smooths

Smooths are a broad concept used across mathematics, statistics, and computer science to describe processes or objects that reduce high-frequency variation to yield a smoother result. The aim is to suppress noise or irregularities while preserving essential structure. In mathematics, a function is called smooth if it has derivatives of all orders, typically denoted as C∞. Smooth functions include polynomials, exponentials, and trigonometric functions, and they underpin much of analysis, differential equations, and geometry.

In statistics and data analysis, smoothing methods estimate a smooth curve or surface from noisy data. Common approaches include moving averages, kernel smoothers (e.g., Gaussian kernels), and local regression such as LOESS. Smoothing involves a bandwidth or smoothing parameter that balances bias and variance. Applications include time-series analysis, density estimation, and regression.

In image processing and computer graphics, smoothing reduces noise and produces visually pleasing images and meshes. Techniques include Gaussian blur, median filters, and edge-preserving filters; in geometry processing, Laplacian smoothing and subdivision surfaces create smoother models.

Practical considerations: choosing methods and parameters depends on the data and goal. Oversmoothing can erase important features, while undersmoothing leaves noise. Diagnostics and cross-validation or information criteria aid selection.