arXiv subjects

Monika Dörfler

Publications and source records attributed to Monika Dörfler.

17 recordsLinked to original sources

arXiv2025

Approximation properties of operator coorbit spaces and sparsity classes

Extensions of coorbit spaces for functions to operators have been introduced by two different groups in \cite{doelumcskr24} and \cite{köbaLOC25}, where one is based on the coorbit theory of Feichtinger-Gröchening while the other is based on the theory of localized frames. We show that for certain Gabor g-frames the co-orbit spaces in \cite{köbaLOC25} conincide with the ones in \cite{doelumcskr24} and we refer to this class of operators as operator coorbit spaces. Based on the description of operator coorbit spaces in terms of Gabor g-frames we provide operator dictionaries for these spaces that allow us to define sparsity classes in this setting. We establish that these sparsity classes also coincide with the operator coorbit spaces, which holds, in particular for all Feichtinger operators, a nice class of mixed states. Numerical examples confirm the expected approximation quality by few terms for appropriately chosen operators.

math.FA↗

arXiv2024

Redistributor: Transforming Empirical Data Distributions

We present an algorithm and package, Redistributor, which forces a collection of scalar samples to follow a desired distribution. When given independent and identically distributed samples of some random variable $S$ and the continuous cumulative distribution function of some desired target $T$, it provably produces a consistent estimator of the transformation $R$ which satisfies $R(S)=T$ in distribution. As the distribution of $S$ or $T$ may be unknown, we also include algorithms for efficiently estimating these distributions from samples. This allows for various interesting use cases in image processing, where Redistributor serves as a remarkably simple and easy-to-use tool that is capable of producing visually appealing results. For color correction it outperforms other model-based methods and excels in achieving photorealistic style transfer, surpassing deep learning methods in content preservation. The package is implemented in Python and is optimized to efficiently handle large datasets, making it also suitable as a preprocessing step in machine learning. The source code is available at https://github.com/paloha/redistributor.

cs.CV↗

arXiv2023

Time-Frequency Analysis and Coorbit Spaces of Operators

We introduce an operator valued Short-Time Fourier Transform for certain classes of operators with operator windows, and show that the transform acts in an analogous way to the Short-Time Fourier Transform for functions, in particular giving rise to a family of vector-valued reproducing kernel Banach spaces, the so called coorbit spaces, as spaces of operators. As a result of this structure the operators generating equivalent norms on the function modulation spaces are fully classified. We show that these operator spaces have the same atomic decomposition properties as the function spaces, and use this to give a characterisation of the spaces using localisation operators.

math.FA↗

arXiv2019

Signal Analysis using Born-Jordan-type Distribution

In this note we exhibit recent advances in signal analysis via time-frequency distributions. New members of the Cohen class, generalizing the Wigner distribution, reveal to be effective in damping artefacts of some signals. We will survey their main properties and drawbacks and present open problems related to such phenomena.

math.FA↗

arXiv2019

Gabor frames and deep scattering networks in audio processing

This paper introduces Gabor scattering, a feature extractor based on Gabor frames and Mallat's scattering transform. By using a simple signal model for audio signals specific properties of Gabor scattering are studied. It is shown that for each layer, specific invariances to certain signal characteristics occur. Furthermore, deformation stability of the coefficient vector generated by the feature extractor is derived by using a decoupling technique which exploits the contractivity of general scattering networks. Deformations are introduced as changes in spectral shape and frequency modulation. The theoretical results are illustrated by numerical examples and experiments. Numerical evidence is given by evaluation on a synthetic and a "real" data set, that the invariances encoded by the Gabor scattering transform lead to higher performance in comparison with just using Gabor transform, especially when few training samples are available.

cs.SD↗

arXiv2019

On orthogonal projections for dimension reduction and applications in augmented target loss functions for learning problems

The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, preservation of variance and of pairwise relative distances. Investigations of their asymptotic correlation as well as numerical experiments show that a projection does usually not satisfy both objectives at once. In a standard classification problem we determine projections on the input data that balance the objectives and compare subsequent results. Next, we extend our application of orthogonal projections to deep learning tasks and introduce a general framework of augmented target loss functions. These loss functions integrate additional information via transformations and projections of the target data. In two supervised learning problems, clinical image segmentation and music information classification, the application of our proposed augmented target loss functions increase the accuracy.

math.NA↗

arXiv2019

Learning Complex Basis Functions for Invariant Representations of Audio

Learning features from data has shown to be more successful than using hand-crafted features for many machine learning tasks. In music information retrieval (MIR), features learned from windowed spectrograms are highly variant to transformations like transposition or time-shift. Such variances are undesirable when they are irrelevant for the respective MIR task. We propose an architecture called Complex Autoencoder (CAE) which learns features invariant to orthogonal transformations. Mapping signals onto complex basis functions learned by the CAE results in a transformation-invariant "magnitude space" and a transformation-variant "phase space". The phase space is useful to infer transformations between data pairs. When exploiting the invariance-property of the magnitude space, we achieve state-of-the-art results in audio-to-score alignment and repeated section discovery for audio. A PyTorch implementation of the CAE, including the repeated section discovery method, is available online.

