How Can a Quadratic Formula Reconstruct Functions with Sudden Jumps?

Imagine you need to rebuild a curve that jumps abruptly, like a stepped wave, using only a few numbers. A recent paper suggests a trick: instead of approximating the function f(x) directly, we build three variable coefficients a(x), b(x) and c(x) that depend on x. With them we form the quadratic equation a(x)·f² – b(x)·f – c(x)=0. Solving it with the quadratic formula gives f(x). The key is picking the right sign in the root, guided by an “index function” that tells which of the two solutions is correct.
The coefficients are obtained with a simple least‑squares method, so they are smooth and easy to compute. The result? A degree‑2 representation in f that outperforms classic orthogonal‑polynomial approximations (degree 0) and rational approximations (degree 1), especially when the function has sharp transitions or discontinuities.
A practical use is data denoising. Instead of costly “edge‑preserving” algorithms, one can denoise the index function with statistical or clustering techniques, keeping the signal’s shape on the algebraic variety defined by the quadratic equation. Tests show the method cleans data while preserving sharp edges.
The work also raises theoretical questions: how to ensure numerical stability, which algorithms are most efficient for computing a(x), b(x), c(x), and how to compactly represent the index function. In short, a quadratic formula becomes a fresh lens for viewing and cleaning complex signals.

The proposed approach relies on an algebraic parametrisation of the target function f(x) via a variety defined by the equation a(x)f^2 - b(x)f - c(x)=0. The coefficients a(x), b(x) and c(x) are polynomials of chosen degree (typically ≤ 3) obtained by minimising \(\sum_{i=1}^N (a(x_i)f_i^2 - b(x_i)f_i - c(x_i))^2\) over a discrete sample \{(x_i,f_i)\}. The quadratic solution yields two roots \(f_{\pm}(x)=\frac{b(x)\pm\sqrt{b(x)^2+4a(x)c(x)}}{2a(x)}\). The index function I(x)∈\{+,-\} selects the physically consistent root, e.g., preserving continuity or satisfying boundary conditions.
Numerical stability hinges on evaluating the discriminant \(Δ(x)=b(x)^2+4a(x)c(x)\). In regions where a(x)≈0 the quadratic formula degenerates to a linear form, requiring regularisation or local reformulations. Ridge regression or gradient penalties on a, b, c can damp oscillations.
For denoising, I(x) can be treated as a discrete signal on \([-1,1]\) and processed with Markov Random Field filters or clustering methods (k‑means, Gaussian Mixture Models) to remove outliers. Since a, b, c are smooth, projecting noisy data onto the manifold defined by the equation reduces the mean‑square error compared with TV‑based methods.
Algorithmically, coefficient construction can exploit SVD decompositions to solve the over‑determined linear system, guaranteeing a minimum‑norm solution. Global optimisation may include an annealing phase for I(x), minimising \(\sum_i \|f_i - f_{I(x_i)}(x_i)\|^2\).
Finally, the index function can be compressed via entropy coding or lightweight neural networks, balancing bitrate and reconstruction accuracy. This opens the way to efficient nonlinear representations for signals with sharp edges, with potential extensions to higher dimensions and inverse physical modelling problems.