图像处理与傅里叶变换
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图像处理与傅里叶变换
一维傅里叶变换
- 连续域
F ( w ) = ∫ − ∞ ∞ f ( x ) e − j 2 π w x d x F(w) = \\int_-\\infty^\\infty f(x) e^-j2\\pi wx dx F(w)=∫−∞∞f(x)e−j2πwxdx
f ( x ) = ∫ − ∞ ∞ F ( w ) e j 2 π w x d w f(x) = \\int_-\\infty^\\infty F(w)e^j2\\pi wxdw f(x)=∫−∞∞F(w)ej2πwxdw
- 离散域
F ( k ) = 1 N ∑ x = 0 N − 1 f ( x ) e − j 2 π k x / N , k = 0 , 1 , 2 , . . . N − 1 F(k) = \\frac1N\\sum_x=0^N-1f(x)e^-j2\\pi kx/N, \\quad k=0,1,2,...N-1 F(k)=N1x=0∑N−1f(x)e−j2πkx/N,k=0,1,2,...N−1
f ( x ) = ∑ k = 0 N − 1 F ( k ) e j 2 π k x / N , x = 0 , 1 , 2 , . . . N − 1 f(x) = \\sum_k=0^N-1F(k)e^j2\\pi kx/N, \\quad x=0,1,2,...N-1 f(x)=k=0∑N−1F(k)ej2πkx/N,x=0,1,2,...N−1
二维傅里叶变换
- 连续域
F ( u , v ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 j π ( x u + y v ) d x d y F(u, v) = \\int_-\\infty^\\infty \\int_-\\infty^\\infty f(x, y) e^-2j \\pi (xu + yv) dxdy F(u,v)=∫−∞∞∫−∞∞f(x,y)e−2jπ(xu+yv)dxdy
f ( x , y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ F ( u , v ) e 2 j π ( x u + y v ) d u d v f(x, y) = \\int_-\\infty^\\infty \\int_-\\infty^\\infty F(u, v) e^2j \\pi (xu + yv) dudv f(x,y)=∫−∞∞∫−∞∞F(u,v)e2jπ(xu+yv)dudv
- 离散域
F ( u , v ) = 1 M N ∑ x = 0 N − 1 ∑ y = 0 M − 1 f ( x , y ) e − 2 j π [ x u N + y v M ] F(u, v) = \\frac1MN\\sum_x=0^N-1 \\sum_y=0^M-1f(x, y) e^-2j \\pi [\\fracxuN + \\fracyvM] F(u,v)=MN1x=0∑N−1y=0∑M−1f(x,y)e−2jπ[Nxu+Myv]
f ( x , y ) = ∑ x = 0 N − 1 ∑ y = 0 M − 1 F ( u , v ) e 2 j π [ x u N + y v M ] f(x, y) = \\sum_x=0^N-1\\sum_y=0^M-1 F(u, v) e^2j \\pi [\\fracxuN + \\fracyvM] f(x,y)=x=0∑N−1y=0∑M−1F(u,v)e2jπ[Nxu+Myv]
x = 0 , 1 , 2 , . . . N − 1 , y = 0 , 1 , 2 , . . . M − 1 u = 0 , 1 , 2 , . . . N − 1 , v = 0 , 1 , 2 , . . . M − 1 x = 0, 1, 2, ... N-1, \\quad y = 0, 1, 2,... M-1 \\\\ u = 0, 1, 2, ... N-1, \\quad v = 0, 1, 2,... M-1 x=0,1,2,...N−1,y=0,1,2,...M−1u=0,1,2,...N−1,v=0,1,2,...M−1
# -*- coding: utf-8 -*-
"""
Created on Sun Oct 10 16:05:40 2021
@author: shiyi
"""
import numpy as np
import matplotlib.pyplot as plt
PI = np.pi
N = 256
x = np.linspace(0, 1, N)
y = np.linspace(0, 1, N)
X,Y = np.meshgrid(x, y)
wx = 10 * PI
wy = 0 * PI
Z = np.sin(wx * X + wy * Y)
fft_ = np.fft.fft2(Z)
fshift_ = np.fft.fftshift(fft_)
mag_fft = (np.abs(fshift_))
plt.figure(figsize=(10, 8))
plt.subplot(3, 2, 1)
plt.imshow(Z, cmap='gray')
plt.xlabel('wx=10PI, yx=0')
plt.title('Z = sin(wx * X + wy * Y)')
plt.subplot(3, 2, 2)
plt.imshow(mag_fft, cmap='gray')
plt.title('FT of Z')
wx = 0 * PI
wy = 10 * PI
Z = np.sin(wx * X + wy * Y)
fft_ = np.fft.fft2(Z)
fshift_ = np.fft.fftshift(fft_)
mag_fft = (np.abs(fshift_))
plt.subplot(3, 2, 3)
plt.imshow(Z, cmap='gray')
plt.xlabel('wx=0, yx=10PI')
# plt.title('Z = sin(wx * X + wy * Y)')
plt.subplot(3, 2, 4)
plt.imshow(mag_fft, cmap='gray')
# plt.title('FT of Z')
wx = 10 * PI
wy = 10 * PI
Z = np.sin(wx * X + wy * Y)
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