Orthogonal%20Transforms

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Orthogonal Transforms

• Fourier Walsh Hadamard

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Review

• Introduce the concepts of base functions
• For Reed-Muller, FPRM
• For Walsh
• Linearly independent matrix
• Non-Singular matrix
• Examples
• Butterflies, Kronecker Products, Matrices
• Using matrices to calculate the vector of
  spectral coefficients from the data vector

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Orthogonal Functions
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Illustrate it for Walsh and RM
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Mean Square Error
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Important result
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• We want to minimize this kinds of errors.
• Other error measures are also used.

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Unitary Transforms

• Unitary Transformation for 1-Dim. Sequence
• Series representation of
• Basis vectors
• Energy conservation

Here is the proof
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• Unitary Transformation for 2-Dim. Sequence
• Definition
• Basis images
• Orthonormality and completeness properties
• Orthonormality
• Completeness

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• Unitary Transformation for 2-Dim. Sequence
• Separable Unitary Transforms
• separable transform reduces the number of
  multiplications and additions from to
• Energy conservation

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Properties of Unitary Transform
transform
Covariance matrix
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Example of arbitrary basis functions being
rectangular waves
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This determining first function determines next
functions
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1
0
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Small error with just 3 coefficients
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This slide shows four base functions multiplied
by their respective coefficients
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This slide shows that using only four base
functions the approximation is quite good
End of example
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Orthogonal and separable Image Transforms
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Extending general transforms to 2-dimensions
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Forward transform
inverse transform
separable
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Fourier Transform
separable
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Extension of Fourier Transform to two dimensions
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Discrete Fourier Transform (DFT)
New notation
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Fast Algorithms for Fourier Transform
Task for students Draw the butterfly for these
matrices, similarly as we have done it for Walsh
and Reed-Muller Transforms
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Pay attention to regularity of kernels and order
of columns corresponding to factorized matrices
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Fast Factorization Algorithms are general and
there is many of them
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• 1-dim. DFT (cont.)
• Calculation of DFT Fast Fourier Transform
  Algorithm (FFT)
• Decimation-in-time algorithm

Derivation of decimation in time
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• 1-dim. DFT (cont.)
• FFT (cont.)
• Decimation-in-time algorithm (cont.)

Butterfly for Derivation of decimation in time
Please note recursion
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• 1-dim. DFT (cont.)
• FFT (cont.)
• Decimation-in-frequency algorithm (cont.)

Derivation of Decimation-in-frequency algorithm
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Decimation in frequency butterfly shows recursion

• 1-dim. DFT (cont.)
• FFT (cont.)
• Decimation-in-frequency algorithm (cont.)

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Conjugate Symmetry of DFT

• For a real sequence, the DFT is conjugate symmetry

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• Use of Fourier Transforms for fast convolution

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Calculations for circular matrix
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By multiplying
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W ? Cw
In matrix form next slide
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w ? Cw
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Here is the formula for linear convolution, we
already discussed for 1D and 2D data, images
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Linear convolution can be presented in matrix
form as follows
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As we see, circular convolution can be also
represented in matrix form
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Important result
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Inverse DFT of convolution
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• Thus we derived a fast algorithm for linear
  convolution which we illustrated earlier and
  discussed its importance.
• This result is very fundamental since it allows
  to use DFT with inverse DFT to do all kinds of
  image processing based on convolution, such as
  edge detection, thinning, filtering, etc.

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2-D DFT
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Circular convolution works for 2D images
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Circular convolution works for 2D images So we
can do all kinds of edge-detection, filtering etc
very efficiently

• 2-Dim. DFT (cont.)
• example

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• 2-Dim. DFT (cont.)
• Properties of 2D DFT
• Separability

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• 2-Dim. DFT (cont.)
• Properties of 2D DFT (cont.)
• Rotation

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• 2-Dim. DFT (cont.)
• Properties of 2D DFT
• Circular convolution and DFT
• Correlation

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• 2-Dim. DFT (cont.)
• Calculation of 2-dim. DFT
• Direct calculation
• Complex multiplications additions
• Using separability
• Complex multiplications additions
• Using 1-dim FFT
• Complex multiplications additions ???

Three ways of calculating 2-D DFT
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Karhunen-Leove Transform(KLT)
Covariance matrix
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• What happens if ??1?

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This happens if ??1
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Category of transforms
These are what I called earlier transforms with
standard butterflies
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Discrete Cosine Transform (DCT)
This is DCT
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DCT is an orthogonal transformm so its inverse
kernel is the same as forward kernel
This is inverse DCT
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DCT can be obtained from DFT
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• Discrete Cosine Transform is asymptotically
  equivalent to Karhunen-Loeve

We take a 2N-point DFT
This is why guys in industry believe that only
DCT is worth their work.
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Properties of DCT real, orthogonal,
energy-compacting, eigenvector-based
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The eigenvectors of R and Qc are very close
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• Basis Functions for 1-dim. DCT(N 16)

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There are many DCT fast algorithms and hardware
designs.
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There are many DCT fast algorithms and hardware
designs.
Many fast algorithms are available fast algorithm
Lee(1-D), Lee-Cho(2-D) VLSI algorithm
regularity, local interconnection,
moduality ref 1.Nam Ik Cho and Sang Uk Lee,
Fast algorithm and implementation of 2-D
DCT. IEEE Trans Circuits and Systems, vol.
38, no.3, pp.297-305, March 1991. 2.Nam Ik Cho
and Sang Uk Lee, DCT algorithm for VLSI parallel
implementation IEEE Trans. ASSP, vol.
ASSP-38, no. 1, pp.121-127, Jan, 1990.
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• Discrete Sine Transform(DST)

Similar to DCT.
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• 1-dim. DST (cont.)
• Basis Functions for 1-dim. DST (N16)

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Walsh Transform
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Here we calculate the matrix of Walsh coefficients
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Here we calculate the matrix of Walsh coefficients
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Here we calculate the matrix of Walsh coefficients
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We have done it earlier in different ways
Here we calculate the matrix of Walsh coefficients
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Symmetry of Walsh
Think about other transforms that you know, are
they symmetric?
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Two-Dimensional Walsh Transform
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Two-dimensional Walsh
Inverse Two-dimensional Walsh
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Properties of Walsh Transforms
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Here is the separable 2-Dim Inverse Walsh
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Example for N4
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odd
even
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Discuss the importance of this figure
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Hadamard Transform
We will go quickly through this material since it
is very similar to Walsh
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separable
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Example of calculating Hadamard coefficients
analogous to what was before
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Standard Trivial Functions for Hadamard
One change
two changes
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Discrete Walsh-Hadamard transform
Now we meet our old friend in a new light again!
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Relationship between Walsh-ordered and
Hadamard-ordered
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Nonsinusoidal orthogonal function
Haar Transform

• Haar transform
• Haar function (1910, Haar) periodic,
  orthonormal, complete

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Slant transform
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SVD(Singular Value Decomposition)
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2-D linear processing technique
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Questions to Students

1. You do not have to remember derivations but you
   have to understand the main concepts.
2. Much software for all discussed transforms and
   their uses is available on internet and also in
   Matlab, OpenCV, and similar packages.

1. How to create an algorithm for edge detection
   based on FFT?
2. How to create a thinning algorithm based on DCT?
3. How to use DST for convolution show example.
4. Low pass filter based on Hadamard.
5. Texture recognition based on Walsh