Localized Regularization of the Discretized Fredholm Integral Equation of the First Kind with the Laplace Kernel Using the Balancing Principle

Myelin forms a lipid-rich insulating layer around nerve axons, crucial for signal transmission in the central nervous system. Its degradation is a hallmark of neurodegenerative diseases, including Alzheimer's disease. Magnetic resonance relaxometry (MRR) can be used to map myelin distribution by measuring the short transverse relaxation time (T₂) of relatively immobile water trapped within myelin sheaths with a multi-echo MRI sequence. Recovering the T₂ distribution function (DF) involves solving an ill-posed inverse problem defined by the Fredholm integral equation of the first kind with a Laplace kernel, typically using Tikhonov regularization (TR) with a single global regularization parameter λ. For DFs with diverse features, this approach often leads to recoveries with over-smoothing in some areas and insufficient regularization in others. To improve this, we introduce localized regularization (LocReg), which determines a vector-valued λ that varies with T₂ across the DF. By adapting to local features, LocReg improves the reconstruction of the desired DF. In simulations, LocReg outperforms conventional single- λ TR methods in problematic regions of parameter space and can sometimes even surpass oracle-based TR. We initially apply LocReg for myelin mapping in the brain.