Image Wavelet Transform and Multi-resolution Analysis

Wavelet transform is a cutting-edge image processing technology that has garnered significant attention in recent years. It has spurred the development of numerous innovative methods in areas such as image compression, feature detection, and texture analysis, including multi-resolution analysis, time-frequency analysis, and pyramid algorithms, all of which fall under the wavelet transform umbrella.

From Fourier Transform to Wavelet Transform

Concept of Wavelets

A wavelet is a mathematical function that undergoes changes within a limited time frame and has an average value of zero. It possesses two key characteristics: (1) it has a finite duration with sudden changes in frequency and amplitude, and (2) within this limited time frame, its average value is zero. The result of a wavelet transform is various wavelet coefficients composed of scale and displacement functions.

Intuitive Understanding of Wavelet Transform

The Fourier transform has long been the most widely used and effective analysis tool in signal processing, serving as a means to convert between time and frequency domains. However, while the Fourier transform can provide frequency information across the entire time domain, it fails to offer frequency information for specific local time segments.

For instance, two signals that appear very different in the time domain may look quite similar in the frequency domain when analyzed using the Fourier transform.

python

import numpy as np
import matplotlib.pyplot as plt
from scipy.fftpack import fft

# Function for Chinese font display
def set_ch():
    from pylab import mpl
    mpl.rcParams['font.sans-serif'] = ['FangSong']
    mpl.rcParams['axes.unicode_minus'] = False

set_ch()
t = np.linspace(0, 1, 400, endpoint=False)
cond = [t < 0.25, (t >= 0.25) & (t < 0.5), t >= 0.5]
f1 = lambda t: np.cos(2 * np.pi * 10 * t)
f2 = lambda t: np.cos(2 * np.pi * 50 * t)
f3 = lambda t: np.cos(2 * np.pi * 100 * t)
y1 = np.piecewise(t, cond, [f1, f2, f3])
y2 = np.piecewise(t, cond, [f2, f1, f3])
Y1 = abs(fft(y1))
Y2 = abs(fft(y2))

plt.figure(figsize=(12, 9))

plt.subplot(221)
plt.plot(t, y1)
plt.title('Signal 1 Time Domain')
plt.xlabel('Time/s')

plt.subplot(222)
plt.plot(range(400), Y1)
plt.title('Signal 1 Frequency Domain')
plt.xlabel('Frequency/Hz')

plt.subplot(223)
plt.plot(t, y2)
plt.title('Signal 2 Time Domain')
plt.xlabel('Time/s')

plt.subplot(224)
plt.plot(range(400), Y2)
plt.title('Signal 2 Frequency Domain')
plt.xlabel('Frequency/Hz')

plt.show()

To address this limitation, windowing (Short-Time Fourier Transform) can be employed, dividing long-duration signals into shorter, equal-length segments and then performing Fourier transforms on each window to obtain frequency changes over time.

python

import numpy as np
import matplotlib.pyplot as plt
import pywt

# Function for Chinese font display
def set_ch():
    from pylab import mpl
    mpl.rcParams['font.sans-serif'] = ['FangSong']
    mpl.rcParams['axes.unicode_minus'] = False

set_ch()
t = np.linspace(0, 1, 400, endpoint=False)
cond = [t < 0.25, (t >= 0.25) & (t < 0.5), t >= 0.5]

f1 = lambda t: np.cos(2 * np.pi * 10 * t)
f2 = lambda t: np.cos(2 * np.pi * 50 * t)
f3 = lambda t: np.cos(2 * np.pi * 100 * t)

y1 = np.piecewise(t, cond, [f1, f2, f3])
y2 = np.piecewise(t, cond, [f2, f1, f3])

cwtmatr1, freqs1 = pywt.cwt(y1, np.arange(1, 200), 'cgau8', 1 / 400)
cwtmatr2, freqs2 = pywt.cwt(y2, np.arange(1, 200), 'cgau8', 1 / 400)

plt.figure(figsize=(12, 9))

plt.subplot(221)
plt.plot(t, y1)
plt.title('Signal 1 Time Domain')
plt.xlabel('Time/s')

plt.subplot(222)
plt.contourf(t, freqs1, abs(cwtmatr1))
plt.title('Signal 1 Time-Frequency Relationship')
plt.xlabel('Time/s')
plt.ylabel('Frequency/Hz')

plt.subplot(223)
plt.plot(t, y2)
plt.title('Signal 2 Time Domain')
plt.xlabel('Time/s')

plt.subplot(224)
plt.contourf(t, freqs2, abs(cwtmatr2))
plt.title('Signal 2 Time-Frequency Relationship')
plt.xlabel('Time/s')
plt.ylabel('Frequency/Hz')

plt.tight_layout()
plt.show()

Wavelet transform overcomes the limitations of Fourier transform by providing both time and frequency information. It can identify not only the frequency components of a signal but also when these components occur.

Simple Wavelet Examples

Haar Wavelet Construction

The Haar wavelet has the following characteristics:

It is compactly supported in the time domain, with a non-zero interval of [0,1)
It belongs to orthogonal wavelets
It is symmetric, which helps eliminate phase distortion
It takes only +1 and -1, making calculations simple
It is a discontinuous wavelet with certain limitations in practical signal analysis and processing

Image Multi-resolution Analysis

Wavelet Multi-resolution

Multi-resolution analysis is a core concept in wavelet analysis. It represents a function as a combination of low-frequency components and high-frequency components at different resolutions. The properties of multi-resolution analysis include:

Monotonicity
Scalability
Translation invariance
Riesz basis

Image Pyramid

Image pyramids are used in machine vision or image compression, representing an image as a collection of gradually downsampled images. In multi-resolution analysis, each level contains an approximation image and a residual image, forming what is known as an image pyramid when combined across resolutions.

Image Subband Coding

In subband coding, an image is decomposed into multiple subbands, each of which is a frequency-band limited component. These subbands can be combined to reconstruct the original image without distortion, with each subband obtained by applying a bandpass filter to the input image.