Image Processing Reference
In-Depth Information
Table B.1 Different rational numbers and three different ways of converting to integers
x
Floor of x
Ceiling of x
Round of x
3 . 14
3
4
3
0 . 7
0
1
1
4 . 5
4
5
5
3 . 14
4
3
3
0 . 7
1
0
1
4 . 5
5
4
4
B.3
Converting a Rational Number to an Integer
Sometimes we want to convert a rational number into an integer. This can be done
in different ways, where the three most common are:
Floor simply rounds a rational number to the nearest smaller integer. For example:
Floor of 4 . 2
=
4. Mathematically it is denoted
4 . 2
=
4. In C-programming a
build-in function exists: floor() .
Ceiling is the opposite of floor and rounds off to the nearest bigger integer. For
example: Ceiling of 4 . 2
=
5. Mathematically it is denoted
4 . 2
=
5. In C-
programming a build-in function exists: ceil() .
Round finds the nearest integer, i.e., Round of 4 . 2
=
4 and Round of 4 . 7
=
5. In
terms of C-code the following expression is often used: int(x
+
0 . 5 ) . That is, we
add 0.5 to the number and then typecast it to an integer.
In Table B.1 some examples are provided.
B.4
Summation
Say you want to add the first 12 positive integers:
1
+
2
+
3
+
4
+
5
+
6
+
7
+
8
+
9
+
10
+
11
+
12
=
78
(B.2)
This is no problem writing down, but what if you want to add the first 1024
positive integers? This will be dreadful to write down. Luckily there exists a more
compact way of writing this using summation , which is denoted as . Adding the
first 1024 positive integers can now be written as
1024
i
(B.3)
i =
1
where i is the summation index. Below the summation sign we have i
1, which
means that the first value of i is 1. Above the summation sign we have 1024. This
actually means i
=
. Either way, it means that
the last value of i is 1042. You can think of i as a counter going from 1 to 1042 in
=
1042, but we virtually always skip i
=
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