Blind Identification of Invertible Graph Filters with Multiple Sparse Inputs

This paper deals with problem of blind identification of a graph filter and its sparse input signal, thus broadening the scope of classical blind deconvolution of temporal and spatial signals to irregular graph domains. While the observations are bilinear functions of the unknowns, a mild requirement on invertibility of the filter enables an efficient convex formulation, without relying on matrix lifting that can hinder applicability to large graphs. On top of scaling, it is argued that (non-cyclic) permutation ambiguities may arise with some particular graphs. Deterministic sufficient conditions under which the proposed convex relaxation can exactly recover the unknowns are stated, along with those guaranteeing identifiability under the Bernoulli-Gaussian model for the inputs. Numerical tests with synthetic and real-world networks illustrate the merits of the proposed algorithm, as well as the benefits of leveraging multiple signals to aid the (blind) localization of sources of diffusion

Enhancing Geometric Deep Learning via Graph Filter Deconvolution

In this paper, we incorporate a graph filter deconvolution step into the classical geometric convolutional neural network pipeline. More precisely, under the assumption that the graph domain plays a role in the generation of the observed graph signals, we pre-process every signal by passing it through a sparse deconvolution operation governed by a pre-specified filter bank. This deconvolution operation is formulated as a group-sparse recovery problem, and convex relaxations that can be solved efficiently are put forth. The deconvolved signals are then fed into the geometric convolutional neural network, yielding better classification performance than their unprocessed counterparts. Numerical experiments showcase the effectiveness of the deconvolution step on classification tasks on both synthetic and real-world settings.Comment: 5 pages, 8 figures, to appear in the proceedings of the 2018 6th IEEE Global Conference on Signal and Information Processing, November 26-29, 2018, Anaheim, California, US

Image Restoration Using Convolutional Auto-encoders with Symmetric Skip Connections

Image restoration, including image denoising, super resolution, inpainting, and so on, is a well-studied problem in computer vision and image processing, as well as a test bed for low-level image modeling algorithms. In this work, we propose a very deep fully convolutional auto-encoder network for image restoration, which is a encoding-decoding framework with symmetric convolutional-deconvolutional layers. In other words, the network is composed of multiple layers of convolution and de-convolution operators, learning end-to-end mappings from corrupted images to the original ones. The convolutional layers capture the abstraction of image contents while eliminating corruptions. Deconvolutional layers have the capability to upsample the feature maps and recover the image details. To deal with the problem that deeper networks tend to be more difficult to train, we propose to symmetrically link convolutional and deconvolutional layers with skip-layer connections, with which the training converges much faster and attains better results.Comment: 17 pages. A journal extension of the version at arXiv:1603.0905

Focused blind deconvolution

We introduce a novel multichannel blind deconvolution (BD) method that extracts sparse and front-loaded impulse responses from the channel outputs, i.e., their convolutions with a single arbitrary source. A crucial feature of this formulation is that it doesn't encode support restrictions on the unknowns, unlike most prior work on BD. The indeterminacy inherent to BD, which is difficult to resolve with a traditional L1 penalty on the impulse responses, is resolved in our method because it seeks a first approximation where the impulse responses are: "maximally white" -- encoded as the energy focusing near zero lag of the impulse-response auto-correlations; and "maximally front-loaded" -- encoded as the energy focusing near zero time of the impulse responses. Hence we call the method focused blind deconvolution (FBD). The focusing constraints are relaxed as the iterations progress. Note that FBD requires the duration of the channel outputs to be longer than that of the unknown impulse responses. A multichannel blind deconvolution problem that is appropriately formulated by sparse and front-loaded impulse responses arises in seismic inversion, where the impulse responses are the Green's function evaluations at different receiver locations, and the operation of a drill bit inputs the noisy and correlated source signature into the subsurface. We demonstrate the benefits of FBD using seismic-while-drilling numerical experiments, where the noisy data recorded at the receivers are hard to interpret, but FBD can provide the processing essential to separate the drill-bit (source) signature from the interpretable Green's function

Blind Identification of Graph Filters

Network processes are often represented as signals defined on the vertices of a graph. To untangle the latent structure of such signals, one can view them as outputs of linear graph filters modeling underlying network dynamics. This paper deals with the problem of joint identification of a graph filter and its input signal, thus broadening the scope of classical blind deconvolution of temporal and spatial signals to the less-structured graph domain. Given a graph signal $\mathbf{y}$ modeled as the output of a graph filter, the goal is to recover the vector of filter coefficients $\mathbf{h}$, and the input signal $\mathbf{x}$ which is assumed to be sparse. While $\mathbf{y}$ is a bilinear function of $\mathbf{x}$ and $\mathbf{h}$, the filtered graph signal is also a linear combination of the entries of the lifted rank-one, row-sparse matrix $\mathbf{x} \mathbf{h}^T$. The blind graph-filter identification problem can thus be tackled via rank and sparsity minimization subject to linear constraints, an inverse problem amenable to convex relaxations offering provable recovery guarantees under simplifying assumptions. Numerical tests using both synthetic and real-world networks illustrate the merits of the proposed algorithms, as well as the benefits of leveraging multiple signals to aid the blind identification task

