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Basic ideas of Image Transforms are derived from those showed earlier. 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.

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image transforms
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
spatial frequency or fourier transform
Spatial FrequencyorFourier Transform

Fourier face in Fourier Transform Domain

Jean Baptiste Joseph Fourier

another formula for two dimensional fourier
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

slide8

Fourier Transform of a spot

Original image

Fourier Transform

slide9

Transform Results

image

transform

spectrum

slide11

Filtering in Frequency Domain

… will be covered in a separate lecture on spectral approaches…..

slide12

H(u,v) for various values of u and v

  • These are standard trivial functions to compose the image from
slide14

<

<

image

..and its spectrum

slide18

Convolution Theorem

Let g(u,v) be the kernel

Let h(u,v) be the image

G(k,l) = DFT[g(u,v)]

H(k,l) = DFT[h(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.

slide19

Convolution Theorem

Let g(u,v) be the kernel

Let h(u,v) be the image

G(k,l) = DFT[g(u,v)]

H(k,l) = DFT[h(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.

slide20

v

Image

u

Spectrum

Noise and its spectrum

Noise filtering

slide21

Spectrum

Image

v

u

slide22

Image x(u,v)

v

u

Spectrum log(X(k,l))

l

k

slide23

Image x(u,v)

v

u

Spectrum log(X(k,l))

l

k

Image of cow with noise

slide24

white noise

white noise spectrum

kernel spectrum (low pass filter)

red noise

red noise spectrum

slide26

Discrete Cosine Transform (DCT)

  • Used in JPEG and MPEG
  • Another Frequency Transform, with Different Set of Basis Functions
slide27

Discrete Cosine Transform in Matlab

trucks

Two-dimensional Discrete Cosine Transform

Two dimensional spectrum of tracks. Nearly all information in left top corner

absolute

slide28

“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?
slide29

“Statistical” Filters Continued

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

Now we can do this quickly in spectral domain

slide30

thinning

growing

  • Thinning
  • Growing
slide31

Pixellate Examples

Original image

After pixellate

Noise added

dct used in compression and recognition
DCT used in compression and recognition

1 2 3 4 5

1

2

3

4

5

Can be used for face recognition, tell my story from Japan.

Fringe Pattern

DCT Coefficients

Zonal Mask

DCT

(1,1)

(1,2)

(2,1)

(2,2)

.

.

.

Artificial

Neural

Network

Feature

Vector

noise removal
Noise Removal

Transforms for Noise Removal

Image with Noise Transform been removed

Image reconstructed as the noise has been removed

image segmentation recall edge detection
Image Segmentation Recall: Edge Detection

Gradient

Mask

-1

-1

-2

0

-1

1

f(x,y)

fe(x,y)

0

0

0

2

0

-2

0

2

1

1

1

-1

Now we do this in spectral domain!!

image moments
Image Moments

2-D continuous function f(x,y), the moment of order (p+q) is:

Moments were found by convolutions

Central moment of order (p+q) is:

image moments contd
Image Moments (contd.)

Normalized central moment of order (p+q) 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!!

image textures
Image Textures

Grass Sand Brick wall

Now we do texture analysis like this in spectral domain!!

The USC-SIPI Image Database

http://sipi.usc.edu/

problems
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
sources
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/lab4.html

image compression

Image Compression

Please visit the website

http://www.cs.sfu.ca/CourseCentral/365/li/material/notes/Chap4/Chap4.html

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