Limits of CNR and MSE in medical image quality

Contrast-to-noise ratio (CNR) and mean-squared error (MSE) are often treated as compact surrogates for image quality. The purpose of this tutorial is to revisit why these scalar summaries are incomplete for detection and decision tasks unless the task, ensemble, and observer that make them meaningful are stated.

We use reproducible numerical demonstrations, beginning with simple examples where CNR agrees with visual intuition and then showing how CNR fails when signal size changes. The Rose model is used as an intermediary between CNR and task-based assessment because it includes contrast, noise, and signal area while explicitly stating its assumptions. We then discuss failures caused by background variability, signal variability, noise correlation, and reconstruction-dependent texture and relate these failures to observer-based figures of merit.

CNR agrees with detectability only under restrictive conditions: a specified detection task, fixed known signal support, known background, effectively white noise, and an observer matched to the relevant image features. The Rose model corrects the most immediate omission by including signal area, and under its assumptions, it is ideal-observer-like. When signal extent, spatial frequency content, noise correlation, anatomical variability, location uncertainty, nonlinear processing, or distortion type changes, images can have equal CNR or equal MSE and substantially different detectability.

CNR and MSE are useful descriptive statistics, but they are not general image quality figures of merit in the Barrett sense unless the task, ensemble, and observer that make them meaningful are stated. In particular, low-contrast detectability should not be used as a synonym for CNR unless the detection task and observer model that make the equivalence valid have been specified. For medical imaging optimization, CNR and MSE should be interpreted alongside, or replaced by, task-based measures such as ideal observer, Hotelling observer, channelized Hotelling observer, receiver operating characteristic, or estimation-task performance.