We propose a generalized equation to represent a continuum of surface reconstruction solutions of a given non-integrable gradient field. We show that common approaches such as Poisson solver and Frankot-Chellappa algorithm are special cases of this generalized equation. For a N x N pixel grid, the subspace of all integrable gradient fields is of dimension N 2 - 1. Our framework can be applied to derive a range of meaningful surface reconstructions from this high dimensional space. The key observation is that the range of solutions is related to the degree of anisotropy in applying weights to the gradients in the integration process. While common approaches use isotropic weights, we show that by using a progression of spatially varying anisotropic weights, we can achieve significant improvement in reconstructions. We propose (a) α-surfaces using binary weights, where the parameter a allows trade off between smoothness and robustness, (b) M-estimators and edge preserving regularization using continuous weights and (c) Diffusion using affine transformation of gradients. We provide results on photometric stereo, compare with previous approaches and show that anisotropic treatment discounts noise while recovering salient features in reconstructions.

What Is the Range of Surface Reconstructions from a Gradient Field

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images--a known disadvantage of the ROF model--while being simple and efficiently numerically solvable.

A Combined First and Second Order Variational Approach for Image Reconstruction

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.

Scale space and variational methods in computer vision : Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009 : proceedings

Cross-sectional imaging of an electrical conductivity distribution inside the human body has been an active research goal in impedance imaging. By injecting current into an electrically conducting object through surface electrodes, we induce current density and voltage distributions. Based on the fact that these are determined by the conductivity distribution as well as the geometry of the object and the adopted electrode configuration, electrical impedance tomography (EIT) reconstructs cross-sectional conductivity images using measured current-voltage data on the surface. Unfortunately, there exist inherent technical difficulties in EIT. First, the relationship between the boundary current-voltage data and the internal conductivity distribution bears a nonlinearity and low sensitivity, and hence the inverse problem of recovering the conductivity distribution is ill posed. Second, it is difficult to obtain accurate information on the boundary geometry and electrode positions in practice, and the inverse problem is sensitive to these modeling errors as well as measurement artifacts and noise. These result in EIT images with a poor spatial resolution. In order to produce high-resolution conductivity images, magnetic resonance electrical impedance tomography (MREIT) has been lately developed. Noting that injection current produces a magnetic as well as electric field inside the imaging object, we can measure the induced internal magnetic flux density data using an MRI scanner. Utilization of the internal magnetic flux density is the key idea of MREIT to overcome the technical difficulties in EIT. Following original ideas on MREIT in early 1990s, there has been a rapid progress in its theory, algorithm and experimental techniques. The technique has now advanced to the stage of human experiments. Though it is still a few steps away from routine clinical use, its potential is high as a new impedance imaging modality providing conductivity images with a spatial resolution of a few millimeters or less. This paper reviews MREIT from the basics to the most recent research outcomes. Focusing on measurement techniques and experimental methods rather than mathematical issues, we summarize what has been done and what needs to be done. Suggestions for future research directions, possible applications in biomedicine, biology, chemistry and material science are discussed.

https://iopscience.iop.org/article/10.1088/0967-3334/29/10/R01/pdf

Magnetic resonance electrical impedance tomography (MREIT) for high-resolution conductivity imaging

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. A high order nonlinear partial differential equation (PDE) needs to be solved, and the gradient descent method usually takes high computational cost. In this paper, we propose a fast and efficient numerical algorithm to solve minimization problems related to Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

In this paper, we propose a general form of TV-Stokes models and provide an efficient and fast numerical algorithm based on the augmented Lagrangian method. The proposed model and numerical algorithm can be used for a number of applications such as image inpainting, image decomposition, surface reconstruction from sparse gradient, direction denoising, and image denoising. Comparing with properties of different norms in regularity term and fidelity term, various results are investigated in applications. We numerically show that the proposed model recovers jump discontinuities of a data and discontinuities of the data gradient while reducing stair-case effect.

Augmented Lagrangian Method for Generalized TV-Stokes Model

In this paper, we propose an orientation-matching functional minimization for image denoising and image inpainting. Following the two-step TV-Stokes algorithm (Rahman et al. in Scale space and variational methods in computer vision, pp. 473---482, Springer, Heidelberg, 2007; Tai et al. in Image processing based on partial differential equations, pp. 3---22, Springer, Heidelberg, 2006; Bertalmio et al. in Proc. conf. comp. vision pattern rec., pp. 355---362, 2001), a regularized tangential vector field with zero divergence condition is first obtained. Then a novel approach to reconstruct the image is proposed. Instead of finding an image that fits the regularized normal direction from the first step, we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction. This functional yields a new nonlinear partial differential equation (PDE) for reconstructing denoised and inpainted images. The equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field, providing reconstructed images which have sharp edges and smooth regions. The additive operator splitting (AOS) scheme is used for discretizing Euler-Lagrange equations. We present the results of various numerical experiments that illustrate the improvements obtained with the new functional.

https://link.springer.com/content/pdf/10.1007%2Fs11263-010-0371-5.pdf

Orientation-Matching Minimization for Image Denoising and Inpainting

Magnetic resonance electrical impedance tomography (MREIT) is designed to produce high resolution conductivity images of an electrically conducting subject by injecting current and measuring the longitudinal component, B/sub z/, of the induced magnetic flux density B = (B/sub x/,B/sub y/,B/sub z/). In MREIT, accurate measurements of B/sub z/ are essential in producing correct conductivity images. However, the measured B/sub z/ data may contain fundamental defects in local regions where MR magnitude image data are small. These defective B/sub z/ data result in completely wrong conductivity values there and also affect the overall accuracy of reconstructed conductivity images. Hence, these defects should be appropriately recovered in order to carry out any MREIT image reconstruction algorithm. This paper proposes a new method of recovering B/sub z/ data in defective regions based on its physical properties and neighboring information of B/sub z/. The technique will be indispensable for conductivity imaging in MREIT from animal or human subjects including defective regions such as lungs, bones, and any gas-filled internal organs.

Conductivity image reconstruction from defective data in MREIT: numerical Simulation and animal experiment

Variable splitting schemes for the function space version of the image reconstruction problem with total variation regularization (TV-problem) in its primal and pre-dual formulations are considered. For the primal splitting formulation, while existence of a solution cannot be guaranteed, it is shown that quasi-minimizers of the penalized problem are asymptotically related to the solution of the original TV-problem. On the other hand, for the pre-dual formulation, a family of parametrized problems is introduced and a parameter dependent contraction of an associated fixed point iteration is established. Moreover, the theory is validated by numerical tests. Additionally, the augmented Lagrangian approach is studied, details on an implementation on a staggered grid are provided and numerical tests are shown.

/pdf/functional-analytic-and-numerical-issues-in-splitting-43bui9li5u.pdf

Jooyoung Hahn

Papers

A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

Augmented Lagrangian Method for Generalized TV-Stokes Model

Orientation-Matching Minimization for Image Denoising and Inpainting

Conductivity image reconstruction from defective data in MREIT: numerical Simulation and animal experiment

Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction