| Image
Processing Articles
Inde |
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| Boolean
Alegebra |
| Logical
operators are derived
from the boolean algebra,
which is the mathematical
way of representing
the concepts without
much bothering about
what the concepts
generally means. For
example, boolean algebra
can represent the
concept: |
| "The
sky is high and blue." |
| in
the following way:
X and Y |
| here,
X represents the concept
that "the sky
is high" and
Y represents the concept
that "the sky
is blue" and
the notation can be
fully represented
in the mathematical
way by the notation
( X and Y ). Here,
the notation is only
true when both X and
Y are true, else the
notation is false.
Now, to understand
this take another
example: |
| "Ali
is playing cricket
or studying at home." |
| it
will take the following
notation in the mathematical
domain: |
| X
xor Y |
| here
X and Y represent
"Ali is playing
cricket" and
"Ali is studying
at home" respectively.
And the notation X
xor Y represents that
"Ali is playing
cricket or studying
at home." Here
X or Y can not represent
the desired concept,
because Ali can not
be playing cricket
and studying at home
at the same time. |
|
| The
truth value of a concept
in Boolean value can
have just one of two
possible values: true
or false. And the
truth values of a
combination of two
concepts combined
using a certain operator
are shown by the help
of the truth tables.
For example, the truth
table for "and"
operator is as follows: |
|
 |
|
| The
left hand table shows
each of the possible
combinations of truth
values of X and Y,
and the the resulting
truth value of X xor
Y. The truth tables
of the each operator
can be seen in the
respective articles. |
|
| Imeplementing
Logical Operators
on Binary Images |
| Binary
Image represent the
pixel data in the
form of two intensity
levels, that may be
0 and 1. But in the
real case the binary
image low intensity
level is normally
0 and the high intensity
level is 255. In this
case, the 255 can
be taken as logical
1 when applying the
logical operators
on the binary images.
Using this convention
we can carry out logical
operations on images
simply by applying
the truth-table combination
rules to the pixel
values from a pair
of input images (or
a single input image
in the case of NOT).
Normally, corresponding
pixels from each of
two identically sized
binary input images
are compared to produce
the output image,
which is another binary
image of the same
size. As with other
image arithmetic operations,
it is also possible
to logically combine
a single input image
with a constant logical
value, in which case
each pixel in the
input image is compared
to the same constant
in order to produce
the corresponding
output pixel. See
the individual logical
operator descriptions
for examples of these
operations. |
|
| Implementing
Logical Operators
on Graylevel Images |
Logical
operations can also
be carried out on
images with integer
pixel values. In this
extension the logical
operations are normally
carried out in bitwise
fashion on binary
representations of
those integers, comparing
corresponding bits
with corresponding
bits to produce the
output pixel value.
For instance, suppose
that we wish to ORing
the integers 167 and
211 together using
8-bit integers. 167
is 10100111 in binary
and 255 is 11010011.
ORing these together
in bitwise fashion,
we have 11110111 in
binary or 247 in decimal.
This is not the only
implementation of
the logical operators
on the binary images,
rather it can be applied
simply by taking the
0 pixel value as the
logical 0 and the
non-zero pixel values
as the logical 1 value. |
|
| Implementing
Binary Logical Operators
on a Single Image |
| The
binary logical operators
can be applied on
the single image.
It can be achieved
by using the image
as one input and the
other input can be
a structuring element
in a single pass through
the image. During
the pass the operation
between the image
and the operator is
applied on each pixel,
by imposing the structuring
the origin of the
structuring element
on that particular
pixel. |
|
| Binary
Operator List |
AND
& NAND
OR & NOR
Inversion
XOR & XNOR |
|
| The
details can be found
on the relative pages. |
|
| Combined
Applications of the
Binary Operators |
| Thresholding
through ANDing |
| Thresholding
is the operation,
when applied to an
image the image is
transformed into a
binary image. This
operation can be performed
using the AND operator
on an image with the
other input containing
the threshold value
for the thresholding
operation. For example,
you want to threshold
the image at a value
128. So, first input
to the AND operation
is the image and the
other input is the
threshold value i.e
128 ( = 10000000 ).
Then the ouput image
only contains two
values either 0 or
128 ( threshold value
). From there you
can set any value
for the binary intensity
stages. For example,
set the high intensity
value to 255. For
clarification take
the following image: |
|
 |
|
| After
ANDing with 128, it
yeilds: |
|
 |
|
| If
the threshold value
contains more than
one binary 1 in its
binary equivalent,
then in the output
image contains 0 as
the low binary intensity,
but contains many
high intensities which
have to converted
into the binary high
intensity value. This
can be achieved by
setting all the non-zero
intensities at 255,
for example. |
|
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