Definition
The one-dimensional Gaussian filter has an impulse response given by
and the frequency response is given by (eq.206 Fourier transform)
with the ordinary frequency. These equations can also be expressed with the standard deviation as parameter
and the frequency response is given by
By writing as a function of with the two equations for and as a function of with the two equations for it can be shown that the product of the standard deviation and the standard deviation in the frequency domain is given by
- ,
where the standard deviations are expressed in their physical units, e.g. in the case of time and frequency in seconds and Hertz.
In two dimensions, it is the product of two such Gaussians, one per direction:
where x is the distance from the origin in the horizontal axis, y is the distance from the origin in the vertical axis, and σ is the standard deviation of the Gaussian distribution.
Read more about this topic: Gaussian Filter
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)