# C02Digital Representations of Manifold Data

Whenever data is processed using computers, analog-to-digital (A/D) conversion, that is, representing the data using just 0s and 1s, is a crucial ingredient of the procedure. This project will mathematically study approaches to this problem for data with geometric structural constraints.

#### Mission-

Investigating how well geometric objects can be represented using bit streams

#### Scientific Details+

Many results in mathematical data analysis follow the paradigm of increasing efficiency by incorporating structural assumptions on the underlying data. The most prominent models today are sparsity models and manifold models in high dimensional data analysis. One main goal in these scenarios is a faithful analog to digital conversion, which not only requires a discretization, but also a digitalization step, often termed quantization, as only a finite number of bits can be processed.

A key aspect in finding suitable methods for digitally representing data is a careful balance between the resolution of this quantization step and the redundancy of the underlying discretization. Arguably the most popular class of suitable quantization schemes for highly redundant settings are so-called Sigma-Delta modulators. The underlying idea is to quantize recursively and then deduce recovery guarantees from the stability of the resulting discrete dynamical system. Variants are available for bandlimited signals, for frame representations, and for compressed sensing. For manifold models, however, there is very little quantization literature available. Project C02 aims to fill this gap, attempting a systematic study of quantization under manifold constraints.

The following three viewpoints shall be investigated:

• Functions on manifolds: This work package is devoted to quantized representations of functions whose domain is a manifold. We will study bandlimited functions and also discuss applications to digital halftoning on manifolds.
• Data lying on a manifold: We aim to study redundancies resulting from the fact that high dimensional data lies on a or close to a known manifold. We aim to incorporate quantization into the compressed sensing methodology for manifold models, as they have recently received attention in the literature.

Geometric and topological properties: Our question here is to what extent the manifold as a whole or its important geometric and topological properties can be recovered despite quantization. This project part will closely interact with project C04, which studies a related problem without quantization.

#### Publications+

##### Papers
###### One-Bit Unlimited Sampling

Authors: Graf, Olga and Bhandari, Ayush and Krahmer, Felix
Journal: preprint
Date: Nov 2018

###### One-Bit Sigma-Delta Modulation on a Closed Loop

Authors: Krause-Solberg, Sara and Graf, Olga and Krahmer, Felix
Journal: 2018 IEEE Statistical Signal Processing Workshop (SSP)
Date: Aug 2018

###### Level-Dependent Interpolatory Hermite Subdivision Schemes and Wavelets

Authors: Mariantonia Cotronei, Caroline Moosmüller, Tomas Sauer and Sissouno, Nada
Journal: Constructive Approximation
Date: Jul 2018
DOI: 10.1007/s00365-018-9444-4

###### On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds

Authors: Iwen, Mark A. and Krahmer, Felix and Krause-Solberg, Sara and Maly, Johannes
Journal: preprint
Date: Jul 2018

###### Compressed Sensing for Analog Signals

Authors: Bodmann, B. and Flinth, A. and Kutyniok, G.
Note: preprint
Date: Mar 2018

###### A Haar Wavelet-Based Perceptual Similarity Index for Image Quality Assessment

Authors: Reisenhofer, R. and Bosse, S. and Kutyniok, G. and Wiegand, T.
Journal: Signal Proc. Image Comm., 61:33--43
Date: Feb 2018
DOI: 10.1016/j.image.2017.11.001

###### Compressed sensing for finite-valued signals

Authors: Keiper, S. and Kutyniok, G. and Lee, D. G. and Pfander, G. E.
Journal: Linear Algebra and its Applications, 532:570--613
Date: Nov 2017
DOI: 10.1016/j.laa.2017.07.006

###### $\ell^1$-Analysis Minimization and Generalized \mbox{(Co-)}Sparsity: {W}hen Does Recovery Succeed?

Authors: Genzel, M. and Kutyniok, G. and März, M.
Note: preprint
Date: Oct 2017

##### Posters
###### One-bit sigma-delta modulation on a closed loop

Author: Sara Krause-Solberg, Olga Graf, Felix Krahmer
Journal: poster
Date: Nov 2018

#### Prof. Dr. Felix Krahmer    +

Projects: C02
University: TU München
E-Mail: Felix.Krahmer[at]ma.tum.de

#### Prof. Dr. Gitta Kutyniok   +

Projects: C03, C02
University: TU Berlin
E-Mail: kutyniok[at]math.tu-berlin.de
Website: http://www.math.tu-berlin.de/?108957

#### Olga Graf   +

Projects: C02
University: TU München
E-Mail: graf[at]ma.tum.de