Md Shiplu Hawlader Roll: SH-224
Apr 01, 2015
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