20-12-2012, 11:52 AM
IMAGE ENHANCEMENT IN FREQUENCY DOMAIN
IMAGE ENHANCEMENT.pptx (Size: 949.7 KB / Downloads: 26)
INTRODUTION
Frequency-the number of times that a periodic function repeats the same sequence of values during a unit variation of the independent variable.
Filter- a device or material for suppressing or minimizing waves or oscillation of certain frequencies.
In the following discussion we will focus on-
How Fourier transform and the frequency domain can be used for image filtering.
Connection between image characteristic and the mathematical tools used to represent them.
Impact of Fourier transform on the images.
4. Brief history of Fourier series and transform-
The French mathematician jean batiste Joseph fourier,born in 1768,is most remembered for his book “the analytic theory of heat”.
Basically ,Fourier ‘s contribution in this field states that any periodic function can be expressed as the sum of sines and/or cosines of different frequencies ,each multiplied by a different coefficient.
It doesn’t matter how complicated the function is ,if it is periodic and satisfies some mild mathematical conditions, it can be represented
by such a sum.
Even functions that are not periodic (but whose area is infinite) can be expressed as the integral of sines and/or cosines multiplied by a weighing functions. The formulation in this case is “ Fourier transform”.
The function expressed in either a Fourier series or transform can be reconstructed (recovered) completely via an inverse process ,with no loss of information .
“one of the most important characteristic of these representation is-it allow us to work in Fourier domain and then return to the original domain of function without losing any information”
FREQUENCY SHIFTING PROPERTY
According to this property ,multiplication of a function f(t) by expJ wt) is equivalent to shifting its Fourier transform F(w) in the positive direction by an amount w i.e. spectrum F(w) is translated by an amount w.Therefore,this theorem is known as “frequency translation theorem”
CONVOLUTION
Convolution of two functions involves flipping (rotating by 180°) one function about its origin and sliding it past the other.
At each displacement in the sliding process we perform a computation ,such as sum of products.
THE SAMPLING THEOREM
A function f(t) whose fourier transform is zero for values of frequencies outside a finite band [-μmax,μmin],about origin is called a band-limited function .
A lower value of 1/ΔT would cause the period in sampled function to