Several inhibitory and excitatory factors regulate the beating of the heart. Consequently, the interbeat intervals (IBIs) vary around a mean value. Various statistics have been proposed to capture heart rate variability (HRV) to give a glimpse into this balance. However, these statistics require accurate estimation of IBIs as a first step, which can be challenging especially for signals recorded in ambulatory conditions. We propose a lightweight state-space filter that models the IBIs as samples of an inverse Gaussian distribution with time-varying parameters. We make the filter robust against outliers by adapting the probabilistic data association filter to the setup. We demonstrate that the resulting filter can accurately identify outliers and the parameters of the tracked distribution can be used to compute a specific HRV statistic (standard deviation of normal-to-normal intervals, SDNN) without further analysis.

We consider a separation problem where the observation consists of the sum of a high amplitude smooth signal and a low amplitude transient signal. We propose a method for decomposition that relies on solving instances of a `constrained filtering problem,' which is posed as a convex minimization problem. We provide a fast algorithm for solving the minimization problem, and demonstrate the potential of the scheme on a vital signs monitoring experiment using radar.

We consider the problem of recovering a signal from the norms of its linear transforms. We formulate the recovery as a minimization problem, and propose algorithms for solving the formulation. We discuss the convergence of the algorithms and compare their behaviors. We demonstrate the usefulness of the formulation/algorithm on the problem of recovering positions of points given their pairwise distances.

Recently, penalty functions promoting signals that are sparse within and across groups have been proposed. In this letter, we propose a modified penalty function that offers additional flexibility in forming groups. We study the properties of the penalty function and propose a new algorithm that can be used in energy minimization formulations that employ it. We demonstrate the effects of using the penalty function on a simple linear inverse problem.

We consider the problem of reconstructing an object given magnitudes of linear measurements. We follow the `lifting' approach, but unlike previous work which use convex relaxations of the unit rank constraint, we use a weakly-convex matrix penalty. We derive a convergent algorithm and show that it is computationally more feasible than those obtained under convex relaxations. We demonstrate numerically that when the signal to noise ratio is high, the proposed algorithm can achieve almost error-free reconstruction with fewer measurements than when convex relaxation is employed.

We derive the mapping that takes an observation vector to the minimizer of a bivariate cost consisting of the sum of a quadratic data fidelity term and an ℓ1norm. The derived mapping is useful for accelerating convergence of iterative algorithms that aim to solve ℓ1regularized problems. We discuss how to use the mapping in practice and demonstrate the improvement in convergence rate with numerical experiments.

We consider the problem of constructing a penalty function associated with a given monotone function. We provide a construction that allows the monotone function to be discontinuous or bounded. We show that, although the penalty may be non-convex, it is weakly convex. We briefly discuss the implications for iterative solutions of linear inverse problems.

We consider a minimization problem where the cost function consists of the sum of a quadratic data fidelity term and a penalty term. The quadratic involves a matrix H that can be embedded into a larger matrix H̃ where multiplication with the inverse of I + αH̃TH̃ can be efficiently performed. We discuss how to take advantage of this property when the Douglas-Rachford algorithm is utilized.

Total variation (TV) denoising is an effective noise suppression method when the derivative of the underlying signal is known to be sparse. TV denoising is defined in terms of a convex optimization problem involving a quadratic data fidelity term and a convex regularization term. A non-convex regularizer can promote sparsity more strongly, but generally leads to a non-convex optimization problem with non-optimal local minima. This letter proposes the use of a non-convex regularizer constrained so that the total objective function to be minimized maintains its convexity. Conditions for a non-convex regularizer are given that ensure the total TV denoising objective function is convex. An efficient algorithm is given for the resulting problem.

We consider the estimation of an audio source from multiple noisy observations, where the correlation between noise in the different observations is low. We propose a two-stage method for this estimation problem. The method does not require any information about noise and assumes that the signal of interest has a sparse time-frequency representation. The first stage uses this assumption to obtain the best linear combination of the observations. The second stage estimates the amount of remaining noise and applies a post-filter to further enhance the reconstruction. We discuss the optimality of this method under a specific model and demonstrate its usefulness on synthetic and real data.

