Image Digitization

1. Why Image digitization is Necessary ?2. What is meant by Signal Bandwidth ? 3. How to select the Sampling Frequency of given Signal.4. How to Re-construct Image from Sampled Value ?

Why Image digitization is Necessary ?

The Image can be viewed as a 2D Dimension function given in the form of f(x,y).

This image has certain length (L) and certain height (H).

Any point in this Image will identify the Image co-ordinates x and y.

Here x-axis is taken as Height and y-axis is taken as Length.

Every co-ordinate in this 2D image have a limit as following, 0 < x < H and 0 < y < L.

F(x,y) which is any point in an image is actually the product of two terms,

r(x,y) and i(x,y) r(x,y) represents the reflectance of the surface point of which particular point f(x,y) corresponds.

and its value vary from 0 to 1. i(x,y) represents the intensity of the light that is falling on the object surface and its value vary

from 0 to Infinity (theoretically) but practically it is, Imin < f(x,y) < Imax

By Sufiyan Ghori

The above plot shows the variation in the value of intensity along with the minimum and maximum values.

The problem is ,

When x varies from 0 to H there is infinite number of values between 0 and H (according to the definition of real number) similarly for y. This means if we want to represent Image in a computer there should be an infinite number of points and Secondly, If we consider the intensity value at particular point we will see that the intensity value f(x,y) varies from certain minimum Imin and certain maximum Imax.

Again If we take these two min and max as to be maximum and minimum values possible then the problem is the number of intensity values that can be between min and max is again infinite in number, which means if we want to represent an Intensity value in a Digital computer then we have to have infinite number of bits which is practically impossible.

The solution is,Instead of taking all possible values we take some discrete points and those set of points are decided by grid and represented in the form of finite 2-D matrix as follows,

By Sufiyan Ghori

Where M = number of Rows and N = number of Columns

Here every matrix value represents the intensity value at the corresponding point of an image location and these Intensity values can be infinite with in certain minimum and maximum.

So far we’ve learned that, an Image Digitization means representing an the form of 2-D finite Matrix which is called Sampling and each Matrix element must be represented in one of the finite set of discrete values and this process is called Quantization.

So mainly we can say that Digitization includes two Process the first is Sampling while the second is Quantization.

Sampling in detail,

We know that when we are processing the image, the image is in the Digital form but when we want to see the output Image it must be converted back into Analog form, which means everything done during Image processing must be reversed eventually

Sampling => Quantization => Digital Computer => D to A Converter => Display.

To understand sampling let us take an example of 1-D signal,

Where ts is the Sampling interval

By Sufiyan Ghori

The problem with this sampling process is that most of the information in the signal is lost due to the low sampling rate, the solution is either to increase the Sampling Frequency i.e more samples in a given time, or decrease the sampling interval as shown below,

Here we’ve decreased the Sampling interval by half which means the sampling frequency is automatically doubled. This way we’ve recover more information than the previous process.

In order to go further and to understand how samples process takes place, first we should have an idea of Dirac Delta function,

“The Dirac delta function, or δ function, is (informally) a generalized function on the real number line that is zero everywhere except at zero. In the context of signal processing it is often referred to as the unit impulse symbol (or function)”

Now look at this digital signal,

The spacing between the two consecutive samples is represented as ∆t, these kind of function is is called a Comb function.

By Sufiyan Ghori

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T, some textbook authors in electrical engineering and circuit theory, refer to it as the Shah function (possibly because its graph resembles the shape of the Cyrillic letter sha Ш).

Comb(t; ∆t ) => The comb function t at an interval of ∆t.

-4 to 4 => represents values of m.

We know that Dirac delta function is zero everywhere except zero. This means that if both t and m∆t are equal or if t-m∆t = 0, the Dirac delta function will be equal to 1 for example,

if t=m=∆t=1 then,

δ(1-(1*1)) δ (0) = 1

By Sufiyan Ghori

On the left size the signal X(t) is our original continuous signal and after sampling we’ve got the signal as shown on the right side.

These signal (Xs) can now be represented as the product of X(t) and the Dirac Comb function, which means the Samples will be generated for those values of X(t) where the Dirac Comb function is equal to 1 or in simple words, those values of original signal will retain in Xs for which the value of Dirac delta is 1.

By Sufiyan Ghori