Understanding Image Transforms & Filters in Frequency Domain, Slides of Computer Science

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Basic ideas of Image

Transforms are

derived from those

showed earlier

Image Transforms

• Fast Fourier
  • 2-D Discrete Fourier Transform
• Fast Cosine
  • 2-D Discrete Cosine Transform
• Radon Transform
• Slant
• Walsh, Hadamard, Paley, Karczmarz
• Haar
• Chrestenson
• Reed-Muller

Spatial Frequency

or

Fourier Transform

Jean Baptiste Joseph Fourier

Fourier face in Fourier Transform Domain Docsity.com

Examples of

Fourier 2D Image

Transform

Another formula for Two-Dimensional

Fourier

A cos(x⋅ 2 πi/N) B cos(y⋅ 2 πj/M)

f x = u = i/N, f y = v =j/M

Image is function of x and y

Now we need two cosinusoids for each point, one for x and one for y

Lines in the figure correspond to real value 1

Now we have waves in two directions and they have frequencies and amplitudes Docsity.com

Fourier Transform of a spot

Original image Fourier Transform

Two Dimensional Fast Fourier in Matlab

Filtering in

Frequency

Domain

… will be covered in a separate

lecture on spectral

approaches…..

< < image

..and its spectrum

Image and its spectrum

Image and its spectrum

Let g ( u,v ) be the kernel Let h ( u,v ) be the image G ( k , l ) = DFT [ g ( u,v )] H ( k , l ) = DFT [ h ( u,v )]

Then

DFT −^1 [^ G H ⋅ ] = g h ∗

where means multiplication and means convolution.

⋅ ∗

Convolution Theorem

Instead of doing convolution in spatial domain we can do multiplication In frequency domain

Convolution in spatial domain

Multiplication in spectral domain

v

u

Image

Spectrum (^) Noise and its spectrum

Noise filtering