Image Transformation in MATLAB: Fourier Transform and Inverse Fourier Transform

目录
2.1 Definition and Properties of Fourier Transform

2.1.1 Definition of Fourier Transform
2.1.2 Properties of Fourier Transform

2. Fourier Transform Theory

2.1 Definition and Properties of Fourier Transform

2.1.1 Definition of Fourier Transform
2.1.2 Properties of Fourier Transform

2.2 Applications of Fourier Transform

2.2.1 Image Enhancement
2.2.2 Image Denoising

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2.1.1 Definition of Fourier Transform
The Fourier Transform is a mathematical transformation that converts time-domain signals (images) into frequency-domain representations. For a two-dimensional image f(x, y), its Fourier Transform F(u, v) is defined as:
F(u, v) = ∫∫ f(x, y) e^(-2πi(ux + vy)) dx dy登录后复制
Here, (u, v) represents the frequency domain coordinates, and i is the imaginary unit.
2.1.2 Properties of Fourier Transform
The Fourier Transform has several important properties:
*Linearity: The Fourier Transform is linear, meaning for any two images f(x, y) and g(x, y), and constants a and b, the following holds true:
F(a f(x, y) + b g(x, y)) = a F(f(x, y)) + b F(g(x, y))登录后复制
*Translation Invariance: The Fourier Transform is invariant to translations of the image. For an image translated by (a, b) as f(x - a, y - b), its Fourier Transform is:
F(f(x - a, y - b)) = F(f(x, y)) e^(-2πi(ua + vb))登录后复制
*Rotation Invariance: The Fourier Transform is invariant to rotations of the image. For an image rotated by θ degrees as f(x cos θ - y sin θ, x sin θ + y cos θ), its Fourier Transform is:
F(f(x cos θ - y sin θ, x sin θ + y cos θ)) = F(f(x, y)) e^(-2πiθ(u + v))登录后复制
2. Fourier Transform Theory
The Fourier Transform is a mathematical transform that converts a time-domain signal (an image) into a frequency-domain representation. It decomposes the image into a series of sine and cosine waves, each with its own frequency and amplitude.
2.1 Definition and Properties of Fourier Transform
2.1.1 Definition of Fourier Transform
For a continuous function f(x), its Fourier Transform F(u) is defined as:
F(u) = ∫_{-\infty}^{\infty} f(x) e^(-2πiux) dx登录后复制
Where:

u is the frequency variable
i is the imaginary unit

2.1.2 Properties of Fourier Transform
The Fourier Transform has the following properties:
*Linearity: F(af(x) + bg(x)) = aF(f(x)) + bF(g(x))
*Time Shift: F(f(x - a)) = e^(-2πiua) F(f(x))
*Frequency Shift: F(f(x)e^(2πiax)) = F(f(x)) * e^(-2πiua)
*Convolution: F(f(x) * g(x)) = F(f(x)) * F(g(x))
*Autocorrelation: F(f(x) * f(-x)) = |F(f(x))|^2
2.2 Applications of Fourier Transform
The Fourier Transform has a wide range of applications in image processing, including:
2.2.1 Image Enhancement
The Fourier Transform can be used to enhance the contrast, brightness, and sharpness of an image. By adjusting the amplitude and phase of different frequency components in the frequency domain, specific features in the image can be enhanced.
2.2.2 Image Denoising
The Fourier Transform can be utilized to remove noise from an image. Noise usually manifests as an increase in the high-frequency components in the frequency domain. By filtering out these high-frequency components, noise can be effectively removed.
Code Block:
import numpy as npimport cv2# Reading the imageimage = cv2.imread('image.jpg')# Fourier TransformF = np.fft.fft2(image)# Shift the zero frequency component to the center of the spectrumF_shifted = np.fft.fftshift(F)# Filter out high-frequency noiseF_filtered = F_shifted.copy()F_filtered[np.abs(F_shifted) < 10] = 0# Shift the spectrum back to the original positionF_unshifted = np.fft.ifftshift(F_filtered)# Inverse Fourier Transformimage_denoised = np.fft.ifft2(F_unshifted)# Display the denoised imagecv2.imshow('Denoised Image', np.abs(image_denoised))cv2.waitKey(0)登录后复制
Logical Analysis: