Scalespace

Scalespace, or scale-space, is a framework for representing signals and images across a continuum of spatial or temporal scales. In image processing, it yields a family of representations {L(x, y; t)} obtained by smoothing the input image with Gaussian kernels at increasing scales. This produces progressively coarser versions of the image that preserve essential structures while removing finer details.

Mathematically, the smoothing at scale t is expressed as L(x, y; t) = G(x, y; t) * f(x, y), where G is a Gaussian kernel with variance t and f is the original image. The parameter t controls the degree of smoothing: small t preserves detail, large t removes fine structure. The Gaussian kernel is connected to the heat equation, with L evolving as ∂L/∂t = (1/2)∇^2L, making the Gaussian the fundamental solution. The smoothing operation is associative, so L(t1 + t2) = G(t1) * L(t2).

A key property of scale-space representations is that increasing the scale tends to remove fine details without introducing new structures; in particular, coarse scales do not create new local extrema. This causality underpins many multi-scale analysis techniques and justifies using derivatives of L with respect to space and scale for feature detection.

Applications include multi-scale feature detection and blob localization (via Laplacian of Gaussian or Difference of Gaussians), edge and corner detection, texture analysis, and motion or video analysis with spatio-temporal scale-space. Extensions cover spatio-temporal scale-space for video and the broader scale-space axioms developed by researchers such as Koenderink and Lindeberg.