Introduction

2. ₤ 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
Image
Input Output

3. ₤ 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.

4. ₤ 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.

5. ₤ 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

6. ₤ 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

7. ₤ 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

8. ₤ Sinusoidal pattern Single Fourier term that
encodes
╬ The spatial frequency,
╬ The magnitude (positive or negative),
╬ The phase.

9. ₤ 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

10. Plausibility
Original Magnitude Phase

11. Magnitude Phase

12. Brightness Fourier Inverse
Image Transform Transformed