Compressed sensing for practical optical imaging systems: a tutorial

Abstract

Abstract. The emerging field of compressed sensing has potentially powerful implications for the design of optical imaging devices. In particular, compressed sensing theory suggests that one can recover a scene at a higher resolution than is dictated by the pitch of the focal plane array. This rather remarkable result comes with some important caveats however, especially when practical issues associated with physical implementation are taken into account. This tutorial discusses compressed sensing in the context of optical imaging devices, emphasizing the practical hurdles related to building such devices, and offering suggestions for overcoming these hurdles. Examples and analysis specifically related to infrared imaging highlight the challenges associated with large format focal plane arrays and how these challenges can be mitigated using compressed sensing ideas. C 2011 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/1.3596602] Subject terms: compressed sensing; sampling; image reconstruction; inverse problems; computational imaging; infrared; coded aperture imaging; optimization.Paper 100978TR received Nov. 29, 2010; revised manuscript received May 8, 2011; accepted for publication May 12, 2011; published online Jul. 6, 2011. IntroductionThis tutorial describes new methods and computational imagers for increasing system resolution based on recently developed compressed sensing (CS, also referred to as compressive sampling) 1, 2 techniques. CS is a mathematical framework with several powerful theorems that provide insight into how a high resolution image can be inferred from a relatively small number of measurements using sophisticated computational methods. For example, in theory a 1 mega-pixel array could potentially be used to reconstruct a 4 mega-pixel image by projecting the desired high resolution image onto a set of low resolution measurements (via spatial light modulators, for instance) and then recovering the 4 mega-pixel scene through sparse signal reconstruction software. However, it is not immediately clear how to build a practical system that incorporates these theoretical concepts. This paper provides a tutorial on CS for optical engineers which focuses on 1. a brief overview of the main theoretical tenets of CS, 2. physical systems designed with CS theory in mind and the various tradeoffs associated with these systems, and 3. an overview of the state-of-the-art in sparse reconstruction algorithms used for CS image formation. There are several other tutorials on CS available in the literature which we highly recommend; 3-7 however, these papers do not address important technical issues related to optical systems, including a discussion of the tradeoffs associated with non-negativity, photon noise, and the practicality of implementation in real imaging systems.Although the CS theory and methods we describe in this paper can be applied to many general imaging systems, we concentrate on infrared (IR) technology as a specific example to highlight the challenges associated...

Sparsity based denoising of spectral domain optical coherence tomography images

“…The basic idea of compressive sensing is that when for a class of images the majority of elements in vector α corresponding to a particular dictionary are zeros or very small (sparsity), relatively few well-chosen measurements suffice to reconstruct the images in this class [21]. …”

“…The rest of the B-scans are captured fast at the nominal SNR. Utilizing compressive sensing principles [15–21], we learn a sparse representation dictionary for each of these high-SNR images and utilize these dictionaries to denoise the neighboring low-SNR B-scans. The rationale for this approach is that neighboring B-scans, in common SDOCT volumes, are expected to have similar texture and noise pattern, as illustrated in Fig.…”

Temporal derivatives in compressed video sensing

“…The goal is to reduce such burdens by directly observing compressed samples that are collected independently from each frame without a compression protocol [8], [3]. However, this is an ill-posed problem, since the full samples of a frame are not available for reconstruction.…”

“…The samples are provided individually from each image sequence (frame) and hence the measurements are utilized in two-dimensional tensor analysis. Since the consecutive frames are considered to be correlated with each other, such correlation is encoded by frame difference approach [8], [9], [10], [11], [12], [3], [13]. The benefit of deploying such methods is to avoid computational complexities, where the analysis is done in 2D tensor calculations.…”

“…Such a continuity can be addressed in similarities between consecutive frames and hence the frames are correlated [2]. So, the video data can be interpreted as a sequence of correlated images, which corresponds to a particular scene movement [3].…”