Marius Lysaker

Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time

In this paper, we introduce a new method for image smoothing based on a fourth-order PDE model. The method is tested on a broad range of real medical magnetic resonance images, both in space and time, as well as on nonmedical synthesized test images. Our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail, even at poor signal-to-noise ratio. We have also compared our method with related PDE models.

A binary level set model and some applications to Mumford-Shah image segmentation

In this paper, we propose a PDE-based level set method. Traditionally, interfaces are represented by the zero level set of continuous level set functions. Instead, we let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e., it can only be 1 or -1; thus, our method is related to phase-field methods. Some of the properties of standard level set methods are preserved in the proposed method, while others are not. Using this new method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth. We show numerical results using the method for segmentation of digital images.

Scale Space and Variational Methods in Computer Vision

Segmentation and Detection.- Graph Cut Optimization for the Piecewise Constant Level Set Method Applied to Multiphase Image Segmentation.- Tubular Anisotropy Segmentation.- An Unconstrained Multiphase Thresholding Approach for Image Segmentation.- Extraction of the Intercellular Skeleton from 2D Images of Embryogenesis Using Eikonal Equation and Advective Subjective Surface Method.- On Level-Set Type Methods for Recovering Piecewise Constant Solutions of Ill-Posed Problems.- The Nonlinear Tensor Diffusion in Segmentation of Meaningful Biological Structures from Image Sequences of Zebrafish Embryogenesis.- Composed Segmentation of Tubular Structures by an Anisotropic PDE Model.- Extrapolation of Vector Fields Using the Infinity Laplacian and with Applications to Image Segmentation.- A Schrodinger Equation for the Fast Computation of Approximate Euclidean Distance Functions.- Semi-supervised Segmentation Based on Non-local Continuous Min-Cut.- Momentum Based Optimization Methods for Level Set Segmentation.- Optimization of Divergences within the Exponential Family for Image Segmentation.- Convex Multi-class Image Labeling by Simplex-Constrained Total Variation.- Geodesically Linked Active Contours: Evolution Strategy Based on Minimal Paths.- Validation of Watershed Regions by Scale-Space Statistics.- Adaptation of Eikonal Equation over Weighted Graph.- A Variational Model for Interactive Shape Prior Segmentation and Real-Time Tracking.- Image Enhancement and Reconstruction.- A Nonlinear Probabilistic Curvature Motion Filter for Positron Emission Tomography Images.- Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion Imaging.- Bregman-EM-TV Methods with Application to Optical Nanoscopy.- PDE-Driven Adaptive Morphology for Matrix Fields.- On Semi-implicit Splitting Schemes for the Beltrami Color Flow.- Multi-scale Total Variation with Automated Regularization Parameter Selection for Color Image Restoration.- Multiplicative Noise Cleaning via a Variational Method Involving Curvelet Coefficients.- Projected Gradient Based Color Image Decomposition.- A Dual Formulation of the TV-Stokes Algorithm for Image Denoising.- Anisotropic Regularization for Inverse Problems with Application to the Wiener Filter with Gaussian and Impulse Noise.- Locally Adaptive Total Variation Regularization.- Basic Image Features (BIFs) Arising from Approximate Symmetry Type.- An Anisotropic Fourth-Order Partial Differential Equation for Noise Removal.- Enhancement of Blurred and Noisy Images Based on an Original Variant of the Total Variation.- Coarse-to-Fine Image Reconstruction Based on Weighted Differential Features and Background Gauge Fields.- Edge-Enhanced Image Reconstruction Using (TV) Total Variation and Bregman Refinement.- Nonlocal Variational Image Deblurring Models in the Presence of Gaussian or Impulse Noise.- A Geometric PDE for Interpolation of M-Channel Data.- An Edge-Preserving Multilevel Method for Deblurring, Denoising, and Segmentation.- Fast Dejittering for Digital Video Frames.- Sparsity Regularization for Radon Measures.- Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage.- Anisotropic Smoothing Using Double Orientations.- Image Denoising Using TV-Stokes Equation with an Orientation-Matching Minimization.- Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model.- The Convergence of a Central-Difference Discretization of Rudin-Osher-Fatemi Model for Image Denoising.- Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering.- L 0-Norm and Total Variation for Wavelet Inpainting.- Total-Variation Based Piecewise Affine Regularization.- Image Denoising by Harmonic Mean Curvature Flow.- Motion Analysis, Optical Flow, Registration and Tracking.- Tracking Closed Curves with Non-linear Stochastic Filters.- A Multi-scale Feature Based Optic Flow Method for 3D Cardiac Motion Estimation.- A Combined Segmentation and Registration Framework with a Nonlinear Elasticity Smoother.- A Scale-Space Approach to Landmark Constrained Image Registration.- A Variational Approach for Volume-to-Slice Registration.- Hyperbolic Numerics for Variational Approaches to Correspondence Problems.- Surfaces and Shapes.- From a Single Point to a Surface Patch by Growing Minimal Paths.- Optimization of Convex Shapes: An Approach to Crystal Shape Identification.- An Implicit Method for Interpolating Two Digital Closed Curves on Parallel Planes.- Pose Invariant Shape Prior Segmentation Using Continuous Cuts and Gradient Descent on Lie Groups.- A Non-local Approach to Shape from Ambient Shading.- An Elasticity Approach to Principal Modes of Shape Variation.- Pre-image as Karcher Mean Using Diffusion Maps: Application to Shape and Image Denoising.- Fast Shape from Shading for Phong-Type Surfaces.- Generic Scene Recovery Using Multiple Images.- Scale Space and Feature Extraction.- Highly Accurate PDE-Based Morphology for General Structuring Elements.- Computational Geometry-Based Scale-Space and Modal Image Decomposition.- Highlight on a Feature Extracted at Fine Scales: The Pointwise Lipschitz Regularity.- Line Enhancement and Completion via Linear Left Invariant Scale Spaces on SE(2).- Spatio-Featural Scale-Space.- Scale Spaces on the 3D Euclidean Motion Group for Enhancement of HARDI Data.- On the Rate of Structural Change in Scale Spaces.- Transitions of a Multi-scale Image Hierarchy Tree.- Local Scale Measure for Remote Sensing Images.

