuse Optical Tomography (DOT) is an emerging technology in medical imaging which employs light in the NIR spectrum to estimate the distribution of optical coefficients in biological tissues for diagnostic and monitoring purposes. DOT reconstruction implies the solution of a severely ill-posed inverse problem, for which regularization techniques are mandatory in order to achieve reasonable results. Traditionally, regularization techniques put a variance prior on the desired solution/gradient via regularization parameters, whose choice requires a fine tuning, specific for each case. In this w…
The paper surveys variational approaches for image reconstruction in dynamic inverse problems. Emphasis is on methods that rely on parametrised temporal models. These are here encoded as diffeomorphic deformations with time dependent parameters, or as motion constrained reconstruction where the motion model is given by a partial differential equation. The survey also includes recent development in integrating deep learning for solving these computationally demanding variational methods. Examples are given for 2D dynamic tomography, but methods apply to general inverse problems.
In this paper we propose a motion-aware variational approach to reconstruct moving objects from sparse dynamic data. The motivation of this work stems from X-ray imaging of plants perfused with a liquid contrast agent, aimed at increasing the contrast of the images and studying the phloem transport in plants over time. The key idea of our approach is to deploy 3D shearlets as a space-temporal prior, treating time as the third dimension. The rationale behind this model is that a continuous evolution of a cartoon-like object is well suited for the use of 3D shearlets. We provide a basic mathemat…
We propose a novel automatic parameter selection strategy for variational imaging problems under Poisson noise corruption. The selection of a suitable regularization parameter, whose value is crucial in order to achieve high quality reconstructions, is known to be a particularly hard task in low photon-count regimes. In this work, we extend the so-called residual whiteness principle originally designed for additive white noise to Poisson data. The proposed strategy relies on the study of the whiteness property of a standardized Poisson noise process. After deriving the theoretical properties t…
University of Applied Sciences Würzburg-Schweinfurt（维尔茨堡-施韦因富特应用科学大学）
We consider reconstructing multi-channel images from measurements performed by photon-counting and energy-discriminating detectors in the setting of multi-spectral X-ray computed tomography (CT). Our aim is to exploit the strong structural correlation that is known to exist between the channels of multi-spectral CT images. To that end, we adopt the multi-channel Potts prior to jointly reconstruct all channels. This prior produces piecewise constant solutions with strongly correlated channels. In particular, edges are enforced to have the same spatial position across channels which is a benefit…
In this paper, we consider the problem of feature reconstruction from incomplete x-ray CT data. Such problems occurs, e.g., as a result of dose reduction in the context medical imaging. Since image reconstruction from incomplete data is a severely ill-posed problem, the reconstructed images may suffer from characteristic artefacts or missing features, and significantly complicate subsequent image processing tasks (e.g., edge detection or segmentation). In this paper, we introduce a novel framework for the robust reconstruction of convolutional image features directly from CT data, without the …
Huawei Noah’s Ark Lab, Huawei Technologies（华为诺亚方舟实验室，华为）；University College London（伦敦大学学院）
Learned image reconstruction techniques using deep neural networks have recently gained popularity, and have delivered promising empirical results. However, most approaches focus on one single recovery for each observation, and thus neglect the uncertainty information. In this work, we develop a novel computational framework that approximates the posterior distribution of the unknown image at each query observation. The proposed framework is very flexible: It handles implicit noise models and priors, it incorporates the data formation process (i.e., the forward operator), and the learned recon…
A New Variational Model for Joint Image Reconstruction and Motion Estimation in Spatiotemporal Imaging
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（LSEC，ICMSEC，数学系统科学研究所，中国科学院）；Department of Mathematics, KTH–Royal Institute of Technology（数学系，皇家理工学院）
