Md Shiplu Hawlader Roll: SH-224. Fourier Series Theorem Fourier Transform Discrete Fourier Transform Fast Fourier Transform.

Md Shiplu HawladerRoll: SH-224

Fourier Series Theorem Fourier Transform Discrete Fourier Transform Fast Fourier Transform

Any periodic function can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency

Transforms a signal (i.e., function) from the spatial domain to the frequency domain.

where

Forward DFT

Inverse DFT

Typically, we visualize |F(u,v)| The dynamic range of |F(u,v)| is typically

very large Apply stretching: (c

is const)

original image before scaling after scaling

magnitude phase

Reconstructed image using magnitude only (i.e., magnitude determines the contribution of each component!)

Reconstructed image using phase only(i.e., phase determineswhich components are present!)

Easier to remove undesirable frequencies.

Faster perform certain operations in the frequency domain than in the spatial domain.

noisy signal frAequencies

remove highfrequencies

reconstructedsignal

To remove certainfrequencies, set theircorresponding F(u)coefficients to zero!

Low frequencies correspond to slowly varying information (e.g., continuous surface).

High frequencies correspond to quickly varying information (e.g., edges)

Original Image Low-passed

1. Take the FT of f(x):

2. Remove undesired frequencies:

3. Convert back to a signal:

The FFT is an efficient algorithm for computing the DFT

The FFT is based on the divide-and-conquer paradigm: If n is even, we can divide a polynomial

into two polynomials

and we can write

The running time is O(n log n)

Fourier Transform has multitude of applications in all the field of engineering but has a tremendous contribution in image processing fields like image enhancement and restoration.

Image Processing, Analysis and Machine Vision, chapter 6.2.3. Chapman and Hall, 1993

The Image Processing Handbook, chapter 4. CRC Press, 1992

Fundamentals of Electronic Image Processing, chapter 8.4. IEEE Press, 1996

http://en.wikipedia.org/wiki/Fourier_transform