Spectral audio denoising methods usually make use of the magnitudes of a time-frequency representation of the signal. However, if the time-frequency frame consists of quadrature pairs of atoms (as in the short-time Fourier transform), then the phases of the coefficients also follow a predictable pattern, for which simple models are viable. In this paper, we propose a scheme that takes into account the phase information of the signals for the audio denoising problem. The scheme requires to minimize a cost function composed of a diagonally weighted quadrature data term and a fused-lasso type penalty. We formulate the problem as a saddle point search problem and propose an algorithm that numerically finds the solution. Based on the optimality conditions of the problem, we present a guideline on how to select the parameters of the problem. We discuss the performance and the influence of the parameters through experiments.

The fused lasso problem involves the minimization of the sum of a quadratic, a TV term and an ℓ1term. The solution can be obtained by applying a TV denoising filter followed by soft-thresholding. However, soft-thresholding introduces a certain bias to the non-zero coefficients. In order to prevent this bias, we propose to replace the ℓ1penalty with a non-convex penalty. We show that the solution can similarly be obtained by applying a modified thresholding function to the result of the TV-denoising filter.

We propose a simple prior for restoration problems involving oscillatory signals. The prior makes use of an underlying analytic frame decomposition with narrow subbands. Other than this, the prior does not have any other parameters, which makes it simple to use and apply. We demonstrate the utility of the proposed prior through some real audio restoration experiments.

This paper introduces a “directional total variation” (TV) where the gradients are weighted depending on their direction. The introduced directional TV has increased (and tunable) sensitivity to variations at a selected direction. In order to demonstrate the utility of the directional TV, we consider an image denoising formulation. This formulation requires the realization of the “proximal map” of the directional TV. Therefore, it is relevant for more general inverse problem settings as well. We derive an algorithm that solves the problem and use the algorithm to study the effects of the parameters of the directional TV.

In a number of signal processing applications, problem formulations based on the ℓ1norm as a sparsity inducing signal prior lead to simple algorithms with good performance. However, ℓ1norm is not flexible enough to handle certain signal structures that are represented using a few groups of coefficients. Formulations that make use of mixed norms provide an alternative that can handle such signals by forcing sparsity on a group level and allowing non-sparse distributions within the groups. However, conventional mixed norms allow only non-overlapping groups - a restriction that leads to characteristics unlikely for natural signals. In this paper, we investigate mixed norms with overlapping groups. We consider a simple denoising formulation that gives a convex optimization problem and provide an algorithm that solves the problem. We use the algorithm to evaluate the performance of mixed norms with overlapping groups as signal priors.

Modern deconvolution algorithms are often specified as minimization problems involving a non-quadratic regularization functional. When the latter is a wavelet-domain ℓ1-norm that favors sparse solutions, the problem can be solved by a simple iterative shrinkage/thresholding algorithm (ISTA). This approach provides state-of-the-art results in 2-D, but is harder to deploy in 3-D because of its slow convergence. In this paper, we propose an acceleration scheme that turns wavelet-regularized deconvolution into a competitive solution for 3-D fluorescence microscopy. A significant speed-up is achieved though a synergistic combination of subband-adapted thresholds and sequential TwIST updates. We provide a theoretical justification of the procedure together with an experimental evaluation, including the application to real 3-D fluorescence data.

In this letter, we expand upon the method of Tay for the design of orthonormal Q-shift filters for the dual-tree complex wavelet transform. The method of Tay searches for good Hilbert-pairs in a one-parameter family of conjugate-quadrature filters that have one vanishing moment less than the Daubechies conjugate-quadrature filters (CQFs). In this letter, we compute feasible sets for one- and two-parameter families of CQFs by employing the trace parameterization of nonnegative trigonometric polynomials and semidefinite programming. This permits the design of CQF pairs that define complex wavelets that are more nearly analytic, yet still have a high number of vanishing moments.