Scale-space axioms

Scale space
Scale-space axioms
Scale-space implementation
Feature detection
Edge detection
Blob detection
Corner detection
Ridge detection
Interest point detection
Scale selection
Affine shape adaptation
Scale-space segmentation
Axiomatic theory of receptive fields

In image processing and computer vision, a scale space framework can be used to represent an image as a family of gradually smoothed images. This framework is very general and a variety of scale space representations exist. A typical approach for choosing a particular type of scale space representation is to establish a set of scale-space axioms, describing basic properties of the desired scale-space representation and often chosen so as to make the representation useful in practical applications. Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters.

A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision.

The linear scale space representation ${\displaystyle L(x,y,t)=(T_{t}f)(x,y)=g(x,y,t)*f(x,y)}$ of signal ${\displaystyle f(x,y)}$ obtained by smoothing with the Gaussian kernel ${\displaystyle g(x,y,t)}$ satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation:

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