Fourier transform
Post on 20-Jan-2015
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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
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