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**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 Functions

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Orthogonal 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 Square Error

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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

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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**

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

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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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**Orthogonality and separability**

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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 Transforms in new notations**

We emphasize generality Matrices

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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 2 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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**Decimation in Time versus Decismation in Frequency**

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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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2-D DFT

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**Circular convolution works for 2D images**

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**2-Dim. DFT (cont.) example Circular convolution works for 2D images**

So we can do all kinds of edge-detection, filtering etc very efficiently 2-Dim. DFT (cont.) example (a) Original Image (b) Magnitude (c) Phase

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

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**(d) resulting spectrum**

2-Dim. DFT (cont.) Properties of 2D DFT (cont.) Rotation (a) a sample image (b) its spectrum (c) rotated image (d) resulting spectrum

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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 Using 1-dim FFT Complex multiplications & additions : ??? Three ways of calculating 2-D DFT

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Questions to Students You do not have to remember derivations but you have to understand the main concepts. Much software for all discussed transforms and their uses is available on internet and also in Matlab, OpenCV, and similar packages. How to create an algorithm for edge detection based on FFT? How to create a thinning algorithm based on DCT? How to use DST for convolution – show example. Low pass filter based on Hadamard. Texture recognition based on Walsh

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