Piecewise Constant Mumford-Shah Model Research Articles

Automation of hemocompatibility analysis using image segmentation and supervised classification

The hemocompatibility of blood-contacting medical devices remains one of the major challenges in biomedical engineering and makes research in the field of new and improved materials inevitable. However, current in-vitro test and analysis methods are still lacking standardization and comparability, which impedes advances in material design. For example, the optical platelet analysis of material in-vitro hemocompatibility tests is carried out manually or semi-manually by each research group individually.As a step towards standardization, this paper proposes an automation approach for the optical platelet count and analysis. To this end, fluorescence images are segmented using Zach’s convexification of the multiphase-phase piecewise constant Mumford–Shah model. The non-background components then need to be classified as platelet or no platelet. For this purpose, a supervised random forest is applied to feature vectors derived from the components using features like area, perimeter and circularity. With an overall high accuracy (>93%) and low error rates (≤ 5%), the random forest achieves reliable results. This is supported by high areas under the receiver–operator characteristic curve (≥ 0.94) and the prediction–recall curve (≥ 0.77), respectively.We developed a novel method for a fast, user-independent and reproducible analysis of material hemocompatibility tests. The automatized analysis method overcomes the current obstacles in the way of standardized in-vitro material testing and is therefore a unique and powerful tool for advances in biomaterial research.

Smoothing for signals with discontinuities using higher order Mumford–Shah models

Minimizing the Mumford-Shah functional is frequently used for smoothing signals or time series with discontinuities. A significant limitation of the standard Mumford-Shah model is that linear trends -- and in general polynomial trends -- in the data are not well preserved. This can be improved by building on splines of higher order which leads to higher order Mumford-Shah models. In this work, we study these models in the univariate situation: we discuss important differences to the first order Mumford-Shah model, and we obtain uniqueness results for their solutions. As a main contribution, we derive fast minimization algorithms for Mumford-Shah models of arbitrary orders. We show that the worst case complexity of all proposed schemes is quadratic in the length of the signal. Remarkably, they thus achieve the worst case complexity of the fastest solver for the piecewise constant Mumford-Shah model (which is the simplest model of the class). Further, we obtain stability results for the proposed algorithms. We complement these results with a numerical study. Our reference implementation processes signals with more than 10,000 elements in less than one second.

Comparison of minimization methods for nonsmooth image segmentation

Abstract Segmentation is a typical task in image processing having as main goal the partitioning of the image into multiple segments in order to simplify its interpretation and analysis. One of the more popular segmentation model, formulated by Chan-Vese, is the piecewise constant Mumford-Shah model restricted to the case of two-phase segmentation. We consider a convex relaxation formulation of the segmentation model, that can be regarded as a nonsmooth optimization problem, because the presence of the l1-term. Two basic approaches in optimization can be distinguished to deal with its non differentiability: the smoothing methods and the nonsmoothing methods. In this work, a numerical comparison of some first order methods belongs of both approaches are presented. The relationships among the different methods are shown, and accuracy and efficiency tests are also performed on several images.

Model-based learning of local image features for unsupervised texture segmentation.

Features that capture well the textural patterns of a certain class of images are crucial for the performance of texture segmentation methods. The manual selection of features or designing new ones can be a tedious task. Therefore, it is desirable to automatically adapt the features to a certain image or class of images. Typically, this requires a large set of training images with similar textures and ground truth segmentation. In this work, we propose a framework to learn features for texture segmentation when no such training data is available. The cost function for our learning process is constructed to match a commonly used segmentation model, the piecewise constant Mumford-Shah model. This means that the features are learned such that they provide an approximately piecewise constant feature image with a small jump set. Based on this idea, we develop a two-stage algorithm which first learns suitable convolutional features and then performs a segmentation. We note that the features can be learned from a small set of images, from a single image, or even from image patches. The proposed method achieves a competitive rank in the Prague texture segmentation benchmark, and it is effective for segmenting histological images.

