Image Transformation in MATLAB: Fourier Transform and Inverse Fourier Transform

# 2.1 Definition and Properties of Fourier Transform

### 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

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))

***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))

***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))

# 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

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 np
import cv2

# Reading the image
image = cv2.imread('image.jpg')

# Fourier Transform
F = np.fft.fft2(image)

# Shift the zero frequency component to the center of the spectrum
F_shifted = np.fft.fftshift(F)

# Filter out high-frequency noise
F_filtered = F_shifted.copy()
F_filtered[np.abs(F_shifted) < 10] = 0

# Shift the spectrum back to the original position
F_unshifted = np.fft.ifftshift(F_filtered)

# Inverse Fourier Transform
image_denoised = np.fft.ifft2(F_unshifted)

# Display the denoised image
cv2.imshow('Denoised Image', np.abs(image_denoised))
cv2.waitKey(0)

**Logical Analysis:**

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