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Quantitative MRI methods that estimate multiple physical parameters simultaneously often require the fitting of a computational complex signal model defined through the Bloch equations. Repeated Bloch simulations can be avoided by matching the measured signal with a precomputed signal dictionary on a discrete parameter grid, as used in MR Fingerprinting. However, accurate estimation requires discretizing each parameter with a high resolution and consequently high computational and memory costs for dictionary generation, storage, and matching. Here, we reduce the required parameter resolution by approximating the signal between grid points through B-spline interpolation. The interpolant and its gradient are evaluated efficiently which enables a least-squares fitting method for parameter mapping. The resolution of each parameter was minimized while obtaining a user-specified interpolation accuracy. The method was evaluated by phantom and in-vivo experiments using fully-sampled and undersampled unbalanced (FISP) MR fingerprinting acquisitions. Bloch simulations incorporated relaxation effects ($T_1,T_2$), proton density ($PD$), receiver phase ($\phi_0$), transmit field inhomogeneity ($B_1^+$), and slice profile. Parameter maps were compared with those obtained from dictionary matching, where the parameter resolution was chosen to obtain similar signal (interpolation) accuracy. For both the phantom and the in-vivo acquisition, the proposed method approximated the parameter maps obtained through dictionary matching while reducing the parameter resolution in each dimension ($T_1,T_2,B_1^+$) by, on average, an order of magnitude. In effect, the applied dictionary was reduced from 1.47 GB to 464 KB. Dictionary fitting with B-spline interpolation reduces the computational and memory costs of dictionary-based methods and is therefore a promising method for multi-parametric mapping., Comment: 14 pages, 9 figures, accepted for publication in IEEE Transaction on Medical Imaging (TMI)

Membership functions are a key concept in fuzzy set theory and their correctness and precision are essential for the accuracy of obtained results. This article discusses the use of Bezier curve to construct a membership function. Based on the frequency distribution of data by minimization, the coordinate formulas of the control points that define the curve are derived. Use of the described membership function is illustrated by an example. These formulas are applied to bispectral index data sets in order to compare with other published method.