SHORT COMMUNICATION A New Methodology to Extend the Canonical Piecewise-Linear Model from One to Two Dimensions V. M. Jimenez-Fernandez 1 • M. Jimenez-Fernandez 2 • H. Vazquez-Leal 1 • U. A. Filobello-Nino 1 • C. H. Castan ˜eda-Roldan 3 • V. M. Tlapa-Carrera 3 Received: 25 October 2018 / Revised: 27 February 2020 / Accepted: 7 March 2020 Ó The National Academy of Sciences, India 2020 Abstract In this paper, it is shown that an extension of the formulas used to calculate the parameters of the one-di- mensional canonical piecewise-linear model leads to a generalized methodology which can be also applicable to two dimensions. The proposal is based on an approach by which a two-dimensional function is reconstructed by the collection of parallel projections of one-dimensional functions. In order to verify the performance of this pro- posal, numerical simulations are contrasted to those obtained from the standard high-level canonical piecewise- linear methodology. The comparison reveals that our pro- posal allows the construction of the function not only requiring a shorter run time but also demanding less memory. Keywords Canonical Á Piecewise-linear Á Function Á Two-dimensional Mathematics Subject Classification 65D10 Á 65D15 Á 65D05 One of the most popular piecewise-linear models is the so- called canonical piecewise-linear (CPWL) which is defined in its more general form by the following theorem [1–3]: Theorem 1 Any single-valued n-dimensional piecewise- linear function f ðxÞ with r breakpoints can be represented by the following explicit expression f ðxÞ¼ a þ Bx þ X r i¼1 c i a i ; x h iÀ b i j j ð1Þ where a, x, c i , are n-dimensional vectors, B is an ðn  nÞ matrix , b i is a scalar, and ‘‘ ; hi’’ denotes the inner product of two vectors. Although (1) is an n-dimensional representation, it has been mainly exploited in its one-dimensional form. In fact, in the literature it can be found analytical expressions to compute its model parameters. Such expressions are sum- marized in Theorem 2 [4]. Theorem 2 Any one-dimensional piecewise-linear func- tion with r breakpoints (b i , for i ¼ 1; 2; ...r) can be rep- resented by f ðxÞ¼ a þ bx þ X r i¼1 c i x À b i j j ð2Þ where a; b; c i are real numbers that can be calculated as follows: b ¼ J ð1Þ þ J ðrþ1Þ 2 ; ð3Þ c i ¼ J ðiþ1Þ À J ðiÞ 2 ; ð4Þ a ¼ f ð0ÞÀ X r i¼1 c i jb i j ð5Þ & V. M. Jimenez-Fernandez vicjimenez@uv.mx 1 Facultad de Instrumentacio ´n Electro ´nica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltra ´n s/n, zona universitaria, P.O. 91000, Xalapa, Veracruz, Mexico 2 Instituto de Ciencias Ba ´sicas, Universidad Veracruzana, Av. Dr. Luis Castelazo, Industrial Las Animas, P.O. 91190, Xalapa, Veracruz, Mexico 3 Instituto de Fı ´sica y Matema ´ticas, Universidad Tecnolo ´gica de la Mixteca, Carretera Acatlima Km. 2.5, P.O. 69000, Huajuapan de Leo ´n, Oaxaca, Mexico 123 Natl. Acad. Sci. Lett. https://doi.org/10.1007/s40009-020-00970-8