Basic ideas of Image Transforms are derived from those showed earlier

1
Basic ideas of Image Transforms are derived from
those showed earlier
2
Image Transforms

• Fast Fourier
• 2-D Discrete Fourier Transform
• Fast Cosine
• 2-D Discrete Cosine Transform
• Radon Transform
• Slant
• Walsh, Hadamard, Paley, Karczmarz
• Haar
• Chrestenson
• Reed-Muller

3
Methods for Digital Image Processing
4
Spatial FrequencyorFourier Transform
Fourier face in Fourier Transform Domain
Jean Baptiste Joseph Fourier
5
Examples of Fourier 2D Image Transform
6
Fourier 2D Image Transform
7
Another formula for Two-Dimensional Fourier
Image is function of x and y
A cos(x?2?i/N) B cos(y?2?j/M) fx u i/N, fy
v j/M
Lines in the figure correspond to real value 1
Now we need two cosinusoids for each point, one
for x and one for y
Now we have waves in two directions and they have
frequencies and amplitudes
8
Fourier Transform of a spot
Original image
Fourier Transform
9
Transform Results
image
transform
spectrum
10
Two Dimensional Fast Fourier in Matlab
11
Filtering in Frequency Domain
will be covered in a separate lecture on
spectral approaches..
12

• H(u,v) for various values of u and v
• These are standard trivial functions to compose
  the image from

13
(No Transcript)
14
lt
lt
image
..and its spectrum
15
Image and its spectrum
16
Image and its spectrum
17
Image and its spectrum
18
Convolution Theorem
Let g(u,v) be the kernel Let h(u,v) be the
image G(k,l) DFTg(u,v) H(k,l)
DFTh(u,v) Then
This is a very important result
where means multiplication and means
convolution.
This means that an image can be filtered in the
Spatial Domain or the Frequency Domain.
19
Convolution Theorem
Let g(u,v) be the kernel Let h(u,v) be the
image G(k,l) DFTg(u,v) H(k,l)
DFTh(u,v) Then
Instead of doing convolution in spatial domain
we can do multiplication In frequency domain
Multiplication in spectral domain
Convolution in spatial domain
where means multiplication and means
convolution.
20
v
Image
u
Spectrum
Noise and its spectrum
Noise filtering
21
Image
v
u
22
(No Transcript)
23
Image of cow with noise
24
white noise
white noise spectrum
kernel spectrum (low pass filter)
red noise
red noise spectrum
25
Filtering is done in spectral domain. Can be very
complicated
26
Discrete Cosine Transform (DCT)

• Used in JPEG and MPEG
• Another Frequency Transform, with Different Set
  of Basis Functions

27
Discrete Cosine Transform in Matlab
trucks
Two-dimensional Discrete Cosine Transform
Two dimensional spectrum of tracks. Nearly all
information in left top corner
absolute
28
Statistical Filters

• Median Filter also eliminates noise
• preserves edges better than blurring
• Sorts values in a region and finds the median
• region size and shape
• how define the median for color values?

29
Statistical Filters Continued

• Minimum Filter (Thinning)
• Maximum Filter (Growing)
• Pixellate Functions

Now we can do this quickly in spectral domain
30
thinning
growing

• Thinning
• Growing

31
Pixellate Examples
Original image
Noise added
After pixellate
32
DCT used in compression and recognition
Can be used for face recognition, tell my story
from Japan.
Fringe Pattern
DCT Coefficients
DCT
Zonal Mask
(1,1) (1,2) (2,1) (2,2) . . .
Artificial Neural Network
Feature Vector
33
Noise Removal
Transforms for Noise Removal
Image with Noise Transform
been removed
Image reconstructed as the noise has been removed
34
Image Segmentation Recall Edge Detection
Now we do this in spectral domain!!
35
Image Moments
2-D continuous function f(x,y), the moment of
order (pq) is
Moments were found by convolutions
Central moment of order (pq) is
36
Image Moments (contd.)
Normalized central moment of order (pq) is
convolutions are now done in spectral domain
A set of seven invariant moments can be derived
from gpq
Now we do this in spectral domain!!
37
Image Textures
Now we do texture analysis like this in spectral
domain!!
The USC-SIPI Image Database http//sipi.usc.edu/
38
Problems

• There is a lot of Fourier and Cosine Transform
  software on the web, find one and apply it to
  remove some kind of noise from robot images from
  FAB building.
• Read about Walsh transform and think what kind of
  advantages it may have over Fourier
• Read about Haar and Reed-Muller transform and
  implement them. Experiment

39
Sources

• Howard Schultz, Umass
• Herculano De Biasi
• Shreekanth Mandayam
• ECE Department, Rowan University
• http//engineering.rowan.edu/shreek/fall01/dip/

http//engineering.rowan.edu/shreek/fall01/dip/la
b4.html
40
Image Compression

• Please visit the website
• http//www.cs.sfu.ca/CourseCentral/365/li/material
  /notes/Chap4/Chap4.html