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In the name of AllahIn the name of Allah
the compassionate, the merciful
Digital Image Processing
S. Kasaei
Room CE307, SUT
E-Mail: skasaei@sharif eduE Mail: skasaei@sharif.eduHome Page: http://ce.sharif.edu
http://ipl.ce.sharif.eduhttp://sharif.edu/~skasaei
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ttp //s a edu/ s asae
Chapter 5
Image Transforms
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Image Transform
Motivation: Represents a block of image pixels as the superposition of– Represents a block of image pixels as the superposition of some typical basic patterns (transform basic functions).
General process:Forward transformation– Forward transformation
– Process on transform coefficients– Inverse transformation
+t1 t t t
Transform Basis
Transform Coefficients
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+t1 t 2 t 3 t 4
Basis Vectors & Images
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Introduction
The term image transforms usually refers to a class of unitary matrices used fora class of unitary matrices used for representing images.
Just as a 1-D signal can be represented by an orthogonal series of basis functions, an image can also be expanded in terms of a discrete set of basis arrays called basis images
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images.
Introduction
These basis images can be generated by it t iunitary matrices.
An image transform provides a set of coordinates of basis vectors to form the vector spacevector space.
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Introduction
Series coeff.
Series rep.
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1-D Basis Vectors
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2-D Unitary Transforms
Transform coeff.
Series rep.
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2-D Unitary Transforms
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2-D Unitary Transforms
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2-D Unitary Transforms
T
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2-D Unitary Transforms
Basis image
Projection of U on A
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Projection of U on A
2-D Unitary Transforms
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2-D Unitary Transforms
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2-D Unitary Transforms
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2-D Unitary Transforms
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2-D Unitary Transforms
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2-D Unitary Transforms
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2-D Unitary Transforms
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Properties of Unitary Transforms
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Properties of Unitary Transforms
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Properties of Unitary Transforms
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Properties of Unitary Transforms
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Properties of Unitary Transforms
About (5.33):}{}{ BATrABTr =
IAA T =∗
⇒ }{}{}{ uuTT
u RTrARATrAARTr == ∗∗
About (5.34):
]|)()([|)( 22 nnvEn µδ =
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]|)()([|)( nnvEn vv µδ −=
Properties of Unitary Transforms
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Properties of Unitary Transforms
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Properties of Unitary Transforms
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Properties of Unitary Transforms
A
A+B=2A=B
B
A’+B’=2A’>B’A >B
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Properties of Unitary Transforms
Cov[v(0),v(1)]Cov[v(0),v(1)]
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Discrete Fourier Transform (DFT)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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DFT (cntd)
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2-D DFT
Kasaei45A 2-D image and its Fourier spectrum.
2-D DFT
Kasaei46An image, its phase only image, and its contrast enhanced image.
2-D DFT
Kasaei47An image and its Fourier spectrum’s magnitude and phase.
2-D DFT
Kasaei48Shifted image and its Fourier spectrum’s magnitude and phase.
2-D DFT
Kasaei49Rotated image and its Fourier spectrum’s magnitude.
2-D DFT
Kasaei50 An image and its Fourier spectrum’s magnitude.
2-D DFT
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2-D DFT (cntd)
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2-D DFT (cntd)
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2-D DFT (cntd)
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2-D DFT (cntd)
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Discrete Cosine TransformDiscrete Cosine Transform(DCT)
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DCT (cntd)
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DCT (cntd)
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DCT (cntd)
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DCT (cntd)
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DCT (cntd)
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DCT (cntd)
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DCT (cntd)
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Discrete Sine TransformDiscrete Sine Transform (DST)
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DST (cntd)
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Hadamard TransformHadamard Transform(HT)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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HT (cntd)
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Harr TransformHarr Transform (HrT)
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HrT (cntd)
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HrT (cntd)
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HrT (cntd)
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HrT (cntd)
Haar transform takes differences of the l diff f l l fsamples or differences of local average of
the samples of the input vector.
2-D Haar transform coefficients (except for k=l=0) are the differences along rows &k=l=0), are the differences along rows & columns of the local averages of the pixels in the image
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the image.
HrT (cntd)
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Slant TransformSlant Transform (ST)
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ST (cntd)
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ST (cntd)
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ST (cntd)
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Karhunen-Loeve Transform (KLT)
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R G B
KLT (cntd)
Kasaei89PCA1 PCA2 PCA3
KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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KLT (cntd)
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100
KLT (cntd)
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KLT (cntd)
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The End
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Random Signals
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Random Signals
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Recommended