A new augmented Lagrangian primal dual algorithm for elastica regularization

In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

We propose a novel fourth order dual method for the minimization of the non-smooth semi-norm ∥Δ·∥ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> when in amalgamation with a new staircase reducing texture decomposition model of image processing. The proposed model incorporating this high order energy is a variant of the Chambolle Lions denoising model that additionally utilizes a negative Sobolev norm. We claim that the dual method is faster and more stable than the current gradient descent time marching algorithms often used to minimize such energies. Moreover, a proof of convergence of the proposed method, in conjunction with the new model, will be provided. Lastly, we provide guidelines on how the new energy and proposed framework can be naturally incorporated into many popular texture extraction and restoration models of image processing.

A method based on an augmented Lagrangian formulation is developed which allows coefficients in an elliptic differential equation to be estimated from measurements of the state. This is a hybrid method combining the output-least-squares and the equation-error technique. Seminorm regularization is employed, and convergence and stability properties are discussed. Several aspects of an efficient implementation are described. Finally, the effectiveness of the method is demonstrated by means of one-and two-dimensional examples.

International audience

SUMMARYUp to now, image processing and image analysis techniques have borrowed their basic tools from functional analysis: Fourier filtering, differential and integral calculus, and so on. These tools, however, only realize their efficiency when they are put into a well‐defined algebraic frame, most of the time of a vectorial nature. Unfortunately, the class of functions modelling ‘images’, commonly referred to as ‘grey tone functions’ does not necessarily present this very type of structure. We present here an operation for the ‘addition’ of two images, with a physical justification in the context of transmitted light. Such an addition permits the construction of the family of ‘positive homothetics' of the grey tone function at hand. The vectorial context sought is well defined: The class of images associated with the class of their grey tone functions naturally becomes the positive cone of an ordered real vector space.Furthermore, the proposed model holds for logarithmic imaging and is compatible with what is known about the human visual process. This model has been called ‘LIP’ (logarithmic image processing model).

Recovering high quality surfaces from noisy triangulated surfaces is a fundamental important problem in geometry processing. Sharp features including edges and corners can not be well preserved in most existing denoising methods except the recent total variation (TV) and $\ell_0$ regularization methods. However, these two methods have suffered producing staircase artifacts in smooth regions. In this paper, we first introduce a second order regularization method for restoring a surface normal vector field, and then propose a new vertex updating scheme to recover the desired surface according to the restored surface normal field. The proposed model can preserve sharp features and simultaneously suppress the staircase effects in smooth regions which overcomes the drawback of the first order models. In addition, the new vertex updating scheme can prevent ambiguities introduced in existing vertex updating methods. Numerically, the proposed high order model is solved by the augmented Lagrangian method with a dynamic weighting strategy. Intensive numerical experiments on a variety of surfaces demonstrate the superiority of our method by visually and quantitatively.

With the transmission matrix (TM) of the whole optical system measured, the image of the object behind a turbid medium can be recovered from its speckle field by means of an image reconstruction algorithm. Instead of Tikhonov regularization algorithm (TRA), the total variation minimization by augmented Lagrangian and alternating direction algorithms (TVAL3) is introduced to recover object images. As a total variation (TV)-based approach, TVAL3 allows to effectively damp more noise and preserve more edges compared with TRA, thus providing more outstanding image quality. Different levels of detector noise and TM-measurement noise are successively added to analyze the antinoise performance of these two algorithms. Simulation results show that TVAL3 is able to recover more details and suppress more noise than TRA under different noise levels, thus providing much more excellent image quality. Furthermore, whether it be detector noise or TM-measurement noise, the reconstruction images obtained by TVAL3 at SNR=15 dB are far superior to those by TRA at SNR=50 dB.

In this paper, a new Poisson image denoising model based on fractional-order total variation regularization is proposed. To obtain its global optimal solution, the augmented Lagrangian method, the Chambolle’s dual algorithm and the primal-dual algorithm are introduced. Experimental results are supplied to demonstrate the effectiveness and efficiency of the proposed algorithms for solving our proposed model, with comparison to the total variation Poisson image denoising model.

In this paper we derive a dynamic model of the Delta robot that is well-suited to an object-oriented modeling framework. The approach uses an augmented Lagrangian or Hamiltonian formulation together with Baumgarte's method of index reduction, and results in a singularity-free dynamic model that is well suited to dynamic analysis, control system synthesis and time-domain simulation. The object-oriented structure enables broad application to problems such as coordinated control and robotic assembly. We present several common control algorithms and conduct a dynamic analysis of the Delta robot that shows that the open-loop system is unstable for large volumes of the reachable workspace, which has fundamental implications on closed-loop performance.

