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We present a fundamentally new regularization method for the solution of theFredholm integral equation of the first kind, in which we incorporate solutionscorresponding to a range of Tikhonov regularizers into the end result. Thismethod identifies solutions within a much larger function space, spanned bythis set of regularized solutions, than is available to conventionalregularizaton methods. Each of these solutions is regularized to a differentextent. In effect, we combine the stability of solutions with greater degreesof regularization with the resolution of those that are less regularized. Incontrast, current methods involve selection of a single, or in some casesseveral, regularization parameters that define an optimal degree ofregularization. Because the identified solution is within the span of a set ofdifferently-regularized solutions, we call this method \textit{span ofregularizations}, or SpanReg. We demonstrate the performance of SpanReg througha non-negative least squares analysis employing a Gaussian basis, anddemonstrate the improved recovery of bimodal Gaussian distribution functions ascompared to conventional methods. We also demonstrate that this method exhibitsdecreased dependence of the end result on the optimality of regularizationparameter selection. We further illustrate the method with an application tomyelin water fraction mapping in the human brain from experimental magneticresonance imaging relaxometry data. We expect SpanReg to be widely applicableas an effective new method for regularization of inverse problems.

Solution of Non-negative Least Squares Inverse Problems Using a Span of Regularized Solutions, with Application to Magnetic Resonance Relaxometry