Learning filter functions in regularisers by minimising quotients

Learning approaches have recently become very popular in the field of inverse problems. A large variety of methods has been established in recent years, ranging from bi-level learning to high-dimensional machine learning techniques. Most learning approaches, however, only aim at fitting parametrised models to favourable training data whilst ignoring misfit training data completely. In this paper, we follow up on the idea of learning parametrised regularisation functions by quotient minimisation as established in [3]. We extend the model therein to include higher-dimensional filter functions to be learned and allow for fit- and misfit-training data consisting of multiple functions. We first present results resembling behaviour of well-established derivative-based sparse regularisers like total variation or higher-order total variation in one-dimension. Our second and main contribution is the introduction of novel families of non-derivative-based regularisers. This is accomplished by learning favourable scales and geometric properties while at the same time avoiding unfavourable ones

Necessary and sufficient conditions of solution uniqueness in ℓ1\ell_1 minimization

This paper shows that the solutions to various convex $\ell_1$ minimization problems are \emph{unique} if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other $\ell_1$ models that either minimize $f(Ax-b)$ or impose the constraint $f(Ax-b)\leq\sigma$, where $f$ is a strictly convex function. For these models, this paper proves that, given a solution $x^*$ and defining I=\supp(x^*) and s=\sign(x^*_I), $x^*$ is the unique solution if and only if $A_I$ has full column rank and there exists $y$ such that $A_I^Ty=s$ and $|a_i^Ty|_\infty<1$ for $i\not\in I$. This condition is previously known to be sufficient for the basis pursuit model to have a unique solution supported on $I$. Indeed, it is also necessary, and applies to a variety of other $\ell_1$ models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically.Comment: 6 pages; revised version; submitte

Revisiting Synthesis Model of Sparse Audio Declipper

The state of the art in audio declipping has currently been achieved by SPADE (SParse Audio DEclipper) algorithm by Kiti\'c et al. Until now, the synthesis/sparse variant, S-SPADE, has been considered significantly slower than its analysis/cosparse counterpart, A-SPADE. It turns out that the opposite is true: by exploiting a recent projection lemma, individual iterations of both algorithms can be made equally computationally expensive, while S-SPADE tends to require considerably fewer iterations to converge. In this paper, the two algorithms are compared across a range of parameters such as the window length, window overlap and redundancy of the transform. The experiments show that although S-SPADE typically converges faster, the average performance in terms of restoration quality is not superior to A-SPADE

Non-identical smoothing operators for estimating time-frequency interdependence in electrophysiological recordings

Synchronization of neural activity from distant parts of the brain is crucial for the coordination of cognitive activities. Because neural synchronization varies both in time and frequency, time–frequency (T-F) coherence is commonly employed to assess interdependences in electrophysiological recordings. T-F coherence entails smoothing the cross and power spectra to ensure statistical consistency of the estimate, which reduces its T-F resolution. This trade-off has been described in detail when the cross and power spectra are smoothed using identical smoothing operators, which may yield spurious coherent frequencies. In this article, we examine the use of non-identical smoothing operators for the estimation of T-F interdependence, i.e., phase synchronization is characterized by phase locking between signals captured by the cross spectrum and we may hence improve the trade-off by selectively smoothing the auto spectra. We first show that the frequency marginal density of the present estimate is bound within [0,1] when using non-identical smoothing operators. An analytic calculation of the bias and variance of present estimators is performed and compared with the bias and variance of standard T-F coherence using Monte Carlo simulations. We then test the use of non-identical smoothing operators on simulated data, whose T-F properties are known through construction. Finally, we analyze empirical data from eyes-closed surface electroencephalography recorded in human subjects to investigate alpha-band synchronization. These analyses show that selectively smoothing the auto spectra reduces the bias of the estimator and may improve the detection of T-F interdependence in electrophysiological data at high temporal resolution

Shape description and matching using integral invariants on eccentricity transformed images

Matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision. To keep track of changes inside the breast, for example, it is important for a computer aided detection system to establish correspondences between regions of interest. Shape transformations, computed both with integral invariants (II) and with geodesic distance, yield signatures that are invariant to isometric deformations, such as bending and articulations. Integral invariants describe the boundaries of planar shapes. However, they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure. Conversely, eccentricity transforms (Ecc) can match shapes by signatures of geodesic distance histograms based on information from inside the shape; but they ignore the boundary information. We describe a method that combines the boundary signature of a shape obtained from II and structural information from the Ecc to yield results that improve on them separately