The document discusses the Fourier transform and its applications in image processing. The Fourier transform decomposes an image into its sine and cosine components, representing the spatial frequencies contained in the image. This converts the image from the spatial domain to the frequency domain. Key steps involve transforming the image to the frequency domain, applying image processing techniques, and computing the inverse transform. Applications include image analysis, filtering, reconstruction, and compression.

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