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Wavelet Image Analysis

Wavelet Image Analysis. Engr. Zahid Mehmood. Stationary Signal. Fourier Transform of Stationary Signal. Chirp Signal (non-stationary signals). Chirp Signal (100, 50, 25 & 10 Hz). Fourier Transform of Chirp Signal. Short Time Fourier Transform (STFT).

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Wavelet Image Analysis

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  1. Wavelet Image Analysis Engr. Zahid Mehmood

  2. Stationary Signal

  3. Fourier Transform of Stationary Signal

  4. Chirp Signal (non-stationary signals)

  5. Chirp Signal(100, 50, 25 & 10 Hz)

  6. Fourier Transform of Chirp Signal

  7. Short Time Fourier Transform (STFT) • The signal is divided into small segments. • The signal is assumed to be stationary during these segments. • We use window function to adjust the width of signal where it is assumed to be stationary. • Window is initially kept zero and then increased iteratively

  8. Short Time Fourier Transform

  9. Short Time Fourier Transform(300, 200, 100 & 50 Hz)

  10. Short Time Fourier Transform

  11. Short Time Fourier Transform(Window Function)

  12. Short Time Fourier Transform a = .01

  13. Short Time Fourier Transforma = .0001

  14. Short Time Fourier Transforma = .00001

  15. Multi-Resolution Analysis (MRA) • MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. • Useful when the signal at hand has high frequency components for short durations and low frequency components for long durations. • Fortunately, the signals that are encountered in practical applications are often of this type.

  16. Continuous Wavelet Transform (CWT) • Negative frequencies are not computed • Width of the window is changed for every single spectral component Difference between STFT & CWT

  17. Continuous Wavelet Transform • Translation is related to the location of the window, as the window is shifted through the signal. This term corresponds to time information. • Scale parameter is defined as 1/frequency.

  18. Continuous Wavelet Transform • Wavelet means small wavelet • Smallness refers to the condition that this (window) function is of finite length (compactly supported). • The wave refers to the condition that this function is oscillatory • The term mother implies that the functions with different region of support that are used in the transformation process are derived from one main function, or the mother wavelet.

  19. CWT - Graphical View of Computation

  20. CWT - Graphical View of Computation

  21. CWT - Graphical View of Computation

  22. Non-Stationary Signal(30, 20,10 & 5 Hz)

  23. Wavelets in Image Processing • Computer 2D wavelet transform of an image (Decomposition) • Alter the transform coefficients • Computer the inverse transform (Synthesis)

  24. 2D Fast Wavelet Transform (FWT) filter bank

  25. Image Decomposition

  26. 2D Fast Wavelet Transform (FWT) filter bank

  27. Edge Detection using DWT

  28. Image Smoothing using DWT

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