We propose a new plug-and-play image restoration method based on primal-dual splitting. Existing plug-and-play image restoration methods interpret any off-the-shelf Gaussian denoiser as one step of the so-called alternating direction method of multipliers (ADMM). This makes it possible to exploit the power of such a highly-customized Gaussian denoising method for general image restoration tasks in a plug-and-play fashion. However, the ADMM-based plug-and-play approach (ADMMPnP) has several limitations: 1) it often requires a problem-specific iterative method in solving a subproblem, which results in a computationally expensive inner loop; and 2) it is specialized to handle the formulation of a regularization (plug-and-play) term plus a data-fidelity term, so that it does not allow to impose hard constraints useful for image restoration. Our approach resolves these issues by leveraging the nature of primal-dual splitting, yielding a very flexible plug-and-play image restoration method. Experimental results demonstrate that the proposed method is much more efficient than ADMMPnP with an inner loop, whereas it keeps the same efficiency as ADMMPnP in the case where the subproblem of ADMMPnP can be solved efficiently.

This paper presents a graph signal denoising method with the trilateral filter defined in the graph spectral domain. The original trilateral filter (TF) is a data-dependent filter that is widely used as an edge-preserving smoothing method for image processing. However, because of the data-dependency, one cannot provide its frequency domain representation. To overcome this problem, we establish the graph spectral domain representation of the data-dependent filter, i.e., a spectral graph TF (SGTF). This representation enables us to design an effective graph signal denoising filter with a Tikhonov regularization. Moreover, for the proposed graph denoising filter, we provide a parameter optimization technique to search for a regularization parameter that approximately minimizes the mean squared error w.r.t. the unknown graph signal of interest. Comprehensive experimental results validate our graph signal processing-based approach for images and graph signals.

Using a novel characterization of texture, we propose an image decomposition technique that can effectively decomposes an image into its cartoon and texture components. The characterization rests on our observation that the texture component enjoys a blockwise low-rank nature with possible overlap and shear, because texture, in general, is globally dissimilar but locally well patterned. More specifically, one can observe that any local block of the texture component consists of only a few individual patterns. Based on this premise, we first introduce a new convex prior, named the block nuclear norm (BNN), leading to a suitable characterization of the texture component. We then formulate a cartoon-texture decomposition model as a convex optimization problem, where the simultaneous estimation of the cartoon and texture components from a given image or degraded observation is executed by minimizing the total variation and BNN. In addition, patterns of texture extending in different directions are extracted separately, which is a special feature of the proposed model and of benefit to texture analysis and other applications. Furthermore, the model can handle various types of degradation occurring in image processing, including blur+missing pixels with several types of noise. By rewriting the problem via variable splitting, the so-called alternating direction method of multipliers becomes applicable, resulting in an efficient algorithmic solution to the problem. Numerical examples illustrate that the proposed model is very selective to patterns of texture, which makes it produce better results than state-of-the-art decomposition models.

Minimizing L ₀ gradient, the number of the non-zero gradients of an image, together with a quadratic data-fidelity to an input image has been recognized as a powerful edge-preserving filtering method. However, the L ₀ gradient minimization has an inherent difficulty: a user-given parameter controlling the degree of flatness does not have a physical meaning since the parameter just balances the relative importance of the L ₀ gradient term to the quadratic data-fidelity term. As a result, the setting of the parameter is a troublesome work in the L ₀ gradient minimization. To circumvent the difficulty, we propose a new edge-preserving filtering method with a novel use of the L ₀ gradient. Our method is formulated as the minimization of the quadratic data-fidelity subject to the hard constraint that the L ₀ gradient is less than a user-given parameter α. This strategy is much more intuitive than the L ₀ gradient minimization because the parameter α has a clear meaning: the L ₀ gradient value of the output image itself, so that one can directly impose a desired degree of flatness by α. We also provide an efficient algorithm based on the so-called alternating direction method of multipliers for computing an approximate solution of the nonconvex problem, where we decompose it into two subproblems and derive closed-form solutions to them. The advantages of our method are demonstrated through extensive experiments.

This paper proposes an optimization framework that can efficiently deal with convex data-fidelity constraints onto which the metric projections are difficult to compute. Although such an involved data-fidelity constraint is expected to play an important role in signal recovery under non-Gaussian noise contamination, the said difficulty precludes existing algorithms from solving convex optimization problems with the constraint. To resolve this dilemma, we introduce a fixed point set characterization of involved data-fidelity constraints based on a certain computable quasi-nonexpansive mapping. This characterization enables us to mobilize the hybrid steepest descent method to solve convex optimization problems with such a constraint. The proposed framework can handle a variety of involved data-fidelity constraints in a unified manner, without geometric approximation to them. In addition, it requires no computationally expensive procedure such as operator inversion and inner loop. As applications of the proposed framework, we provide image restoration under several types of non-Gaussian noise contamination with illustrative examples.

Mapping the knowledge structure from word co-occurrences in a collection of academic papers has been widely used to provide insight into the topic evolution in an arbitrary research field. In a traditional approach, the paper collection is first divided into temporal subsets, and then a co-word network is independently depicted in a 2D map to characterize each period’s trend. To effectively map emerging research trends from such a time-series of co-word networks, this paper presents TrendNets, a novel visualization methodology that highlights the rapid changes in edge weights over time. Specifically, we formulated a new convex optimization framework that decomposes the matrix constructed from dynamic co-word networks into a smooth part and a sparse part: the former represents stationary research topics, while the latter corresponds to bursty research topics. Simulation results on synthetic data demonstrated that our matrix decomposition approach achieved the best burst detection performance over four baseline methods. In experiments conducted using papers published in the past 16 years at three conferences in different fields, we showed the effectiveness of TrendNets compared to the traditional co-word representation. We have made our codes available on the Web to encourage scientific mapping in all research fields.