scale space implementation について

However, for implementation, this definition is impractical, since it is continuous. When applying the scale space concept to a discrete signal ''f_D'', different approaches can be taken. This article is a brief summary of some of the most frequently used methods.
==Separability==
Using the ''separability property'' of the Gaussian kernel
:

g_N \left (x_1,\dots, x_N, t \right) = G\left(x_1, t \right) \cdots G\left(x_N, t\right)

the ''N''-dimensional convolution operation can be decomposed into a set of separable smoothing steps with a one-dimensional Gaussian kernel ''G'' along each dimension
:

L(x_1, \cdots, x_N, t) = \int_^ \cdots \int_^ f_C(x_1-u_1, \cdots, x_N-u_N, t) G(u_1, t) \, du_1 \cdots G(u_N, t) \, du_N,

where
:

G(x, t) = \frac }

and the standard deviation of the Gaussian σ is related to the scale parameter ''t'' according to ''t'' = σ².
Separability will be assumed in all that follows, even when the kernel is not exactly Gaussian, since separation of the dimensions is the most practical way to implement multidimensional smoothing, especially at larger scales. Therefore, the rest of the article focuses on the one-dimensional case.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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