Visualization of nonlinearity in image reconstruction using nonlocal means filters for regularization under noisy conditions.

The nonlocal means (NLM) filter is inherently nonlinear because its weighting coefficients depend on image intensities; however, how this nonlinearity manifests when NLM is used as a regularization term in image reconstruction has not been fully clarified. This study aimed to visualize and characterize the influence of NLM-induced nonlinearity on reconstructed images through numerical simulations. We first mathematically demonstrated-using reductio ad absurdum and direct proof-that NLM is a nonlinear filter. A two-dimensional disk source, and the BrainWeb dataset were used as test objects. Ideal parallel-beam projection data and projection data containing Poisson noise were used. Image reconstruction was performed using least squares (LS). NLM-based regularization enforcing the reconstructed image to match the NLM-filtered result (adsorption model). Evaluation metrics included the difference image, the relative L1 norm of the difference image, the profile curve, and the area under the profile curve. LS regularized with NLM exhibited linear approximation behavior in the absence of noise if regularization and NLM parameter carefully chosen. The presence of noise amplified nonlinear effects. These findings indicate that comparing mathematical formula with image-based evaluation metrics is useful for understanding how NLM regularization affects reconstructed images under different input conditions.