This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the l1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the l1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than l1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.

This paper proposes a sampling strategy for bandlimited graph signals over perturbed graph, in which we assume the edge between any pair of the nodes may be deleted randomly. Considering the mismatch between the true graph and the presumed graph, we derive the mean square error (MSE) of the reconstructed bandlimited graph signals. To minimize the MSE, we propose a greedy-based algorithm to obtain the optimal sampling set. Furthermore, we use Neumann series to avoid the pseudo-inverse computing. An efficient algorithm with low-complexity is thus proposed. Finally, numerical results show the superiority of our proposed algorithms over the other existing algorithms.

In this paper, we propose an effective and robust method of spatial feature extraction for acoustic scene analysis utilizing partially synchronized and/or closely located distributed microphones. In the proposed method, a new cepstrum feature utilizing a graph-based basis transformation to extract spatial information from distributed microphones, while taking into account whether any pairs of microphones are synchronized and/or closely located, is introduced. Specifically, in the proposed graph-based cepstrum, the log-amplitude of a multichannel observation is converted to a feature vector utilizing the inverse graph Fourier transform, which is a method of basis transformation of a signal on a graph. Results of experiments using real environmental sounds show that the proposed graph-based cepstrum robustly extracts spatial information with consideration of the microphone connections. Moreover, the results indicate that the proposed method more robustly classifies acoustic scenes than conventional spatial features when the observed sounds have a large synchronization mismatch between partially synchronized microphone groups.

We propose a non-blind deconvolution algorithm of point cloud attributes inspired by multi-Wiener SURE-LET deconvolution for images. The image reconstructed by the SURE-LET approach is expressed as a linear combination of multiple filtered images where the filters are defined on the frequency domain. The coefficients of the linear combination are calculated so that the estimate of mean squared error between the original and restored images is minimized. Although the approach is very effective, it is only applicable to images. Recently we have to handle signals on irregular grids, e.g., texture data on 3D models, which are often blurred due to diffusion or motions of objects. However, we cannot utilize image processing-based approaches straightforwardly since these high-dimensional signals cannot be transformed into their frequency domain. To overcome the problem, we use graph signal processing (GSP) for deblurring the complex-structured data. That is, the SURE-LET approach is redefined on GSP, where the Wiener-like filtering is followed by the subband decomposition with an analysis graph filter bank, and then thresholding for each subband is performed. In the experiments, the proposed method is applied to blurred textures on 3D models and synthetic sparse data. The experimental results show clearly deblurred signals with SNR improvements.

In this paper, we propose a novel design method of two channel critically sampled compactly supported biorthogonal graph wavelet filter banks with half-band kernels. First of all, we use the polynomial half-band kernels to construct a class of biorthogonal graph wavelet filter banks, which exactly satisfy the PR (perfect reconstruction) condition. We then present a design method of the polynomial half-band kernels with the specified degree of flatness. The proposed design method utilizes the PBP (Parametric Bernstein Polynomial), which ensures that the half-band kernels have the specified zeros at λ=2. Therefore the constraints of flatness are satisfied at both of λ=0 and λ=2, and then the resulting graph wavelet filters have the flat spectral responses in passband and stopband. Furthermore, we apply the Remez exchange algorithm to minimize the spectral error of lowpass (highpass) filter in the band of interest by using the remaining degree of freedom. Finally, several examples are designed to demonstrate the effectiveness of the proposed design method.

We propose an edge-preserving multiscale image decomposition method using filters for non-equispaced signals. It is inspired by the domain transform, which is a high-speed edge-preserving smoothing method, and it can be used in many image processing applications. One of the disadvantages of the domain transform is sensitivity to noise. Even though the proposed method is based on non-equispaced filters similar to the domain transform, it is robust to noise since it employs a multiscale decomposition. It uses the Laplacian pyramid scheme to decompose an input signal into the piecewise-smooth components and detail components. We design the filters by using an optimization based on edge-preserving smoothing with a conversion of signal distances and filters taking into account the distances between signal intervals. In addition, we also propose construction methods of filters for non-equispaced signals by using arbitrary continuous filters or graph spectral filters in order that various filters can be accommodated by the proposed method. As expected, we find that, similar to state-of-the-art edge-preserving smoothing techniques, including the domain transform, our approach can be used in many applications. We evaluated its effectiveness in edge-preserving smoothing of noise-free and noisy images, detail enhancement, pencil drawing, and stylization.