Differential Geometric Definition of Ridges and Valleys At A Fixed Scale in A Two-dimensional Image
Let denote a two-dimensional function, and let be the scale-space representation of obtained by convolving with a Gaussian function
- .
Furthermore, let and denote the eigenvalues of the Hessian matrix
of the scale-space representation . With a coordinate transformation (a rotation) applied to local directional derivative operators,
where p and q are coordinates of the rotated coordinate system.
It can be shown that the mixed derivative in the transformed coordinate system is zero if we choose
- ,.
Then, a formal differential geometric definition of the ridges of at a fixed scale can be expressed as the set of points that satisfy
Correspondingly, the valleys of at scale are the set of points
In terms of a coordinate system with the direction parallel to the image gradient
where
it can be shown that this ridge and valley definition can instead be equivalently be written as
where
and the sign of determines the polarity; for ridges and for valleys.
Read more about this topic: Ridge Detection
Famous quotes containing the words fixed, scale, differential, valleys, definition, geometric and/or image:
“Our live experiences, fixed in aphorisms, stiffen into cold epigrams. Our heart’s blood, as we write it, turns to mere dull ink.”
—F.H. (Francis Herbert)
“The poet is the person in whom these powers are in balance, the man without impediment, who sees and handles that which others dream of, traverses the whole scale of experience, and is representative of man, in virtue of being the largest power to receive and to impart.”
—Ralph Waldo Emerson (1803–1882)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“I do like a little romance ... just a sniff, as I call it, of the rocks and valleys.... Of course, bread-and-cheese is the real thing. The rocks and valleys are no good at all, if you haven’t got that.”
—Anthony Trollope (1815–1882)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)
“As death, when we come to consider it closely, is the true goal of our existence, I have formed during the last few years such close relations with this best and truest friend of mankind, that his image is not only no longer terrifying to me, but is indeed very soothing and consoling! And I thank my God for graciously granting me the opportunity ... of learning that death is the key which unlocks the door to our true happiness.”
—Wolfgang Amadeus Mozart (1756–1791)