cs.SD↗

arXiv2019

Improving Machine Hearing on Limited Data Sets

Convolutional neural network (CNN) architectures have originated and revolutionized machine learning for images. In order to take advantage of CNNs in predictive modeling with audio data, standard FFT-based signal processing methods are often applied to convert the raw audio waveforms into an image-like representations (e.g. spectrograms). Even though conventional images and spectrograms differ in their feature properties, this kind of pre-processing reduces the amount of training data necessary for successful training. In this contribution we investigate how input and target representations interplay with the amount of available training data in a music information retrieval setting. We compare the standard mel-spectrogram inputs with a newly proposed representation, called Mel scattering. Furthermore, we investigate the impact of additional target data representations by using an augmented target loss function which incorporates unused available information. We observe that all proposed methods outperform the standard mel-transform representation when using a limited data set and discuss their strengths and limitations. The source code for reproducibility of our experiments as well as intermediate results and model checkpoints are available in an online repository.

cs.SD↗

arXiv2018

Generalized Born--Jordan Distributions and Applications

The quadratic nature of the Wigner distribution causes the appearance of unwanted interferences. This is the reason why engineers, mathematicians and physicists look for related time-frequency distributions, many of them are members of the Cohen class. Among them, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called ghost frequencies are damped quite well, and the noise is in general reduced. The very insight relies on the kernel of such a distribution, which contains the sinus cardinalis (sinc), which can be viewed as the Fourier transform, of the first B-Spline. Replacing the function B-Spline with the spline or order n, on the Fourier side we obtain the n-th power of sinc, whose decay at infinity increases with n. We introduce the corresponding Cohen's Kernel and study the properties of the related time-frequency distribution which we call generalized Born--Jordan distribution.

math.FA↗

arXiv2017

A Phase Vocoder based on Nonstationary Gabor Frames

We propose a new algorithm for time stretching music signals based on the theory of nonstationary Gabor frames (NSGFs). The algorithm extends the techniques of the classical phase vocoder (PV) by incorporating adaptive time-frequency (TF) representations and adaptive phase locking. The adaptive TF representations imply good time resolution for the onsets of attack transients and good frequency resolution for the sinusoidal components. We estimate the phase values only at peak channels and the remaining phases are then locked to the values of the peaks in an adaptive manner. During attack transients we keep the stretch factor equal to one and we propose a new strategy for determining which channels are relevant for reinitializing the corresponding phase values. In contrast to previously published algorithms we use a non-uniform NSGF to obtain a low redundancy of the corresponding TF representation. We show that with just three times as many TF coefficients as signal samples, artifacts such as phasiness and transient smearing can be greatly reduced compared to the classical PV. The proposed algorithm is tested on both synthetic and real world signals and compared with state of the art algorithms in a reproducible manner.

cs.SD↗

arXiv2017

Multi-Window Weaving Frames

In this work we deal with the recently introduced concept of weaving frames. We extend the concept to include multi-window frames and present the first sufficient criteria for a family of multi-window Gabor frames to be woven. We give a Hilbert space norm criterion and a pointwise criterion in phase space. The key ingredient are localization operators in phase space and we give examples of woven multi-window Gabor frames consisting of Hermite functions.

math.FA↗

arXiv2015

Sampling time-frequency localized functions and constructing localized time-frequency frames

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.

math.FA↗

arXiv2013

Frames adapted to a phase-space cover

We construct frames adapted to a given cover of the time-frequency or time-scale plane. The main feature is that we allow for quite general and possibly irregular covers. The frame members are obtained by maximizing their concentration in the respective regions of phase-space. We present applications in time-frequency, wavelet and Gabor analysis.

math.FA↗

arXiv2013

Nonstationary Gabor Frames - Approximately Dual Frames and Reconstruction Errors

Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual frames. In particular, we give constructive examples for so-called almost painless nonstationary frames, that is, frames that are closely related to nonstationary frames with compactly supported windows. The theoretical results are illustrated by concrete computational and numerical examples.

math.FA↗

arXiv2012

A framework for invertible, real-time constant-Q transforms

Audio signal processing frequently requires time-frequency representations and in many applications, a non-linear spacing of frequency-bands is preferable. This paper introduces a framework for efficient implementation of invertible signal transforms allowing for non-uniform and in particular non-linear frequency resolution. Non-uniformity in frequency is realized by applying nonstationary Gabor frames with adaptivity in the frequency domain. The realization of a perfectly invertible constant-Q transform is described in detail. To achieve real-time processing, independent of signal length, slice-wise processing of the full input signal is proposed and referred to as sliCQ transform. By applying frame theory and FFT-based processing, the presented approach overcomes computational inefficiency and lack of invertibility of classical constant-Q transform implementations. Numerical simulations evaluate the efficiency of the proposed algorithm and the method's applicability is illustrated by experiments on real-life audio signals.

math.FA↗

arXiv2012

Nonstationary Gabor Frames - Existence and Construction

Nonstationary Gabor frames were recently introduced in adaptive signal analysis. They represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. In this paper we show a general existence result for this family of frames. We then give a perturbation result for nonstationary Gabor frames and construct nonstationary Gabor frames with non-compactly supported windows from a related painless nonorthogonal expansion. Finally, the theoretical results are illustrated by two examples of practical relevance.

math.FA↗

arXiv2002

Compactness Criteria in Function Spaces

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L^2(R^d), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.

math.FA↗