Logical Operators
Boolean Algebra
Implementing Logical Operators on Binary Images
Implementing Logical Operators on
Gray Level Images
Implementing Binary Logical Operators on a Single Image
Binary Operators List
Combined Applications
of the Binary Operators
Boolean Algebra
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.
Implementing
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
Gray level 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 output 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 yields:
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.