## 1 Introduction  
\(L_{1}\) splines have been shown to provide superior shape preservation for interpolation and approximation of multiscale data, that is, data with sudden large changes in magnitude and/or spacing used for modeling of natural and urban terrain, geophysical features, biological objects, robotic paths and many other irregular surfaces, processes and functions ([1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]). The minimization principles for \(L_{1}\) splines have typically (but not uniformly) been based on the \(L_{1}\) norm of the second derivative (rather than on the \(L_{1}\) norm of expressions involving other levels of derivatives). The resulting non-differentiable convex generalized geometric programs have been solved by active set [3], primal affine [5; 6; 7; 10; 11] and primal-dual [12; 14] algorithms. In the literature, there are a few indications of limitations of the primal affine and primal-dual algorithms for bivariate \(L_{1}\) splines for large data sets [9; 12]. There is also unpublished computational experience of the authors and others who have noticed issues of incomplete convergence or not completely correct convergence of the active set, primal affine and primal-dual algorithms for both univariate and bivariate \(L_{1}\) splines. It is in this context that we seek in this paper a new algorithmic approach for calculating \(L_{1}\) splines. [PERSON], [PERSON] and [PERSON] [16] have developed a subdifferential-based procedure for calculating second-derivative-based \(L_{1}\) splines on 5-point windows. We propose here an algorithm for second-derivative-based 5-point-window \(L_{1}\) splines based on the analysis in Section 2 of [17], which links, via analytical properties of the spline functional, local geometric properties of 5-point windows of the data set with geometric properties of the \(L_{1}\) spline interpolant. In considering second-derivative-based 5-point-window \(L_{1}\) splines, three types of information are needed for full \"situational awareness,\" namely, 1) information about whether use of 5-point windows is superior to use of windows of other sizes and to use of global data sets, 2) information about whether use of the second derivative in the spline functional is superior to use of the first derivative, function value or antiderivative and 3) information about whether the new, analysis-based algorithm mentioned above can achieve computational results superior to those of the primal affine algorithm, which was the previously most widely used algorithm for calculating \(L_{1}\) splines. The third item here has not yet been considered in the literature and the information in the literature on the first and second items is sketchy at best. The computational and analytical results in [2; 16; 17] do suggest that 5-point windows have advantages vs. global calculations. However, it is not yet known whether windows of other sizes might also have advantages. Results presented in [10; 11] indicate that first-derivative-based and function-value-based \(L_{1}\) splines may have advantages over standard second-derivative-based \(L_{1}\) splines. However, these two publications considered only the behavior of \(L_{1}\) splines on the global scale and did not consider behavior on the fine, interval-to-interval scale. The present paper addresses these needs. In Section 2, we give a brief description of the primal affine algorithm that has previously been widely used to calculate \(L_{1}\) splines. In Section 3, we compare \(L_{1}\) splines calculated by minimizing, on 5-point windows, on 7-point windows and on global data sets, four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. These \(L_{1}\) splines are calculated by the primal affine algorithm. The results of this section provide motivation for the development of a new, analysis-based algorithm for calculating 5-point-window, second-derivative-based \(L_{1}\) splines.