A semisupervised segmentation model for collections of images.

In this paper, we consider the problem of segmentation of large collections of images. We propose a semisupervised optimization model that determines an efficient segmentation of many input images. The advantages of the model are twofold. First, the segmentation is highly controllable by the user so that the user can easily specify what he/she wants. This is done by allowing the user to provide, either offline or interactively, some (fully or partially) labeled pixels in images as strong priors for the model. Second, the model requires only minimal tuning of model parameters during the initial stage. Once initial tuning is done, the setup can be used to automatically segment a large collection of images that are distinct but share similar features. We will show the mathematical properties of the model such as existence and uniqueness of solution and establish a maximum/minimum principle for the solution of the model. Extensive experiments on various collections of biological images suggest that the proposed model is effective for segmentation and is computationally efficient.

A primal-dual method for total-variation-based wavelet domain inpainting.

Loss of information in a wavelet domain can occur during storage or transmission when the images are formatted and stored in terms of wavelet coefficients. This calls for image inpainting in wavelet domains. In this paper, a variational approach is used to formulate the reconstruction problem. We propose a simple but very efficient iterative scheme to calculate an optimal solution and prove its convergence. Numerical results are presented to show the performance of the proposed algorithm.

Subspace learning for Mumford-Shah-model-based texture segmentation through texture patches.

In this paper, we develop a robust and effective algorithm for texture segmentation and feature selection. The approach is to incorporate a patch-based subspace learning technique into the subspace Mumford-Shah (SMS) model to make the minimization of the SMS model robust and accurate. The proposed method is fully unsupervised in that it removes the need to specify training data, which is required by existing methods for the same model. We further propose a novel (to our knowledge) pairwise dissimilarity measure for pixels. Its novelty lies in the use of the relevance scores of the features of each pixel to improve its discriminating power. Some superior results are obtained compared to existing unsupervised algorithms, which do not use a subspace approach. This confirms the usefulness of the subspace approach and the proposed unsupervised algorithm.

A variational model for segmentation of overlapping objects with additive intensity value.

We propose a variant of the Mumford-Shah model for the segmentation of a pair of overlapping objects with additive intensity value. Unlike standard segmentation models, it does not only determine distinct objects in the image, but also recover the possibly multiple membership of the pixels. To accomplish this, some a priori knowledge about the smoothness of the object boundary is integrated into the model. Additivity is imposed through a soft constraint which allows the user to control the degree of additivity and is more robust than the hard constraint. We also show analytically that the additivity parameter can be chosen to achieve some stability conditions. To solve the optimization problem involving geometric quantities efficiently, we apply a multiphase level set method. Segmentation results on synthetic and real images validate the good performance of our model, and demonstrate the model's applicability to images with multiple channels and multiple objects.

Fast splitting algorithm for multiframe total variation blind video deconvolution.

We consider the recovery of degraded videos without complete knowledge about the degradation. A spatially shift-invariant but temporally shift-varying video formation model is used. This leads to a simple multiframe degradation model that relates each original video frame with multiple observed frames and point spread functions (PSFs). We propose a variational method that simultaneously reconstructs each video frame and the associated PSFs from the corresponding observed frames. Total variation (TV) regularization is used on both the video frames and the PSFs to further reduce the ill-posedness and to better preserve edges. In order to make TV minimization practical for video sequences, we propose an efficient splitting method that generalizes some recent fast single-image TV minimization methods to the multiframe case. Both synthetic and real videos are used to show the performance of the proposed method.

Semi-supervised subspace learning for Mumford-Shah model based texture segmentation.

We propose a novel image segmentation model which incorporates subspace clustering techniques into a Mumford-Shah model to solve texture segmentation problems. While the natural unsupervised approach to learn a feature subspace can easily be trapped in a local solution, we propose a novel semi-supervised optimization algorithm that makes use of information derived from both the intermediate segmentation results and the regions-of-interest (ROI) selected by the user to determine the optimal subspaces of the target regions. Meanwhile, these subspaces are embedded into a Mumford-Shah objective function so that each segment of the optimal partition is homogeneous in its own subspace. The method outperforms standard Mumford-Shah models since it can separate textures which are less separated in the full feature space. Experimental results are presented to confirm the usefulness of subspace clustering in texture segmentation.

A fast optimization transfer algorithm for image inpainting in wavelet domains.

A wavelet inpainting problem refers to the problem of filling in missing wavelet coefficients in an image. A variational approach was used by Chan et al. The resulting functional was minimized by the gradient descent method. In this paper, we use an optimization transfer technique which involves replacing their univariate functional by a bivariate functional by adding an auxiliary variable. Our bivariate functional can be minimized easily by alternating minimization: for the auxiliary variable, the minimum has a closed form solution, and for the original variable, the minimization problem can be formulated as a classical total variation (TV) denoising problem and, hence, can be solved efficiently using a dual formulation. We show that our bivariate functional is equivalent to the original univariate functional. We also show that our alternating minimization is convergent. Numerical results show that the proposed algorithm is very efficient and outperforms that of Chan et al.

