Sparse signals are commonly expected in remote sensing and Earth observation. E.g. radar images have much less information content than the acquired raw data samples pretend. Another example is hyperspectral unmixing where only few materials are expected in a pixel compared to the prodigious endmember library. Along with the significant development of the compressive sensing theory, exploitation of sparsity in remote sensing became a very relevant and active field. Breakthroughs are brought in different remote sensing problems covering synthetic aperture radar, multispectral and hyperspectral image analysis, LiDAR and cross cutting data fusion. This talk gives a review on recent advances in sparsity exploitation in remote sensing and Earth observation, regarding the theory, applications and future trends. In particular, synthetic aperture radar tomography, image fusion and hyperspectral unmixing will be presented as highlight examples.

Can we see through diffusers? Can we remove reflections when photographing through windows? Can we perform low cost bio-imaging with with game consoles (such as Microsoft Kinect)? Can we infer the geometry of blood cells from pulses of light? In the traditional sense, any imaging system produces a two-dimensional photograph. This is done by accumulating photons over time and in the process, time information is lost. On the contrary, this talk explores the idea: How can we exploit the time dimension in imaging? By considering the speed of light to be finite, the information in time-delays or echoes of light, resulting from the interaction of light and the scene, can be harnessed in unconventional ways. For example, complex, multiple scattering is often ignored or mitigated in literature, leading to applications that work with simplified environments. However, with time-resolved information at hand, one can exploit temporal features of the scene to infer scattering information. In this way, time-resolved imaging fundamentally combines time-stamped photos with computational methods to redefine a "camera." Thus, allowing one to go beyond the conventional barriers in imaging. Of course, observing physical phenomena at the speed of light requires exorbitant sampling rates. However, by exploiting structural (scene) information and using tools from harmonic analysis and sampling theory, we present case studies where a co-design of hardware and mathematical algorithms achieves state-of-art performance in applications linked with different sub-bands of the electromagnetic spectrum. This co-design philosophy also leads to new sensing paradigms. We discuss an example, the Unlimited Sensing Framework, which allows for high-dynamic-range sensing from low-dynamic-range measurements.

Blind deconvolution problems are ubiquitous in many areas of imaging and technology and have been the object of study for several decades. Recently, motivated by the theory of compressed sensing, a new viewpoint has been introduced, motivated by applications in wireless application, where a signal is transmitted through an unknown channel. Namely, the idea is to randomly embed the signal into a higher dimensional space before transmission. Due to the resulting redundancy, one can hope to recover both the signal and the channel parameters. In this talk we give an overview over recent progress on recovery guarantees for this problem. On the one hand, we will discuss convex approaches based on lifting as they have first been studied by Ahmed et al. (2014). We show that one encounters a fundamentally different geometric behavior as compared to generic bilinear measurements. On the other hand we will review recent progress on the study of efficient nonconvex recovery methods and present a nonconvex approach with provable local convergence guarantees under sparsity assumptions. This talk is based on joint works with Jakob Geppert (University of Göttingen), Peter Jung (TU Berlin), Kiryung Lee (Ohio State University), Justin Romberg (GeorgiaTech), and Dominik Stöger (TUM).

The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. The Nyquist theorem also results in a large number of elements in antenna arrays and in wide bandwidths in applications requiring high resolution. In this talk we consider a general framework for sub-Nyquist radar in space, time and frequency which allows to dramatically reduce the number of antenna elements, sampling rates and band occupancy. Sub-Nyquist radars break the link between common radar design trade-offs such as range resolution and transmit bandwidth; dwell time and Doppler resolution; spatial resolution and number of antenna elements; continuous-wave radar sweep time and range resolution. We then show that they also pave the way for cognitive radars which share their transmit spectrum with other communication services, thereby providing a robust solution for coexistence in spectrally crowded environments. Finally, we present state-of-the-art hardware prototypes that demonstrate the real-time feasibility of sub-Nyquist radars.