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Image Processing Transforms

The document discusses the discrete cosine transform (DCT) and Karhunen-Loeve transform (KLT). It explains that the DCT transforms signals into basis functions that are real-valued and has good energy compaction properties. The KLT decorrelates random vectors by transforming them into a new basis defined by the eigenvectors of the covariance matrix. However, the KLT is not often used in practice because its basis depends on the image covariance and it has no fast computational algorithms.

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Marzan Samin Ashrafi
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286 views26 pages

Image Processing Transforms

The document discusses the discrete cosine transform (DCT) and Karhunen-Loeve transform (KLT). It explains that the DCT transforms signals into basis functions that are real-valued and has good energy compaction properties. The KLT decorrelates random vectors by transforming them into a new basis defined by the eigenvectors of the covariance matrix. However, the KLT is not often used in practice because its basis depends on the image covariance and it has no fast computational algorithms.

Uploaded by

Marzan Samin Ashrafi
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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2D DISCRETE COSINE TRANSFORM

Discrete Cosine Transform


■ 1D Discrete Cosine Transform (DCT)

where and

• Inverse DCT
Discrete Cosine Transform
■ 2D Discrete Cosine Transform (DCT)

where and

• Inverse DCT
Discrete Cosine Transform
■ The basis functions of DCT are real. (DFT has complex
basis functions.)
■ DCT has very good energy compaction properties.
■ DCT can be expressed in terms of DFT, therefore, Fast
Fourier Transform implementation can be used.
■ In the case of block-based image compression, (e.g.,
JPEG), DCT produces less artifacts along the
boundaries than DFT does.
DCT and DFT
■ N-point DCT of x[n] can be obtained from 2N-point
DFT of symmetrically extended x[n].

Symmetric extension:

DFT of :

DCT of :
Discrete Cosine Transform
■ Matrix Representation of DCT
Discrete Cosine Transform
■ Matrix Representation of Inverse DCT
Discrete Cosine Transform

■ Inverse DCT matrix is equal to the transpose of DCT


matrix!
Discrete Cosine Transform
■ 2D Discrete Cosine Transform (DCT)

where and

• Inverse DCT
Discrete Cosine Transform

■ For two-dimensional signals:


THE KARHUNEN-LOEVE TRANSFORM (KLT)
IN IMAGE PROCESSING
Eigenvalues and Eigenvectors
The concepts of eigenvalues and eigenvectors are important for understanding the
KL transform.
Vector population

• Consider a population of random vectors of the following form:

• The quantity
•of the image i .

• The population may arise from the formation of the above vectors
for different image pixels.
Example: x vectors could be pixel values
in several spectral bands (channels)
Mean and Covariance Matrix
• The mean vector of the population is defined as:

• The covariance matrix of the population is defined as:

• For M vectors of a random population, where M is large enough


Karhunen-Loeve Transform

• Let A be a matrix whose rows are formed from the eigenvectors of the covariance
matrix C of the population.

• They are ordered so that the first row of A is the eigenvector corresponding to the
largest eigenvalue, and the last row the eigenvector corresponding to the smallest
eigenvalue.

• We define the following transform:

• It is called the Karhunen-Loeve transform.


Karhunen-Loeve Transform
• You can demonstrate very easily that:
Inverse Karhunen-Loeve Transform
Drawbacks of the KL Transform

Despite its favourable theoretical properties, the KLT is not used


in practice for the following reasons.

• Its basis functions depend on the covariance matrix of the


image, and hence they have to recomputed and transmitted
for every image.

• Perfect decorrelation is not possible, since images can rarely be


modelled as realisations of ergodic fields.

• There are no fast computational algorithms for its


implementation.
Example: x vectors could be pixel values
in several spectral bands (channels)
Example of the KLT: Original images

6 spectral images
from an airborne
Scanner.

(Images from Rafael C. Gonzalez and


Richard E.
Wood, Digital Image Processing, 2nd Edition.
Component λ

1 3210
2 931.4
3 118.5
4 83.88
5 64.00
6 13.40

(Images from Rafael C. Gonzalez and


Richard E.
Wood, Digital Image Processing, 2nd Edition.
Six principal components
Original images (channels)
after KL transform
Example: Original Images (left)
and Principal Components (right)

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