With the objective of generating ``shape-preserving'' smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based ${\cal C}^1$-smooth univariate cubic $L_1$ splines. An $L_1$ spline minimizes the $L_1$ norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an $L_1$ spline is a nonsmooth nonlinear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.
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View Geometric Dual Formulation for First-derivative-based Univariate Cubic $ Splines