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In this work, we propose an algorithm for the detection, measurement and classification of discontinuities in signals captured with noise. Our approach is based on the Harten’s subcell-resolution approximation adapted to the presence of noise. This technique has several advantages over other algorithms. The first is that there is a theory that allows us to ensure that discontinuities will be detected as long as we choose a sufficiently small discretization parameter size. The second is that we can consider different types of discretizations such as point values or cell-averages. In this work, we will consider the latter, as it is better adapted to functions with small oscillations, such as those caused by noise, and also allows us to find not only the discontinuities of the function, jumps in functions or edges in images, but also those of the derivative, corners. This also constitutes an advantage over classical procedures that only focus on jumps or edges. We present an application related to heart rate measurements used in sport as a physical indicator. With our algorithm, we are able to identify the different phases of exercise (rest, activation, effort and recovery) based on heart rate measurements. This information can be used to determine the rotation timing of players during a game, identifying when they are in a rest phase. Moreover, over time, we can obtain information to monitor the athlete’s physical progression based on the slope size between zones. Finally, we should mention that regions where heart rate measurements are abnormal indicate a possible cardiac anomaly.

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating C2 subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties.

This paper is devoted to the construction and analysis of some new families of n-point ternary subdivision schemes. Some members of the families were adapted to the presence of discontinuities converging to limit functions without Gibbs oscillations. We present a numerical comparison where we check the theoretical properties.