A Joint Reconstruction and Segmentation Method for Limited-Angle X-Ray Tomography

Limited-angle computed tomography (CT) is common in industrial applications, where incomplete projection data can cause artifacts. For objects made from homogeneous materials, we propose a joint reconstruction and segmentation method that performs joint image reconstruction and segmentation directly on the projection data. We describe an alternating minimization algorithm to solve the resulting optimization problem, and we modify the primal-dual hybrid gradient algorithm for the non-convex piecewise constant Mumford-Shah model, which is a popular approximation model in biomedical image segmentation. The effectiveness of the proposed approach is validated by simulation and by application to actual micro-CT data sets.

Classification of small structures in piecewise-constant Mumford–Shah model

Mumford–Shah model is a very popular variational model in image restoration and image classification. As a simplification, piecewise constant Mumford–Shah model is very useful and has been extensively studied in recent two decades. An interesting topic on Mumford–Shah model is how to choose the weight parameters for implementation. This paper aims at discussing and analyzing the relation between choosing weight parameters and removing/preserving small structures, including noise, for piecewise-constant Mumford–Shah model. The main contributions are: (1) provided a necessary condition on the weight parameter of regularity term for removing a small structure from background. It is proved that whether or not a small structure could be removed from the background in the piecewise-constant Mumford–Shah model depends on two aspects: the ratio of the area to the perimeter for the smaller structure and the intensities of other classes; (2) provided a decision-making strategy on the class that a small structure will be classified to if it does not belong to the background; (3) developed a balanced Mumford–Shah model with which the scale measurements (weights for fidelity terms) can be chosen based on prior knowledge or users’ purposes.

Fast Segmentation From Blurred Data in 3D Fluorescence Microscopy.

We develop a fast algorithm for segmenting 3D images from linear measurements based on the Potts model (or piecewise constant Mumford-Shah model). To that end, we first derive suitable space discretizations of the 3D Potts model, which are capable of dealing with 3D images defined on non-cubic grids. Our discretization allows us to utilize a specific splitting approach, which results in decoupled subproblems of moderate size. The crucial point in the 3D setup is that the number of independent subproblems is so large that we can reasonably exploit the parallel processing capabilities of the graphics processing units (GPUs). Our GPU implementation is up to 18 times faster than the sequential CPU version. This allows to process even large volumes in acceptable runtimes. As a further contribution, we extend the algorithm in order to deal with non-negativity constraints. We demonstrate the efficiency of our method for combined image deconvolution and segmentation on simulated data and on real 3D wide field fluorescence microscopy data.

Variational image segmentation model coupled with image restoration achievements

Image segmentation and image restoration are two important topics in image processing with a number of important applications. In this paper, we propose a new multiphase segmentation model by combining image restoration and image segmentation models. Utilizing aspects of image restoration, the proposed segmentation model can effectively and robustly tackle images with a high level of noise or blurriness, missing pixels or vector values. In particular, one of the most important segmentation models, the piecewise constant Mumford–Shah model, can be extended easily in this way to segment gray and vector-valued images corrupted, for example, by noise, blur or information loss after coupling a new data fidelity term which borrowed from the field of image restoration. It can be solved efficiently using the alternating minimization algorithm, and we prove the convergence of this algorithm with three variables under mild conditions. Experiments on many synthetic and real-world images demonstrate that our method gives better segmentation results in terms of quality and quantity in comparison to other state-of-the-art segmentation models, especially for blurry images and those with information loss.

Gauss-Newton Method for Segmentation assisted Deformable Registration

In this work, we try to develop a fast converging method for segmentation assisted deformable registration. The segmentation step consists of a piece-wise constant Mumford-Shah energy model while reg- istration is driven by the sum of squared distances of both initial images and segmented mask with a diffusion regularization. In order to solve this energy minimization problem, a second order Gauss-Newton opti- mization method is used. For the numerical experiments we used CT data sets from the EMPIRE10 challenge. In this preliminary study, we show high accuracy of our algorithm.

