Fourier transform

Abhishek PachisiaB.Tech – I.T.090102801

Fourier Transform of an Image

Foreword – Image Processing

₤ Image╬ A representation of the external form of a person or

thing in sculpture, painting, etc.

₤ Image Processing╬ The analysis and manipulation of a digitized image, esp.

in order to improve its quality.╬ Study of any algorithm

Input OutputImage

Foreword – Frequency Domain

₤ The rate at which image intensity values are changing in the image

₤ Its domain over which values of F(u) range.u Freq. of component of Transform

₤ Steps:╬ Transform the image to its frequency representation╬ Perform image processing╬ Compute inverse transform.

Foreword – Fourier Transform

₤ Decompose an image into its sine & cosine components.

₤ Sinusoidal variations in brightness across the image.₤ Each point represents a particular frequency contained

in the spatial domain image.

Spatial Domain Freq. Domain (Input) (Output)

₤ Applications╬ Image analysis,╬ Image filtering, ╬ Image reconstruction ╬ Image compression.

Fourier Transform – 1 D

₤ Functions that are NOT periodic BUT with finite area under the curve can be expressed as the integral of sine's and/or cosines multiplied by a weight function

₤ The Fourier transform for f(x) exists iff╬ f(x) is piecewise continuous on every finite interval╬ f(x) is absolutely integrable

Discrete Fourier Transform

₤ Fourier Series is the origin.₤ The DFT is the sampled Fourier Transform₤ 2-D DFT of N*N matrix :

₤ Complexity of 1-D DFT is N2.

₤ Sufficiently accurate

Freq. Domain Procedure

₤ Multiply the input image by (-1)x+y to center the transform

₤ Compute the DFT F(u,v) of the resulting image₤ Multiply F(u,v) by a filter G(u,v)₤ Computer the inverse DFT transform h*(x,y)₤ Obtain the real part h(x,y) of 4₤ Multiply the result by (-1)x+y

Example – 1(i)₤ Sinusoidal pattern Single Fourier

term that encodes ╬ The spatial frequency, ╬ The magnitude (positive or negative),╬ The phase.

Example – 1(ii)₤ The spatial frequency,

╬ Frequency across space

₤ The magnitude (positive or negative),╬ Corresponds to its contrast╬  A negative magnitude represents a contrast-

reversal, i.e. the bright become dark, and vice-versa

₤ The phase.╬ How the wave is shifted relative to the origin

Example - 2

 Original

 Magnitude

 Phase

Plausibility

Magnitude 

Phase

Example - Reconstruct

Brightness Image

Fourier Transform

Inverse Transformed

Example - 3

ThankYou