Scale-Space Theory

The scale-space axioms (causality, homogeneity, rotational symmetry, scale invariance) are mathematical requirements that any scale-space representation should satisfy. Lindeberg proved these axioms uniquely determine Gaussian blur as the only linear scale-space kernel. This result is profound: it says that if you want a well-behaved multi-scale representation with desirable properties, you must use Gaussian blurring—it's not just one choice among many.

A scale-space is a continuous (or fine-grained) representation where each scale corresponds to Gaussian blur at a different sigma. An image pyramid is a discrete sequence of images at downsampled resolutions. Pyramids are more memory-efficient but lose information through downsampling. Scale-space retains all information at the original resolution at each scale, making it more suitable for precise localization.

Scales should span the range of feature sizes in the image. A common approach is geometric progression: σ_i = σ₀ * r^i, where r is typically 1.2–1.4. For objects with unknown sizes, a wide scale range (multiple octaves) is used. For objects of known size, scales can be concentrated around the expected size, saving computation.

A scale-space extremum is a point where a feature detector (e.g., Laplacian) is locally extreme both in space (x, y) and scale (σ). Such extrema often correspond to features at their natural or characteristic scale. For example, a small blob is detected as an extremum at a scale matching its radius, and a large blob at a larger scale. This automatic scale selection is a key strength of scale-space analysis.