Blind Identification of ARX Models with Piecewise Constant Inputs

Blind system identification is known to be a hard ill-posed problem and without further assumptions, no unique solution is at hand. In this contribution, we are concerned with the task of identifying an ARX model from only output measurements. Driven by the task of identifying systems that are turned on and off at unknown times, we seek a piecewise constant input and a corresponding ARX model which approximates the measured outputs. We phrase this as a rank minimization problem and present a relaxed convex formulation to approximate its solution. The proposed method was developed to model power consumption of electrical appliances and is now a part of a bigger energy disaggregation framework. Code will be made available online.Comment: Submitted to the 52nd IEEE Conference on Decision and Contro

A neural network approach for the blind deconvolution of turbulent flows

We present a single-layer feedforward artificial neural network architecture trained through a supervised learning approach for the deconvolution of flow variables from their coarse grained computations such as those encountered in large eddy simulations. We stress that the deconvolution procedure proposed in this investigation is blind, i.e. the deconvolved field is computed without any pre-existing information about the filtering procedure or kernel. This may be conceptually contrasted to the celebrated approximate deconvolution approaches where a filter shape is predefined for an iterative deconvolution process. We demonstrate that the proposed blind deconvolution network performs exceptionally well in the a-priori testing of both two-dimensional Kraichnan and three-dimensional Kolmogorov turbulence and shows promise in forming the backbone of a physics-augmented data-driven closure for the Navier-Stokes equations

Blind Deconvolution using Modulated Inputs

This paper considers the blind deconvolution of multiple modulated signals, and an arbitrary filter. Multiple inputs $\boldsymbol{s}_1, \boldsymbol{s}_2, \ldots, \boldsymbol{s}_N =: [\boldsymbol{s}_n]$ are modulated (pointwise multiplied) with random sign sequences $\boldsymbol{r}_1, \boldsymbol{r}_2, \ldots, \boldsymbol{r}_N =: [\boldsymbol{r}_n]$, respectively, and the resultant inputs $(\boldsymbol{s}_n \odot \boldsymbol{r}_n) \in \mathbb{C}^Q, \ n = [N]$are convolved against an arbitrary input$\boldsymbol{h} \in \mathbb{C}^M$to yield the measurements$\boldsymbol{y}_n = (\boldsymbol{s}_n\odot \boldsymbol{r}_n)\circledast \boldsymbol{h}, \ n = [N] := 1,2,\ldots,N,$where$\odot$, and$\circledast$denote pointwise multiplication, and circular convolution. Given$[\boldsymbol{y}_n]$, we want to recover the unknowns$[\boldsymbol{s}_n]$, and$\boldsymbol{h}$. We make a structural assumption that unknown$[\boldsymbol{s}_n]$are members of a known$K$-dimensional (not necessarily random) subspace, and prove that the unknowns can be recovered from sufficiently many observations using an alternating gradient descent algorithm whenever the modulated inputs$\boldsymbol{s}_n \odot \boldsymbol{r}_n$are long enough, i.e,$Q \gtrsim KN+M$ (to within log factors and signal dispersion/coherence parameters)

Coupled Learning for Facial Deblur

Blur in facial images significantly impedes the efficiency of recognition approaches. However, most existing blind deconvolution methods cannot generate satisfactory results due to their dependence on strong edges, which are sufficient in natural images but not in facial images. In this paper, we represent point spread functions (PSFs) by the linear combination of a set of pre-defined orthogonal PSFs, and similarly, an estimated intrinsic (EI) sharp face image is represented by the linear combination of a set of pre-defined orthogonal face images. In doing so, PSF and EI estimation is simplified to discovering two sets of linear combination coefficients, which are simultaneously found by our proposed coupled learning algorithm. To make our method robust to different types of blurry face images, we generate several candidate PSFs and EIs for a test image, and then, a non-blind deconvolution method is adopted to generate more EIs by those candidate PSFs. Finally, we deploy a blind image quality assessment metric to automatically select the optimal EI. Thorough experiments on the facial recognition technology database, extended Yale face database B, CMU pose, illumination, and expression (PIE) database, and face recognition grand challenge database version 2.0 demonstrate that the proposed approach effectively restores intrinsic sharp face images and, consequently, improves the performance of face recognition

Sparse Blind Deconvolution for Distributed Radar Autofocus Imaging

A common problem that arises in radar imaging systems, especially those mounted on mobile platforms, is antenna position ambiguity. Approaches to resolve this ambiguity and correct position errors are generally known as radar autofocus. Common techniques that attempt to resolve the antenna ambiguity generally assume an unknown gain and phase error afflicting the radar measurements. However, ensuring identifiability and tractability of the unknown error imposes strict restrictions on the allowable antenna perturbations. Furthermore, these techniques are often not applicable in near-field imaging, where mapping the position ambiguity to phase errors breaks down. In this paper, we propose an alternate formulation where the position error of each antenna is mapped to a spatial shift operator in the image-domain. Thus, the radar autofocus problem becomes a multichannel blind deconvolution problem, in which the radar measurements correspond to observations of a static radar image that is convolved with the spatial shift kernel associated with each antenna. To solve the reformulated problem, we also develop a block coordinate descent framework that leverages the sparsity and piece-wise smoothness of the radar scene, as well as the one-sparse property of the two dimensional shift kernels. We evaluate the performance of our approach using both simulated and experimental radar measurements, and demonstrate its superior performance compared to state-of-the-art methods