On robust estimation of the location parameter

Computational geometry

Purpose
Due to the potential risk of inducing cancer, radiation exposure by X-ray CT devices should be reduced for routine patient scanning However, in low-dose X-ray CT, severe artifacts typically occur due to photon starvation, beam hardening, and other causes, all of which decrease the reliability of the diagnosis Thus, a high-quality reconstruction method from low-dose X-ray CT data has become a major research topic in the CT community Conventional model-based de-noising approaches are, however, computationally very expensive, and image-domain de-noising approaches cannot readily remove CT-specific noise patterns To tackle these problems, we want to develop a new low-dose X-ray CT algorithm based on a deep-learning approach

Method
We propose an algorithm which uses a deep convolutional neural network (CNN) which is applied to the wavelet transform coefficients of low-dose CT images More specifically, using a directional wavelet transform to extract the directional component of artifacts and exploit the intra- and inter- band correlations, our deep network can effectively suppress CT-specific noise In addition, our CNN is designed with a residual learning architecture for faster network training and better performance

Results
Experimental results confirm that the proposed algorithm effectively removes complex noise patterns from CT images derived from a reduced X-ray dose In addition, we show that the wavelet-domain CNN is efficient when used to remove noise from low-dose CT compared to existing approaches Our results were rigorously evaluated by several radiologists at the Mayo Clinic and won second place at the 2016 “Low-Dose CT Grand Challenge”

Conclusions
To the best of our knowledge, this work is the first deep-learning architecture for low-dose CT reconstruction which has been rigorously evaluated and proven to be effective In addition, the proposed algorithm, in contrast to existing model-based iterative reconstruction (MBIR) methods, has considerable potential to benefit from large data sets Therefore, we believe that the proposed algorithm opens a new direction in the area of low-dose CT research

https://aapm.onlinelibrary.wiley.com/doi/pdf/10.1002/mp.12344

A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction.

Computational geometry.

We give a methodology-oriented perspective on directional image analysis and rotation-invariant processing. We review the state of the art in the field and make connections with recent mathematical developments in functional analysis and wavelet theory. We unify our perspective within a common framework using operators. The intent is to provide image-processing methods that can be deployed in algorithms that analyze biomedical images with improved rotation invariance and high directional sensitivity. We start our survey with classical methods such as directional-gradient and the structure tensor. Then, we discuss how these methods can be improved with respect to robustness, invariance to geometric transformations (with a particular interest in scaling), and computation cost. To address robustness against noise, we move forward to higher degrees of directional selectivity and discuss Hessian-based detection schemes. To present multiscale approaches, we explain the differences between Fourier filters, directional wavelets, curvelets, and shearlets. To reduce the computational cost, we address the problem of matching directional patterns by proposing steerable filters, where one might perform arbitrary rotations and optimizations without discretizing the orientation. We define the property of steerability and give an introduction to the design of steerable filters. We cover the spectrum from simple steerable filters through pyramid schemes up to steerable wavelets. We also present illustrations on the design of steerable wavelets and their application to pattern recognition.

/pdf/transforms-and-operators-for-directional-bioimage-analysis-a-cz4ketezr8.pdf

Transforms and Operators for Directional Bioimage Analysis: A Survey

Image compression by linear splines over adaptive triangulations

We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted $\ell ^{1}$ minimization (sparse signals).

/pdf/jump-sparse-and-sparse-recovery-using-potts-functionals-if3y3k8uc2.pdf

Jump-Sparse and Sparse Recovery Using Potts Functionals

We consider total variation (TV) minimization for manifold-valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images and interferometric SAR images as well as sphere- and cylinder-valued images. For the class of Cartan--Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer.

Total Variation Regularization for Manifold-Valued Data

A finite dierence scheme is presented for a density-dependent dif- fusion equation that arises in the mathematical modelling of bacterial biofilms. The peculiarity of the underlying model is that it shows degeneracy as the de- pendent variable vanishes, as well as a singularity as the dependent variable approaches its a priori known upper bound. The first property leads to a fi- nite speed of interface propagation if the initial data have compact support, while the second one introduces counter-acting super diusion. This squeez- ing property of this model leads to steep gradients at the interface. Moving interface problems of this kind are known to be problematic for classical nu- merical methods and introduce non-physical and non-mathematical solutions. The proposed method is developed to address this observation. The central idea is a non-local (in time) representation of the diusion operator. It can be shown that the proposed method is free of oscillations at the interface, that the discrete interface satisfies a discrete version of the continuous interface condition and that the eect of interface smearing is quantitatively small.

/pdf/a-finite-difference-scheme-for-a-degenerated-diffusion-1kvorbulgl.pdf

A finite difference scheme for a degenerated diffusion equation arising in microbial ecology.

We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer.

Laurent Demaret

Papers

Image compression by linear splines over adaptive triangulations

Jump-Sparse and Sparse Recovery Using Potts Functionals

Total Variation Regularization for Manifold-Valued Data

A finite difference scheme for a degenerated diffusion equation arising in microbial ecology.

Total variation regularization for manifold-valued data