Integrating clustering with level set method for piecewise constant Mumford-Shah model

In the paper, we present an efficient method to solve the piecewise constant Mumford-Shah (M-S) model for two-phase image segmentation within the level set framework. A clustering algorithm is used to find approximately the intensity means of foreground and background in the image, and so the M-S functional is reduced to the functional of a single variable (level set function), which avoids using complicated alternating optimization to minimize the reduced M-S functional. Experimental results demonstrated some advantages of the proposed method over the well-known Chan-Vese method using alternating optimization, such as robustness to the locations of initial contour and the high computation efficiency.

A Direct Approach Toward Global Minimization for Multiphase Labeling and Segmentation Problems

This paper intends to extend the minimization algorithm developed by Bae, Yuan and Tai [IJCV, 2011] in several directions. First, we propose a new primal-dual approach for global minimization of the continuous Potts model with applications to the piecewise constant Mumford-Shah model for multiphase image segmentation. Different from the existing methods, we work directly with the binary setting without using convex relaxation, which is thereby termed as a direct approach. Second, we provide the sufficient and necessary conditions to guarantee a global optimum. Moreover, we provide efficient algorithms based on a reduction in the intermediate unknowns from the augmented Lagrangian formulation. As a result, the underlying algorithms involve significantly fewer parameters and unknowns than the naive use of augmented Lagrangian-based methods; hence, they are fast and easy to implement. Furthermore, they can produce global optimums under mild conditions.

A Lattice Boltzmann Method for Piece-wise Constant Mumford Shah Model

A Lattice Boltzmann Method for Piece-wise Constant Mumford Shah Model

Image Selective Segmentation Under Geometrical Constraints Using an Active Contour Approach

In this paper we propose a new model for segmentation of an image under some geometrical constraints in order to detect special regions of interest. Our work is based on the recent work by Gout et al. [Numer. Algorithms, 39 (2005), pp. 155-173 and 48 (2008), pp. 105-133] using geodesic active contours models, by combining it with the idea of a piecewise constant Mumford-Shah model as with the non-selective Chan-Vese segmentation. Numerical tests show that our method is more robust than the previous works. AMS subject classifications: 62H35, 65N22, 65N55, 74G65, 74G75

Moving objects segmentation based on piecewise constant Mumford-Shah model solving by additive operator splitting

We propose a moving objects segmentation method for color image sequences based on the piecewise constant Mumford-Shah model (also known as the C-V model) solving by the semi-implicit additive operator splitting (AOS) scheme, which is unconditionally stable, fast, and easy to implement. The method first uses the Gaussian mixture model for background modeling and then subtracts the background to obtain the moving regions that are the handling objects of our method. As a result of the introduction of the AOS scheme, we could use a rather large time step and still maintain the stability of the evolution process. Additionally, the method can easily be parallelized because the AOS scheme decomposes the equations into a sequence of one-dimensional (1-D) systems. The experimental results demonstrate that under real moving objects video tests, the AOS scheme accelerates the evolution of the curve and significantly reduces the number of iterations, and also demonstrates the validity of our method.

A Multiresolution Stochastic Level Set Method for Mumford–Shah Image Segmentation

The Mumford-Shah model is one of the most successful image segmentation models. However, existing algorithms for the model are often very sensitive to the choice of the initial guess. To make use of the model effectively, it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently. While gradient descent based methods are well-known to find a local minimum only, even many stochastic methods do not provide a practical solution to this problem either. In this paper, we consider the computation of a global minimum of the multiphase piecewise constant Mumford-Shah model. We propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess. At the heart of our algorithm is a well-designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods. In our experiments, a very high-quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps. We also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum. Furthermore, we derived a simple but useful theoretical result relating solutions at different spatial resolutions.

Adaptive level set image segmentation using the Mumford and Shah functional

This paper proposes an adaptive 2-phase level set image segmentation algorithm to improve the original level set-based piecewise constant Mumford-Shah model. With the introduction of a multiplicative gain field, the model is adaptive to intensity inhomogeneity, thus tending to obtain the actual boundaries of the objects, as well as a property-seeking and -driven classification algorithm.