Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller,...
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![Page 1: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/1.jpg)
Orthogonal Transforms
Fourier
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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
Our goal is to discuss the best approximation of a function using orthogonal functions
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Orthogonal Orthogonal FunctionsFunctions
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Orthogonal FunctionsOrthogonal Functions
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Note that these are arbitrary functions, we do not assume sinusoids
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Illustrate it for Walsh and RM
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Mean Mean Square Square ErrorError
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Mean Square ErrorMean Square Error
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Important result
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• We want to minimize this kinds of We want to minimize this kinds of errors.errors.
• Other error measures are also used.Other error measures are also used.
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Unitary Unitary TransformsTransforms
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Unitary TransformsUnitary Transforms• Unitary Transformation for 1-Dim. Sequence
– Series representation of
– Basis vectors :
– Energy conservation :
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Here is the proof
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• Unitary Transformation for 2-Dim. SequenceUnitary Transformation for 2-Dim. Sequence– Definition :
– Basis images :– Orthonormality and completeness properties
• Orthonormality :
• Completeness :
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• Unitary Transformation for 2-Dim. SequenceUnitary 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 TransformProperties of Unitary Transformtransform
Covariance matrix
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Example of arbitrary Example of arbitrary basis functions being basis functions being
rectangular wavesrectangular waves
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This determining first function determines next functions
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10
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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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End of example
This slide shows that using only four base functions the approximation is quite good
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Orthogonality Orthogonality and and
separabilityseparability
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Orthogonal and separable Image Orthogonal and separable Image TransformsTransforms
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Extending Extending general transformsgeneral transforms to 2- to 2-dimensionsdimensions
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separable
Forward transform
inverse transform
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Fourier Fourier Transforms in Transforms in new notationsnew notations
1. We emphasize generality2. Matrices
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Fourier TransformFourier Transform
separable
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Extension of Fourier Transform to Extension of Fourier Transform to two dimensionstwo dimensions
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![Page 39: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/39.jpg)
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![Page 40: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/40.jpg)
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Draw the butterfly for these matrices, similarly as we have done it for Walsh and Reed-Muller Transforms
Pay attention to regularity of kernels and order of columns corresponding to factorized matrices
![Page 41: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/41.jpg)
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![Page 42: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/42.jpg)
• 1-dim. DFT (cont.) – Calculation of DFT : Fast Fourier Transform Algorithm
(FFT)
• Decimation-in-time algorithm
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![Page 43: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/43.jpg)
Decimation in Decimation in Time versus Time versus
Decismation in Decismation in FrequencyFrequency
![Page 44: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/44.jpg)
• 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.)
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• 1-dim. DFT (cont.) – FFT (cont.)
• Decimation-in-frequency algorithm (cont.)
Decimation in frequency butterfly shows recursion
![Page 47: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/47.jpg)
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![Page 48: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/48.jpg)
Use of Fourier Use of Fourier Transforms Transforms
for fast for fast convolutionconvolution
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In matrix form next slideW * = Cw*
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matrix diagonal;
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![Page 53: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/53.jpg)
81
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Here is the formula for linear convolution, we already discussed for 1D and 2D data, images
![Page 54: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/54.jpg)
xHy
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Linear convolution can be presented in matrix form as follows:
![Page 55: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/55.jpg)
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As we see, circular convolution can be also represented in matrix form
![Page 56: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/56.jpg)
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![Page 57: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/57.jpg)
DFTfor algorithmfast
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where
and
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Inverse DFT of convolution
![Page 58: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/58.jpg)
padding zero;1,,,0
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then
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y
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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.
![Page 59: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/59.jpg)
2-D DFT2-D DFT
![Page 60: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/60.jpg)
NNNN
and
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notation spacevector
since.1
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),(
1,0,),(1
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is DFTunitary D2
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FF
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2-D DFT2-D DFT
![Page 61: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/61.jpg)
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1
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N+L-1
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MN+L-1
Circular Circular convolution convolution works for 2D works for 2D imagesimages
![Page 62: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/62.jpg)
• 2-Dim. DFT (cont.)– example
image Lena 512512 a of DFT dim2
(a) Original Image (b) Magnitude (c) Phase
Circular convolution works for 2D imagesCircular convolution works for 2D images
So we can do all kinds of edge-detection, filtering etc So we can do all kinds of edge-detection, filtering etc very efficientlyvery efficiently
![Page 63: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/63.jpg)
• 2-Dim. DFT (cont.) – Properties of 2D DFT
• SeparabilitySeparability
1,,0, ,),(11
),(1
0
1
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WN
lkFN
m
N
n
lnN
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1,,0 ,),(11
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0
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WN
nmfN
k
N
l
lnN
kmN
![Page 64: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/64.jpg)
• 2-Dim. DFT (cont.) – Properties of 2D DFT (cont.)
• RotationRotation
sin ,cos ,sin ,cos lkrnrm
),(),( 00 Frf
(a) a sample image (b) its spectrum (c) rotated image (d) resulting spectrum
![Page 65: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/65.jpg)
• 2-Dim. DFT (cont.) – Properties of 2D DFT
• Circular convolution and DFTCircular convolution and DFT
• CorrelationCorrelation
p q
C qnpmgqpfnmgnmf ),(),(),(),(
),(*),(),(),(
),(),(),(*),(
lkGlkFnmgnmf
lkGlkFnmgnmf
p q
Cfg qnpmgqpfnmgnmfnmR ),(),(),(),(),( *
),(),(),(),(
),(),(),(),(*
*
lkGlkFnmgnmf
lkGlkFnmgnmf
![Page 66: Orthogonal Transforms Fourier. Review Introduce the concepts of base functions: –For Reed-Muller, FPRM –For Walsh Linearly independent matrix Non-Singular.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d805503460f94a64a5b/html5/thumbnails/66.jpg)
• 2-Dim. DFT (cont.) – Calculation of 2-dim. DFTCalculation of 2-dim. DFT
• Direct calculation
– Complex multiplications & additions :
• Using separability
– Complex multiplications & additions :
• Using 1-dim FFT – Complex multiplications & additions : ???
10
,10 ,),(),(
2
11
0
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221 2
Nl
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m
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n
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)( 22
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10
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m
kmN
jN
n
lnN
j
))(( 2121 NNNN
Three ways of calculating 2-D DFT
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Questions to StudentsQuestions to Students1. 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