orthogonal transforms fourier. review introduce the concepts of base functions: –for reed-muller,...

Orthogonal Transforms

Fourier

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

Orthogonal Orthogonal FunctionsFunctions

Orthogonal FunctionsOrthogonal Functions

Note that these are arbitrary functions, we do not assume sinusoids

Illustrate it for Walsh and RM

Mean Mean Square Square ErrorError

Mean Square ErrorMean Square Error

Important result

• 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.

Unitary Unitary TransformsTransforms

Unitary TransformsUnitary Transforms• Unitary Transformation for 1-Dim. Sequence

– Series representation of

– Basis vectors :

– Energy conservation :

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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

Example of arbitrary Example of arbitrary basis functions being basis functions being

rectangular wavesrectangular waves

This determining first function determines next functions

10

Small error with just 3 coefficients

This slide shows four base functions multiplied by their respective coefficients

End of example

This slide shows that using only four base functions the approximation is quite good

Orthogonality Orthogonality and and

separabilityseparability

Orthogonal and separable Image Orthogonal and separable Image TransformsTransforms

Extending Extending general transformsgeneral transforms to 2- to 2-dimensionsdimensions

separable

Forward transform

inverse transform

Fourier Fourier Transforms in Transforms in new notationsnew notations

1. We emphasize generality2. Matrices

Fourier TransformFourier Transform

separable

Extension of Fourier Transform to Extension of Fourier Transform to two dimensionstwo dimensions

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Task for students:

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

transformSine Cosine, Fourier,

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• 1-dim. DFT (cont.) – Calculation of DFT : Fast Fourier Transform Algorithm

(FFT)

• Decimation-in-time algorithm

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Decimation in Decimation in Time versus Time versus

Decismation in Decismation in FrequencyFrequency

• 1-dim. DFT (cont.) – FFT (cont.)

• Decimation-in-time algorithm (cont.)Butterfly for Derivation of decimation in time

Please note recursion

• 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.)

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•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.

2-D DFT2-D DFT

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• 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

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

• SeparabilitySeparability

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• RotationRotation

sin ,cos ,sin ,cos lkrnrm

),(),( 00 Frf

(a) a sample image (b) its spectrum (c) rotated image (d) resulting spectrum

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

• Circular convolution and DFTCircular convolution and DFT

• CorrelationCorrelation

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• Direct calculation

– Complex multiplications & additions :

• Using separability

– Complex multiplications & additions :

• Using 1-dim FFT – Complex multiplications & additions : ???

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Three ways of calculating 2-D DFT

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