Image Processing Reference
In-Depth Information
discussed in the previous section. Accordingly, image addition is achieved by adding
the image values at corresponding points, whereas image scaling is achieved by mul-
tiplying the image values with a scalar, e.g., as given by Eqs. (3.8) and (3.9) for gray
images:
A + B =
{
A ( i, j )+ B ( i, j )
}
(3.8)
α A =
{
α A ( i, j )
}
(3.9)
Exercise 3.1. Is the space of images having the same size but taking the gray values
0 , 1 ,
, 255 a vector space?
Extending the number of indices to three in a discrete array
···
A =
{ A ( k, l, m )
}
(3.10)
has also counterparts in the world of images. The most commonly known example is
discrete color images, where the first two indices of A represent the coordinate point
and the third index defines the color components. In other words, at every spatial
position , which is another common name for the coordinate pair k, l ,wehavean
M -dimensional array of scalars (instead of a single scalar). In the case of ordinary
color images, M is 3, representing the color components, typically in the RGB color
space. There are large banks of discrete image data of the earth where M is > 3; these
are multispectral images . Here each “color” represents a specific photon wavelength
range of the light, some visible, many invisible to humans. Such images with scalar
values of A ( k, l, m ) constitute a vector space with the addition and scaling rules:
A + B =
{ A ( k, l, m )+ B ( k, l, m )
}
(3.11)
α A =
{
α A ( k, l, m )
}
(3.12)
For certain image types, there is also a different interpretation of the third index.
In computer tomography images one also obtains an array of discrete data with three
indices, as discussed above. In this case the indices k, l, m represent horizontal, ver-
tical, and depth coordinates of a spatial point, commonly called a voxel . The scalar
A ( k, l, m ) typically represents absorption.
There is yet another interpretation of the third index for black and white motion
images . The first two indices k, l correspond to the spatial position, whereas the
third index m corresponds to the temporal instant. Accordingly, k, l, m are called
the spatio-temporal coordinates . 1 With the same size and scalar image values these
constitute vector spaces too.
Increasing the number of indices to four brings into focus images that are com-
monly produced by consumer electronics, color motion images . These are also called
image sequences . The first three indices are the spatio-temporal coordinates, whereas
the fourth index encodes the color. Evidently, even these images constitute a vector
space, with the addition and scaling rules analogous to Eqs. (3.5)-(3.6).
The list of discrete image types can be made longer, but we stop here to proceed
with an example illustrating what one can do with vector spaces.
1 A suitable name for these coordinates would be stixels , making allusion to the spatio-
temporal nature of the data.
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