In this class, we must express map/image transformations and operations in a rigorous fashion, in order to (a) understand their structure and function, (b) predict consequences of changing transform parameters, and (c) have a unified basis for comparing map manipulation algorithms. Currently, the only rigorous, concise notation that unifies image and signal processing (to which GIS is related) is called image algebra. Some of you may have heard of this notation, which was invented here at UF by Gerhard Ritter. Image algebra is a rigorous, concise notation that unifies linear and nonlinear mathematics in the image domain. Image algebra was developed at UF under DARPA and Air Force support since the early 1980s to provide a simple, powerful notation for describing image and signal processing algorithms in a unified, high- level manner.
The following theory is a small subset of image algebra, since the aim of this class is not to teach the details of image algebra. Rather, we are teaching advanced concepts of GIS within the context of image algebra, for the above-mentioned reasons. We thus provide the following brief theoretical introduction.
NOTE: Bold-italic-face titles or names denote cryptologic terms that we will use in class.
Z2 for purposes
of brevity.
Intervals. Occasionally, we use a concise specification of a
contiguous subset of a given set, called an interval. For
example, [1,4]
Z
denotes the set {1,2,3,4}, while the half-open interval
(1,4]Z denotes
the set {2,3,4}. Similarly,
(2,4]R =
{x
R:
2 < x 
4}.
), intersection
(
),
subtraction (\), complement (´), cardinality (|A|),
choice (choice), infimum (inf), and
supremum (sup) are supported.
Unary arithmetic functions are applicable to subsets of the real
numbers (e.g.,
max or 
,
min or 
,
and abs). Note that
abs denotes absolute value of a set instead of |A|,
which is reserved for cardinality.
Example. The mapping a : X -> F
denotes a generalized image
a
FX, which is a concise way of denoting
the set of all mappings from X to F.
FX, then
X = domain(a) and
F = range(a).
FX is denoted by:
G(a) = {(x, a(x)) :
a(x)
F, x
X} .
We also write a
G(a), where
denotes equivalence. Note that
p1(G(a)) = X =
domain(a) and p2(G(a))
= F = range(a), where pk
denotes projection onto the kth coordinate.
F, we denote a constant image
k
{k}X as
k
{(x, a(x)) :
a(x) = k, x
X}
= {(x,k): x
X} .
Example. The unitary image on X is denoted as 1X, or simply 1, with X understood.
Example. We have the following cases of real and Boolean negation:
-a
{(x, -[a(x)]) :
a(x)
R, x
X}
¬a
{(x, ¬[a(x)]) :
a(x)
B, x
X}
FX
to an element of its value set.
BX, then the operation
a =
a(x)
with
would
tell us if there are any unitary values in a. Example. Given the images a,b
FX,
we have the pointwise arithmetic operations
c = a + b
{(x, c(x)) : c(x) =
a(x) + b(x), where
a(x),b(x)
R,
x X}
c = a · b
{(x, c(x)) : c(x) =
a(x) · b(x), where
a(x),b(x)
R,
x X} .
Example. If a and b are Boolean images on X, then we have the following definition of the exclusive-or function:
c = (a xor b)
{(x, c(x)) : c(x) =
|a(x) - b(x)|,
x X} .
Observation. A wide variety of binary operations can be
described (e.g., ab and logab),
provided that value sets are properly defined. However, in the
case of the logarithm, we must set
range(a), range(b)
R+, the positive reals. Otherwise, the logarithm
is not defined.
FX to a subset
W of its domain X, which is denoted by:
b = a|W
{(x, b(x)) : b(x) =
a(x), xWX} .
FW to a k-valued
constant image on X, the domain of a, is denoted by:
c = b | k
{(x, c(x)) : c(x) =
b(x) if x
W or
c(x) = k(x) if x
X\W} .
FX and
A F,
we have the following notation:
b = a ||A
{(x, b(x)) : b(x) =
a(x) if
a(x)AF,
xX} .
FX. Let S denote a property of a's
range space (for example, "A
F", such that
b =
S(a)
{(x, b(x)) : b(x) =
1 if a(x) satisfies S, or b(x) = 0
otherwise}.
Thus, the characteristic function can be thought of as a logical test with predicate S that returns a Boolean image. Wherever the test condition S is satisfied, a unitary value occurs, with zeroes elsewhere.
Example. The characteristic function can be employed in
thresholding an image a
RX using a real threshold T, as follows:
b =
>T(a)
{(x, b(x)) : b(x) =
1 if a(x) > T, or b(x) = 0
otherwise}.
An important class of operations are called transpostions, in which the domain of a plaintext message is manipulated so that the symbol order is rearranged. Let a denote an image in FX, and define a spatial transformation f: Y -> X. For purposes of simplicity, assume that f is a one-to-one and onto mapping.
The composition of a with f is denoted by:
b = a o f
{(y, b(y)) : b(y) =
a(f(y)), y
Y} .
The most powerful construct in image algebra is called a template, which is defined as a mapping from a point set to a set of images. Thus, a template is an image whose pixel values are images.
(FX)Y is an
F-valued template
from Y to FX.
This means that a template value t(y)
FX, where
yY, is an
image on X. For purposes of notational brevity, we write
ty
t(y).
Note that ty =
{(x,ty(x)):
x X} .
Thus, a value of t's weight
matrix ty, also called a weight of
template t, is expressed as
ty(x)
F. Additionally, we denote the support of
ty as S(ty)
X. Implementationally,
we usually think of the support as the set of domain points
corresponding to the template's nonzero weights. Although this
definition is not 
:
FX ×
(FX)Y ->
FY, which accepts an image and a
template and returns an image.
,
: F × F -> F, an image
a
FX, and a template
t
(FX)X, the
left product is defined as
(a
t)y =
a(x)
ty(x) , y
Y .
a
is symmetrically defined.
Example. When
= + and = ·, we have
the linear convolution
, which is defined
in terms of the left product as
(a
t)y =
a(x) · ty(x) ,
y Y .
t =
t
a.
Example. When
= and
= ·, we have
the multiplicative maximum operation
, and when
= and
= +, we have
the additive maximum operation
. The operations of
multiplicative minimum and additive minimum are symmetrically
defined.
Although we will not use the nonlinear operations
or
extensively in this class,
such operations are frequently employed in image processing.
This concludes our discussion of basic notation for this class. Other notational conventions will be defined as they are introduced in theory development.