Resolution and its Enhancement in Imaging

We present a definition and analysis of the concept of resolution and its enhancement in imaging based on statistical detection and estimation. We also present an overview of the problem of Super-resolution in imaging, which (similar to notions in MIMO communications) involves the reconstruction of high resolution images from a collection of ”diverse” views of the same scene captured either in the presence of relative motion between the sensor and the scene (synthetic aperture), or with multiple nearby cameras (real aperture). c ©Optical Society of America OCIS Codes:100.0100 (Image Processing); 100.6640 (Superresolution) 1 Statistical Analysis of Resolution We know that the optics (lens, aperture, etc.) of an imaging system limit the amount of information – more specifically spatial frequency – that is received by the imager (CCD or CMOS device). Furthermore, the spatial sampling of the image at the focal plane by the imager also involves loss of information, particularly if it results in aliasing. While the above observation is nothing new, the same notions when discussed in the context of noisy measurements are less trivial. In particular, one may ask the question ”What is the highest resolution one can measure with a camera in the presence of noise?” The popular measures provided to the public in terms of so many mega-pixels may sound wonderful in marketing copy, but are not very informative when we try to quantify the performance of these sensors in the presence of noise. But what does resolution really mean? To address this question, we take the canonical case of a simple ”pinhole” camera in 1-D, composed of a slit and a sampler. Reducing the problem to its basic essence, one can define resolution as the ability of the imaging system (or more generally an algorithm acting on an image) to distinguish the presence of two nearby point sources in the presence of noise. In particular, we can define two hypotheses – namely, whether one point source is being observed, or two, where the distance between the point sources is an unknown parameter d. The profile of two point sources imaged through such a system is an incoherent sum of two sinc functions. With proper normalization, in the classical view of resolution according to the Rayleigh limit, the point sources would not be ”resolvable” if d < 1. This rule of thumb does not define the resolution limit of the imaging system. Instead, as we describe below, it is the SNR and the number of samples acquired by the sensor that dictate its ability to resolve. To be more mathematically precise, we can write the ideal scene comprised of two point sources as: α δ(x− d 2 ) + β δ(x+ d 2 ), (1) where in general the distance d, and the amplitudes α and β are unknown. The measured signal is composed of discrete samples of the response of the imaging system to this input (measured at positions xk for k = 1, · · · , N on the image plane), and corrupted with additive (readout) noise. That is to say, the measurement model can be described as: