Interpolationskernel

Interpolationskernel, or interpolation kernel, is a function used to estimate a continuous-valued signal or function from a set of discrete samples. It specifies how surrounding samples contribute to the interpolated value at a given location, typically through a weighted sum or convolution. In practice, the kernel determines the accuracy, smoothness, and local support of the interpolation.

Mathematically, in one dimension, if samples f[n] are known at integer positions n, the interpolated value at x is often written as f_interp(x) = sum_n f[n] φ(x − n), where φ is the interpolation kernel. Commonly, kernels are chosen to be normalized (so the interpolation preserves average level), symmetric, and with a chosen support size to balance accuracy and efficiency. In multi-dimensional settings, kernels can be separable, expressed as a product of one-dimensional kernels for each coordinate.

Several standard kernels are used in practice. The boxcar or nearest-neighbor kernel yields simple but blocky results. The triangular (linear) kernel provides linear interpolation. Cubic kernels, such as Keys or cardinal B-spline, offer smoother results. The Lanczos family, built from a truncated sinc function, can achieve high accuracy with controllable artifacts. In theory, the ideal sinc kernel enables perfect reconstruction of band-limited signals, but its infinite extent makes it impractical, leading to truncated or modified variants.

Interpolationskernel also appears in kernel-based interpolation methods, where the kernel depends on distance and underpins radial basis function approaches. Applications span image and audio resampling, scientific data interpolation, and numerical methods that require reconstructing a continuous field from discrete samples. The choice of kernel involves trade-offs among accuracy, smoothness, artifact suppression, and computational cost.