Duality between noise and spatial resolution in linear systems

Abstract

It is shown that in a broad class of linear systems, including general linear shift-invariant systems, the spatial resolution and the noise satisfy a duality relationship, resembling the uncertainty principle in quantum mechanics. The product of the spatial resolution and the standard deviation of output noise in such systems represents a type of phase-space volume that is invariant with respect to linear scaling of the point-spread function, and it cannot be made smaller than a certain positive absolute lower limit. A corresponding intrinsic "quality" characteristic is introduced and then evaluated for the cases of some popular imaging systems, including computed tomography, generic image convolution and phase-contrast imaging. It is shown that in the latter case the spatial resolution and the noise can sometimes be decoupled, potentially leading to a substantial increase in the imaging quality.

On the noise-resolution duality, Heisenberg uncertainty and Shannon's information

“…These two characteristics, and the interplay between them, attained additional relevance in recent years in the context of biomedical imaging, where the samples are often sensitive to the radiation dose [15], in certain astronomical methods where the detectable photon flux can be extremely low [10], as well as in some other problems, including those related to the foundations of quantum physics [1]. In x-ray medical imaging, in particular, it is critically important to minimize the radiation dose delivered to the patient, while still being able to obtain 2D or 3D images with sufficient spatial resolution and large C3 enough snr to detect the features of interest, such as small tumours [13,14]. In this context, an imaging system (e.g., a ct scanner) must maximize the amount of relevant information extractable from the collected images, while keeping sufficiently low the number of x-ray photons impinging on the patient.…”

“…Gureyev et al [13,14] recently introduced a dimensionless 'intrinsic quality' characteristic Q S which incorporates both the noise propagation and the spatial resolution properties of a linear shift-invariant (lsi) imaging system:…”

“…Gureyev et al [13,14] assumed that the incident fluence corresponds to a spatially stationary and uncorrelated random process with Poisson statistics, hence snr…”

“…Because Q S is normalised with respect to the incident fluence, it may be viewed as 'imaging quality per single incident particle'. In practice, if the incident fluence rate or the exposure time is increased, then the quality of the resultant image is expected to increase too (normally in proportion to F 1/2 in ) [13,14]. However, in applications where the imaging quanta are at premium (e.g., in low-light imaging) or where the irradiation dose delivered to the sample is critical (as in x-ray or electron imaging of biological samples), Q S represents a key performance indicator of the imaging system.…”

“…where C n = 2 n Γ (n/2)n(n + 2)/(n + 4) n/2+1 is the Epanechnikov constant [9,13,7]. More precisely, de Hoog, Schmalz and Gureyev [7] showed that inequality (2) holds and is exact for lsi systems with point spread functions (psfs) T (x) with finite mathematical expectation, variance and energy.…”