Scale Space - Definition

Definition

The notion of scale space applies to signals of arbitrary numbers of variables. The most common case in the literature applies to two-dimensional images, which is what is presented here. For a given image, its linear (Gaussian) scale-space representation is a family of derived signals defined by the convolution of with the Gaussian kernel

such that

where the semicolon in the argument of implies that the convolution is performed only over the variables, while the scale parameter after the semicolon just indicates which scale level is being defined. This definition of works for a continuum of scales, but typically only a finite discrete set of levels in the scale-space representation would be actually considered.

is the variance of the Gaussian filter and as a limit for the filter becomes an impulse function such that that is, the scale-space representation at scale level is the image itself. As increases, is the result of smoothing with a larger and larger filter, thereby removing more and more of the details which the image contains. Since the standard deviation of the filter is, details which are significantly smaller than this value are to a large extent removed from the image at scale parameter, see the following figure and for graphical illustrations.

• Scale-space representation at scale, corresponding to the original image

• Scale-space representation at scale

• Scale-space representation at scale

• Scale-space representation at scale

• Scale-space representation at scale

• Scale-space representation at scale