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.

A primal-dual active-set method for non-negativity constrained total variation deblurring problems.

This paper studies image deblurring problems using a total variation-based model, with a non-negativity constraint. The addition of the non-negativity constraint improves the quality of the solutions, but makes the solution process a difficult one. The contribution of our work is a fast and robust numerical algorithm to solve the non-negatively constrained problem. To overcome the nondifferentiability of the total variation norm, we formulate the constrained deblurring problem as a primal-dual program which is a variant of the formulation proposed by Chan, Golub, and Mulet for unconstrained problems. Here, dual refers to a combination of the Lagrangian and Fenchel duals. To solve the constrained primal-dual program, we use a semi-smooth Newton's method. We exploit the relationship between the semi-smooth Newton's method and the primal-dual active set method to achieve considerable simplification of the computations. The main advantages of our proposed scheme are: no parameters need significant adjustment, a standard inverse preconditioner works very well, quadratic rate of local convergence (theoretical and numerical), numerical evidence of global convergence, and high accuracy of solving the optimality system. The scheme shows robustness of performance over a wide range of parameters. A comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme.

Strategies for identifying statistically significant dense regions in microarray data.

We propose and study the notion of dense regions for the analysis of categorized gene expression data and present some searching algorithms for discovering them. The algorithms can be applied to any categorical data matrices derived from gene expression level matrices. We demonstrate that dense regions are simple but useful and statistically significant patterns that can be used to 1) identify genes and/or samples of interest and 2) eliminate genes and/or samples corresponding to outliers, noise, or abnormalities. Some theoretical studies on the properties of the dense regions are presented which allow us to characterize dense regions into several classes and to derive tailor-made algorithms for different classes of regions. Moreover, an empirical simulation study on the distribution of the size of dense regions is carried out which is then used to assess the significance of dense regions and to derive effective pruning methods to speed up the searching algorithms. Real microarray data sets are employed to test our methods. Comparisons with six other well-known clustering algorithms using synthetic and real data are also conducted which confirm the superiority of our methods in discovering dense regions. The DRIFT code and a tutorial are available as supplemental material, which can be found on the Computer Society Digital Library at http://computer.org/tcbb/archives.htm.

Gene network interconnectedness and the generalized topological overlap measure.

BACKGROUND: Network methods are increasingly used to represent the interactions of genes and/or proteins. Genes or proteins that are directly linked may have a similar biological function or may be part of the same biological pathway. Since the information on the connection (adjacency) between 2 nodes may be noisy or incomplete, it can be desirable to consider alternative measures of pairwise interconnectedness. Here we study a class of measures that are proportional to the number of neighbors that a pair of nodes share in common. For example, the topological overlap measure by Ravasz et al. 1 can be interpreted as a measure of agreement between the m = 1 step neighborhoods of 2 nodes. Several studies have shown that two proteins having a higher topological overlap are more likely to belong to the same functional class than proteins having a lower topological overlap. Here we address the question whether a measure of topological overlap based on higher-order neighborhoods could give rise to a more robust and sensitive measure of interconnectedness. RESULTS: We generalize the topological overlap measure from m = 1 step neighborhoods to m > or = 2 step neighborhoods. This allows us to define the m-th order generalized topological overlap measure (GTOM) by (i) counting the number of m-step neighbors that a pair of nodes share and (ii) normalizing it to take a value between 0 and 1. Using theoretical arguments, a yeast co-expression network application, and a fly protein network application, we illustrate the usefulness of the proposed measure for module detection and gene neighborhood analysis. CONCLUSION: Topological overlap can serve as an important filter to counter the effects of spurious or missing connections between network nodes. The m-th order topological overlap measure allows one to trade-off sensitivity versus specificity when it comes to defining pairwise interconnectedness and network modules.

Dynamic cluster formation using level set methods.

Density-based clustering has the advantages for 1) allowing arbitrary shape of cluster and 2) not requiring the number of clusters as input. However, when clusters touch each other, both the cluster centers and cluster boundaries (as the peaks and valleys of the density distribution) become fuzzy and difficult to determine. We introduce the notion of cluster intensity function (CIF) which captures the important characteristics of clusters. When clusters are well-separated, CIFs are similar to density functions. But, when clusters become closed to each other, CIFs still clearly reveal cluster centers, cluster boundaries, and degree of membership of each data point to the cluster that it belongs. Clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation, which are often oscillatory or oversmoothed. These problems of kernel density estimation are resolved using Level Set Methods and related techniques. Comparisons with two existing density-based methods, valley seeking and DBSCAN, are presented which illustrate the advantages of our approach.