Biphasic contact analysis is essential to obtain a complete understanding of soft tissue biomechanics, and the importance of physiological structure on the joint biomechanics has long been recognised; however, up to date, there are no successful developments of biphasic finite element contact analysis for three-dimensional (3D) geometries of physiological joints. The aim of this study was to develop a finite element formulation for biphasic contact of 3D physiological joints. The augmented Lagrangian method was used to enforce the continuity of contact traction and fluid pressure across the contact interface. The biphasic contact method was implemented in the commercial software COMSOL Multiphysics 4.2® (COMSOL, Inc., Burlington, MA). The accuracy of the implementation was verified using 3D biphasic contact problems, including indentation with a flat-ended indenter and contact of glenohumeral cartilage layers. The ability of the method to model multibody biphasic contact of physiological joints was proved by a 3D knee model. The 3D biphasic finite element contact method developed in this study can be used to study the biphasic behaviours of the physiological joints.

We propose a variational approach for edge-preserving total variation (TV)-based regularization of Q-ball data from high angular resolution diffusion imaging (HARDI). While total variation is among the most popular regularizers for variational problems, its application to orientation distribution functions (ODF), as they naturally arise in Q-ball imaging, is not straightforward. We propose to use an extension that specifically takes into account the metric on the underlying orientation space. The key idea is to write the difference quotients in the TV seminorm in terms of the Wasserstein statistical distance from optimal transport. We combine this regularizer with a matching Wasserstein data fidelity term. Using the Kantorovich-Rubinstein duality, the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm. We demonstrate the effectiveness of the proposed framework on real and synthetic Q-ball data.

Total variation (TV) methods have been proposed to improve the image quality in count-reduced images, by reducing the variation between neighboring pixels. Although very easy to implement and fast to compute, TV-based methods may lead to a loss of texture information when applied to images with complex textures, such as high-resolution abdominal CT images. Here, we investigate the use of another regularization approach in the context of medical images based on multiresolution transformations. One such transformation is the shearlet transform, which is optimally sparse for images that are C2 except for discontinuities along C2 curves, and has better directional sensitivity than most other, related, wavelet transform approaches. We propose to solve the convex problem using the split-Bregman (augmented Lagrangian) approach. One of the primary advantages of the split-Bregman approach, is that the shearlet transform can easily be incorporated into the sparse-view CT reconstruction. The required sparsity prior is the l1 norm of the shearlet coefficients. Results are shown for this method in comparison to the same framework with TV as the regularization term on simulated data. The noise-resolution performance is investigated at different contrast levels. At equal image noise, TV-based regularization outperforms shearlet-based regularization. However, when image texture is analyzed on measured mouse data, shearlets outperform TV, which suffers from staircasing effects. Our results show that there are benefits in using shearlets in CT imaging: texture is reconstructed more accurately compared to when TV is used, without biasing the image towards a piecewise constant image model. However, due to the larger support of the basis functions, our results suggest that uncareful usage of shearlets may lead to wavy artifacts, which can be equally unwanted as staircasing effects.

First-order methods are used widely for large scale optimization problems in signal/image processing and machine learning, because their computation depends mildly on the problem dimension. Nesterov's fast gradient method (FGM) has the optimal convergence rate among first-order methods for smooth convex minimization; its extension to non-smooth case, the fast iterative shrinkage-thresholding algorithm (FISTA), also satisfies the optimal rate; thus both algorithms have gained great interest. We recently introduced a new optimized gradient method (OGM) (for smooth convex functions) having a theoretical convergence speed that is 2× faster than Nesterov's FGM. This paper further discusses the convergence analysis of OGM and explores its fast convergence on an image restoration problem using a smoothed total variation (TV) regularizer. In addition, we empirically investigate the extension of OGM to nonsmooth convex minimization for image restoration with l 1 -sparsity regularization.

We propose a convex, lower semicontinuous, coercive approximation of Euler's elastica energy for images, which is thus very well-suited as a regularizer in image processing. The approximation is not quite the convex relaxation, and we discuss its close relation to the exact convex relaxation as well as the difficulties associated with computing the latter. Interestingly, the convex relaxation of the elastica energy reduces to constantly zero if the total variation part of the elastica energy is neglected. Our convex approximation arises via functional lifting of the image gradient into a Radon measure on the four-dimensional space $\Omega\times S^1\times\mathbb{R}$, $\Omega\subset\mathbb{R}^2$, of which the first two coordinates represent the image domain and the last two the normal and curvature of the image level lines. It can be expressed as a linear program which also admits a predual representation. Combined with a tailored discretization of measures via linear combinations of short line measures, th...

To solve the ill-posed problem in electrical impedance tomography (EIT) and overcome the blur effect made by conventional L2 Tikhonov regularization, a new two-step iterative regularization algorithm is applied for the process reconstruction of lung respiration. This algorithm is based on total variation (TV) regularization method, and uses a two-step iterative method with TV denoising operator, so that it can achieve double regularization effect. This algorithm has advantages in both resolution and robustness. Both simulation and experiment results are executed to prove the effectiveness of TV regularization method.