Iterative Image Restoration Combining Total Variation Minimization and a Second-Order Functional

A noise removal technique using partial differential equations (PDEs) is proposed here. It combines the Total Variational (TV) filter with a fourth-order PDE filter. The combined technique is able to preserve edges and at the same time avoid the staircase effect in smooth regions. A weighting function is used in an iterative way to combine the solutions of the TV-filter and the fourth-order filter. Numerical experiments confirm that the new method is able to use less restrictive time step than the fourth-order filter. Numerical examples using images with objects consisting of edge, flat and intermediate regions illustrate advantages of the proposed model.

Noise removal using smoothed normals and surface fitting

In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a total-variation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper.

On the use of the resting potential and level set methods for identifying ischemic heart disease: An inverse problem

The electrical activity in the heart is modeled by a complex, nonlinear, fully coupled system of differential equations. Several scientists have studied how this model, referred to as the bidomain model, can be modified to incorporate the effect of heart infarctions on simulated ECG (electrocardiogram) recordings. We are concerned with the associated inverse problem; how can we use ECG recordings and mathematical models to identify the position, size and shape of heart infarctions? Due to the extreme CPU efforts needed to solve the bidomain equations, this model, in its full complexity, is not well-suited for this kind of problems. In this paper we show how biological knowledge about the resting potential in the heart and level set techniques can be combined to derive a suitable stationary model, expressed in terms of an elliptic PDE, for such applications. This approach leads to a nonlinear ill-posed minimization problem, which we propose to regularize and solve with a simple iterative scheme. Finally, our theoretical findings are illuminated through a series of computer simulations for an experimental setup involving a realistic heart in torso geometry. More specifically, experiments with synthetic ECG recordings, produced by solving the bidomain model, indicate that our method manages to identify the physical characteristics of the ischemic region(s) in the heart. Furthermore, the ill-posed nature of this inverse problem is explored, i.e. several quantitative issues of our scheme are explored.

Computing the size and location of myocardial ischemia using measurements of ST-segment shift

It is well known that the presence of myocardial ischemiastract can be observed as a shift in the ST segment of an electrocardiogram (ECG) recording. The question we address in this paper is whether or not ST shift can be used to compute approximations of the size and location of the ischemic region. We begin by investigating a cost functional (measuring the difference between synthetic recorded data and simulated values of ST shift) for a parameter identification problem to locate the ischemic region. We then formulate a more flexible representation of the ischemia using a level set framework and solve the associated minimization problem for the size and position of the ischemia. We apply this framework to a set of ECG data generated by the Bidomain model using the cell model of Winslow et al. Based on this data, we show that values of ST shift recorded at the body surface are capable of identifying the position and (roughly) the size of the ischemia.

Computing Ischemic Regions in the Heart With the Bidomain Model—First Steps Towards Validation

We investigate whether it is possible to use the bidomain model and body surface potential maps (BSPMs) to compute the size and position of ischemic regions in the human heart. This leads to a severely ill posed inverse problem for a potential equation. We do not use the classical inverse problems of electrocardiography, in which the unknown sources are the epicardial potential distribution or the activation sequence. Instead we employ the bidomain theory to obtain a model that also enables identification of ischemic regions transmurally. This approach makes it possible to distinguish between subendocardial and transmural cases, only using the BSPM data. The main focus is on testing a previously published algorithm on clinical data, and the results are compared with images taken with perfusion scintigraphy. For the four patients involved in this study, the two modalities produce results that are rather similar: The relative differences between the center of mass and the size of the ischemic regions, suggested by the two modalities, are 10.8% ± 4.4% and 7.1% ± 4.6%, respectively. We also present some simulations which indicate that the methodology is robust with respect to uncertainties in important model parameters. However, in contrast to what has been observed in investigations only involving synthetic data, inequality constraints are needed to obtain sound results.

Piecewise constant level set methods and image segmentation

In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we minimize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation.

A Computationally Efficient Method for Determining the Size and Location of Myocardial Ischemia

The purpose of this paper is to introduce a new method for solving the inverse problem of locating ischemic regions in the heart. The electrical activity in the human heart is modeled by the bidomain equations, which can be expanded to compute the potentials on the body surface. The associated inverse problem is to use ECG recordings to gain information about ischemias. We propose an algorithm for doing this, combining the level set method with a simpler minimization problem. Instead of trying to determine the shape, as in the level set method, we simply make the approximation that the ischemia is spherical. The effects of ischemia on the electrical attributes of heart tissue are examined. The new method is tested with computer simulations on synthetic body surface potential maps (BSPMs) in a realistic geometry, using realistic values for the parameters. It is shown to be, in some respects, superior to the level set approach and may be a step toward a practical algorithm useful in medical diagnostics.