A Novel Bound for Fourier Ring Correlation in Resolution Analysis

This work proposes a novel data-driven approach for computing the resolution number of a given image using the well-established Fourier Ring/Shell Correlation (<sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">frc</small>/<sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fsc</small>) technique. The proposed method eliminates the need for the user to select a threshold criterion in a heuristic way, a requirement in current methodologies. To achieve this, the approach leverages linear algebra-specifically, Niculescu's result-and concepts from information theory, demonstrating that the resolution number is directly linked to the Fisher information derived from each ring or shell in the Fourier domain. As a result, the methodology is entirely data-driven, requiring no prior information about the image under analysis. The mathematical framework's consistency is validated through numerical experiments and tests with real data from x-ray coherent microscopy, tomography, cryo-EM and confocal microscopy, showing that the newly computed resolution numbers align with conventional metrics derived from the classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1/2$</tex-math></inline-formula>-bit and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3\sigma$</tex-math></inline-formula> thresholds. Furthermore, we highlight that the local resolution of images can vary significantly from the single resolution value typically provided by <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">frc</small>/<sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fsc</small>. This observation suggests that resolution maps may provide a more reliable framework for assessing resolution in microscopy.