# Signals, Images, and Data

A large number of signals is collected from a wide variety of physical and biological phenomena, and in a variety of forms, ranging from acoustics, radar, camera images, hyper-spectral images, movies, and many others. These signals are the ones that our own human sensors are used to measuring, and our own brains have evolved to efficiently interpret. Huge data sets of this type are collected daily, from consumer cameras to music studio recordings to satellite images. The large collections of signals need to be analyzed, in order to to order them and to extract useful information. For example, technology companies of all sizes develop and use tools to automatically categorize and recognize objects and faces in images, or in art works; in digital pathology one would like to be able to automatically categorize cell types, or, in neurobiology, cell activity; in applications to agriculture the condition of the crops is monitored by aerial hyper-spectral imaging; in the study of simulations of biological molecules one would like to understand how large molecules move in the high-dimensional space of possible configurations. There are of course many data types that are very different from signals that we are used to hearing and seeing, and require different analysis tools. Examples may include the activity in a social network in time, or a set of transactions in an economic network, or the interactions between gene and proteins through time in a biological network.

Several faculty members in the Duke mathematics department have focused their research on problems related to large-scale processing of signals and data. This includes learning from large collections of signals (sounds, images) and data sets, developing new mathematical tools to explore them, and designing new algorithms to make quantitative assessments and models for large data sets. Ideas from probability, graph theory, harmonic analysis, approximation theory, geometry, topology, computation, among others, all play fundamental roles in these highly interdisciplinary collaborations.

## Faculty

#### William K. Allard

###### Professor Emeritus of Mathematics

#### Paul L Bendich

###### Assistant Research Professor in the Department of Mathematics

**Keywords in this area**

topological data analysis, machine learning, applied topology, data science

#### Robert Calderbank

###### Charles S. Sydnor Professor of Computer Science

**Keywords in this area**

error-correcting codes, wireless communication, data storage, discrete harmonic analysis, algorithms, data compression, source classification

#### Justin Curry

###### Visiting Assistant Professor of Mathematics

**Keywords in this area**

applied algebraic topology, data analysis

#### Ingrid Daubechies

###### James B. Duke Professor of Mathematics and Electrical and Computer Engineering

**Keywords in this area**

wavelets, time-frequency analysis, art conservation

#### Tingran Gao

###### Visiting Assistant Professor in the Department of Mathematics

**Keywords in this area**

Diffusion Maps, Statistical Learning, Geometry Processing, Image Processing

#### John Harer

###### Professor of Mathematics

**Keywords in this area**

Topological Data Analysis,
Geometric Data Analysis

#### Jianfeng Lu

###### Associate Professor of Mathematics

**Keywords in this area**

time-frequency analysis, fast algorithms, optimization, applied harmonic analysis

#### Mauro Maggioni

###### Research Professor of Mathematics

#### Sayan Mukherjee

###### Professor in the Department of Statistical Science

**Keywords in this area**

machine learning, Bayesian statistics

#### Arlie O. Petters

###### Benjamin Powell Professor of Mathematics in Trinity College of Arts and Sciences

**Keywords in this area**

stochastic lensing, mathematical finance

#### Haizhao Yang

###### Visiting Assistant Professor in the Department of Mathematics

**Keywords in this area**

Applied and computational harmonic analysis, high performance optimization, modeling and analyzing data in geophysics, materials science, and art investigation