Part I: Image Transforms

1
Part I Image Transforms

• DIGITAL IMAGE PROCESSING

2
1-D SIGNAL TRANSFORMGENERAL FORM

• Scalar form

• Matrix form

3
1-D SIGNAL TRANSFORM cont.REMEMBER THE 1-D DFT!!!

• General form

• DFT

4
1-D INVERSE SIGNAL TRANSFORMGENERAL FORM

• Scalar form

• Matrix form

5
1-D INVERSE SIGNAL TRANSFORM cont.REMEMBER THE
1-D DFT!!!

• General form

• DFT

6
1-D UNITARY TRANSFORM

• Matrix form

7
SIGNAL RECONSTRUCTION
8
IMAGE TRANSFORMS

• Many times, image processing tasks are best
  performed in a domain other than the spatial
  domain.
• Key steps
• (1) Transform the image
• (2) Carry the task(s) in the transformed domain.
• (3) Apply inverse transform to return to the
  spatial domain.

9
2-D (IMAGE) TRANSFORMGENERAL FORM
10
2-D IMAGE TRANSFORMSPECIFIC FORMS

• Separable
• Symmetric

11

• Separable and Symmetric
• Separable, Symmetric and Unitary

12
ENERGY PRESERVATION

• 1-D
• 2-D

13
ENERGY COMPACTION

• Most of the energy of the original data
  concentrated in only a few of the significant
  transform coefficients remaining coefficients
  are near zero.

14
Why is Fourier Transform Useful?

• Easier to remove undesirable frequencies.
• Faster to perform certain operations in the
  frequency domain than in the spatial domain.
• The transform is independent of the signal.

15
ExampleRemoving undesirable frequencies
frequencies
noisy signal
remove high frequencies
reconstructed signal
16
How do frequencies show up in an image?

• Low frequencies correspond to slowly varying
  information (e.g., continuous surface).
• High frequencies correspond to quickly varying
  information (e.g., edges)

Original Image
Low-passed
17
2-D DISCRETE FOURIER TRANSFORM
18
Visualizing DFT

• Typically, we visualize
• The dynamic range of is typically
  very large
• Apply stretching
  
• ( is constant)

original image
before scaling
after scaling
19
Amplitude and Log of the Amplitude
20
Amplitude and Log of the Amplitude
21
Original and Amplitude
22
DFT PROPERTIES SEPARABILITY

• Rewrite as follows
• If we set
• Then

23
DFT PROPERTIES SEPARABILITY

• How can we compute ?
• How can we compute ?

24
DFT PROPERTIES SEPARABILITY
25
DFT PROPERTIES PERIODICITY

• The DFT and its inverse are periodic with period
  N

26
DFT PROPERTIES SYMMETRY
If is real, then
27
DFT PROPERTIES TRANSLATION

• Translation in spatial domain
• Translation in frequency domain

28
DFT PROPERTIES TRANSLATION

• Warning to show a full period, we need to
  translate the origin of the transform at
  

29
DFT PROPERTIES TRANSLATION
30
DFT PROPERTIES TRANSLATION
31
DFT PROPERTIES ROTATION
32
DFT PROPERTIESADDITION-MULTIPLICATION
33
DFT PROPERTIES SCALE
34
DFT PROPERTIES AVERAGE
35
Original Image Fourier Amplitude Fourier
Phase
36
Magnitude and Phase of DFT

• What is more important?
• Hint use inverse DFT to reconstruct the image
  using magnitude or phase only information

magnitude
phase
37
Magnitude and Phase of DFT
Reconstructed image using magnitude only (i.e.,
magnitude determines the contribution of each
component!)
Reconstructed image using phase only (i.e.,
phase determines which components are present!)
38
Magnitude and Phase of DFT
39
Original Image-Fourier AmplitudeKeep Part of the
Amplitude Around the Origin and Reconstruct
Original Image (LOW PASS filtering)
40
Keep Part of the Amplitude Far from the Origin
and Reconstruct Original Image (HIGH PASS
filtering)
41
(No Transcript)
42
Reconstruction fromphase of one image and
amplitude of the other
43
(No Transcript)
44
Reconstruction fromphase of one image and
amplitude of the other