Housen Li

FDR-control in multiscale change-point segmentation.

Fast multiple change-point segmentation methods, which additionally provide faithful statistical statements on the number, locations and sizes of the segments, have recently received great attention. In this paper, we propose a multiscale segmentation method, FDRSeg, which controls the false discovery rate (FDR) in the sense that the number of false jumps is bounded linearly by the number of true jumps. In this way, it adapts the detection power to the number of true jumps. We prove a non-asymptotic upper bound for its FDR in a Gaussian setting, which allows to calibrate the only parameter of FDRSeg properly. Change-point locations, as well as the signal, are shown to be estimated in a uniform sense at optimal minimax convergence rates up to a log-factor. The latter is w.r.t.

Aggregated motion estimation for real-time MRI reconstruction.

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Justifying Tensor-Driven Diffusion from Structure-Adaptive Statistics of Natural Images

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Aggregated motion estimation for image reconstruction in real-time MRI.

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The Averaged Kaczmarz Iteration for Solving Inverse Problems

, over classes of step functions with bounded jump sizes and either bounded, or possibly increasing, number of change-points. FDRSeg can be efficiently computed by an accelerated dynamic program; its computational complexity is shown to be linear in the number of observations when there are many change-points. The performance of the proposed method is examined by comparisons with some state of the art methods on both simulated and real datasets. An R-package is available online.

Variational multiscale nonparametric regression: Smooth functions

In real‐time MRI serial images are generally reconstructed from highly undersampled datasets as the iterative solutions of an inverse problem. While practical realizations based on regularized nonlinear inversion (NLINV) have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated reconstructions.

Aggregated motion estimation for real-time MRI reconstruction: Aggregated Motion Estimation for Image Reconstruction

Tensor-driven anisotropic diffusion and regularisation have been successfully applied to a wide range of image processing and computer vision tasks such as denoising, inpainting, and optical flow. Empirically it has been shown that anisotropic models with a diffusion tensor perform better than their isotropic counterparts with a scalar-valued diffusivity function. However, the reason for this superior performance is not well understood so far. Moreover, the specific modelling of the anisotropy has been carried out in a purely heuristic way. The goal of our paper is to address these problems. To this end, we use the statistics of natural images to derive a unifying framework for eight isotropic and anisotropic diffusion filters that have a corresponding variational formulation. In contrast to previous statistical models, we systematically investigate structure-adaptive statistics by analysing the eigenvalues of the structure tensor. With our findings, we justify existing successful models and assess the relationship between accurate statistical modelling and performance in the context of image denoising.

The Essential Histogram

In real‐time MRI serial images are generally reconstructed from highly undersampled datasets as the iterative solutions of an inverse problem. While practical realizations based on regularized nonlinear inversion (NLINV) have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated reconstructions.

Frame-constrained Total Variation Regularization for White Noise Regression

We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or nonlinear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz methods. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint subproblem per iterative update. On the other hand, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.

NETT: Solving Inverse Problems with Deep Neural Networks.

For the problem of nonparametric regression of smooth functions, we reconsider and analyze a constrained variational approach, which we call the MultIscale Nemirovski-Dantzig (MIND) estimator. This can be viewed as a multiscale extension of the Dantzig selector (\emph{Ann. Statist.}, 35(6): 2313--51, 2009) based on early ideas of Nemirovski (\emph{J. Comput. System Sci.}, 23:1--11, 1986). MIND minimizes a homogeneous Sobolev norm under the constraint that the multiresolution norm of the residual is bounded by a universal threshold. The main contribution of this paper is the derivation of convergence rates of MIND with respect to