Wavelets and multiresolution - Lecture 15

Wavelets and multiresolution Lecture 15 Milan Gavrilovic [email protected] Centre for Image Analysis Uppsala University

Computer Assisted Image Analysis 2009-05-13

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Today

Today Stationary and non-stationary signals. Short-time Fourier transform. Wavelet transform. Multiresolution processing. I I

Image pyramids. Subband coding.

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Constant Spectra Stationary signal x (t) = cos (5t ∗ 2π) + cos (10t ∗ 2π) + cos (20t ∗ 2π) + cos (50t ∗ 2π) Signal in spatial domain

Fourier transform of signal

(Images taken from http://users.rowan.edu/˜polikar/WAVELETS/WTtutorial.html) M. Gavrilovic (Uppsala University)

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Time Varying Spectra Non-stationary signal Signal in spatial domain

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Fourier transform of signal

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Frequency Components Stationary signal

Non-stationary signal

Both signals constitute of the same frequency components. The signals are undistinguishable in the Fourier domain. Fourier transform is not suitable for analysis of non-stationary signals. M. Gavrilovic (Uppsala University)

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Spectral and Temporal Information

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Short-Time Fourier Transform

Assume some portion of non-stationary signal as stationary. Divide signal into small segments, such that stationary assumption holds. Choose windowing function w of width equal to signal segment. Position window at t = 0, multiply signal with window, then Fourier transform as with any other signal (t - new information). Position window at t = . . . a.s.o. ⇒ one Fourier transform for each time step.

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Short-Time Fourier Transform Short-Time Fourier Transform (STFT) (ω) STFTX

0




Z

t ,f =


 x (t) ω ∗ t − t0 e−j2πft dt

t

STFT of signal ⇒ FT of signal multiplied by window function. Non-stationary signal

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STFT spectra

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Drawback with Short-Time Fourier Transform Window of finite length, covers only portion of signal. I I I

Narrow window ⇒ Good time res, bad freq res. Wide window ⇒ Good freq res, bad time res. Heisenbergs uncertainty principle.

Example of using a wide window.

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Wavelet Transform

Continuous Wavelet Transform CWTxψ

(τ, s) =

Ψψ x

1 (τ, s) = p |s|

Z x (t) ψ

∗



t−τ s

 dt

Function of two variables: I I

τ - translation. s - scale (frequency−1 ).

ψ(t) is the transform function, and is called the mother wavelet.

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Wavelet Transform

Non-stationary input signal.

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Wavelet transform.

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Varied Scale Shifting High frequency (small scale).

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Low frequency (large scale).

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Image Pyramid and System Block Diagram

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Image Pyramid and System Block Diagram Steps

1

Compute reduced-resolution approximation of input image. Done by filtering (approximation filter) and downsampling.

2

Upsample output of (1) and interpolate. This creates a prediction image.

3

Compute the differences between the prediction in (2) and the input image in (1). This difference is called the prediction residual and can later be used to reconstruct the original image.

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Image Pyramids and Statistics

Gaussian pyramid (Gaussian approximation filter). Laplacian pyramid (prediction residual).

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Subband Coding One dimension

Image is decomposed into a set of bandlimited components (subbands). Two-band filter bank for one-dimensional subband coding and decoding. Its spectrum splitting properties. Filter banks satisfying ˆx(n) = x(n) are called biorthogonal.

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Subband Coding Two dimensions

Two-dimensional four-band filter bank for subband image coding.

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Discrete Wavelet Transform (DFT) Haar Basis Functions

The basis functions of the Haar transform are the simplest known orthonormal wavelets. Used as filter banks in subband coding. Discrete wavelet transform using Haar basis functions. Approximations (64 × 64, 128 × 128 and 256 × 256) that can be obtained from wavelet transform.

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Fast Wavelet Transform (FWT)

Analysis filter bank. Resulting decomposition. Synthesis filter bank.

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Wavelets in MATLAB (not mandatory) 1

Start MATLAB.

2

Run wavemenu (part of wavelet toolbox).

3

Select Wavelet-2D.

4

Load an image from /it/sw/matlab/current/... .../toolbox/wavelet/wavedemo/*.mat.

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5

Experiment with different families, types, levels and results of different compression levels.

6

Look at the families of wavelets by going to Display → Wavelet Display.

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Reading Instructions Chapters for next lecture

Chapter 8 in Gonzales-Woods.

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