数字图像处理-傅里叶变换系列编程
问题
问题1
通过计算一维傅里叶变换实现图像二维快速傅里叶变换
代码实现
def dft2D(f):
h,w = f.shape
F = np.zeros(f.shape, dtype=complex)
for i in range(h):
F[i,:] = np.fft.fft(f[i,:])
for i in range(w):
F[:,i] = np.fft.fft(F[:,i])
return F
问题2
图像二维快速傅里叶逆变换
代码实现
def idft2D(F):
h,w = F.shape
F1 = np.conj(F)
f = np.zeros(F1.shape,dtype=complex)
for i in range(h):
f[i,:] = np.fft.fft(F1[i,:])
for i in range(w):
f[:,i] = np.fft.fft(f[:,i])
f = f/(h*w)
f = np.conj(f)
f = np.abs(f)
return f
问题3
测试图像二维快速傅里叶变换与逆变换
代码
import cv2
import numpy as np
#归一化
def normalization(data):
_range = np.max(data) - np.min(data)
return (data - np.min(data)) / _range
def dft2D(f):
h,w = f.shape
F = np.zeros(f.shape, dtype=complex)
for i in range(h):
F[i,:] = np.fft.fft(f[i,:])
for i in range(w):
F[:,i] = np.fft.fft(F[:,i])
return F
def idft2D(F):
h,w = F.shape
F1 = np.conj(F)
f = np.zeros(F1.shape,dtype=complex)
for i in range(h):
f[i,:] = np.fft.fft(F1[i,:])
for i in range(w):
f[:,i] = np.fft.fft(f[:,i])
f = f/(h*w)
f = np.conj(f)
f = np.abs(f)
return f
if __name__ == '__main__':
f1 = cv2.imread("rose512.tif",cv2.IMREAD_GRAYSCALE)
f = normalization(f1)
F = dft2D(f)
g = idft2D(F)
d = np.abs(f-g)
cv2.imshow('original', f1)
cv2.imshow('g', g)
cv2.imshow('d', d)
cv2.waitKey(0)
cv2.destroyAllWindows()
结果展示
问题4
计算图像的中心化二维快速傅里叶变换与谱图像
问题分析
首先合成矩形图像，测试结果如下图所示，其长为 60，宽为 10。
代码
import numpy as np
import cv2
import Q3
if __name__ == '__main__':
img = np.zeros((512,512))
img[512//2-30:512//2+30,512//2-5:512//2+5]=1
cv2.imshow('img', img)
F1 = Q3.dft2D(img)
F1 = Q3.normalization(np.abs(F1))
img_c = np.zeros(img.shape)
cv2.imshow('F1', F1)
for i in range (img.shape[0]):
for j in range (img.shape[1]):
img_c[i,j] = img[i,j]*((-1)**(i+j))
F2 = Q3.dft2D(img_c)
cv2.imshow('F2', Q3.normalization(np.abs(F2)))
F3 = np.log(1+np.abs(F2))
cv2.imshow('F3', Q3.normalization(F3))
cv2.waitKey(0)
cv2.destroyAllWindows()
结果展示
选做题
计算其他 5 幅 图像的二维快速傅里叶变换 ： house.tif, house02.tif, lena_gray_512.tif, lunar_surface.tif, characters_test_pattern.tif。注意，有些图像的尺寸不是 2 的整数次幂，需要进行相应的像素填补处理。如果图像有多个通道可以选择其中的一个通道进行计算。
问题分析
代码实现
import numpy as np
import cv2
import Q3
def isPower(n):
if n < 1:
return False
i = 1
while i <= n:
if i == n:
return True
i <<= 1
return False
def pad(img):
if isPower(img.shape[0]) and isPower(img.shape[1]):
return img
h = w = 0
i = 1
while True:
if 2**i>=img.shape[0] and h==0:
h = 2**i
if 2**i>=img.shape[1] and w==0:
w = 2**i
if h!=0 and w!=0:
break
i=i+1
img2 = np.zeros((h,w))
img2[(h-img.shape[0])//2:h-(h-img.shape[0])//2,(w-img.shape[1])//2:w-(w-img.shape[1])//2] = img
return img2
def center(f):
img_c = np.zeros(f.shape)
for i in range(f.shape[0]):
for j in range(f.shape[1]):
img_c[i, j] = f[i, j] * ((-1) ** (i + j))
return img_c
if __name__ == '__main__':
f1 = cv2.imread("house.tif", cv2.IMREAD_GRAYSCALE)
f2 = cv2.imread("house02.tif", cv2.IMREAD_GRAYSCALE)
f3 = cv2.imread("lena_gray_512.tif",cv2.IMREAD_GRAYSCALE)
f4 = cv2.imread("lunar_surface.tif", cv2.IMREAD_GRAYSCALE)
f5 = cv2.imread("Characters_test_pattern.tif", cv2.IMREAD_GRAYSCALE)
f11 = pad(Q3.normalization(f1))
F11 = Q3.dft2D(f11)
F11 = Q3.normalization(np.abs(F11))
f12 = center(f11)
F12 = Q3.dft2D(f12)
F13 = np.log(1 + np.abs(F12))
cv2.imshow('orign_house', cv2.resize(Q3.normalization(f11), (f11.shape[0]//2,f11.shape[1]//2)) )
cv2.imshow('F1_house',cv2.resize(Q3.normalization(F11) , (F11.shape[0]//2,F11.shape[1]//2)))
cv2.imshow('F2_house', cv2.resize(Q3.normalization(np.abs(F12)), (F12.shape[0]//2,F12.shape[1]//2)))
cv2.imshow('F3_house',cv2.resize( Q3.normalization(F13), (F13.shape[0]//2,F13.shape[1]//2)))
f21 = pad(Q3.normalization(f2))
F21 = Q3.dft2D(f21)
F21 = Q3.normalization(np.abs(F21))
f22 = center(f21)
F22 = Q3.dft2D(f22)
F23 = np.log(1 + np.abs(F22))
cv2.imshow('orign_house02', cv2.resize(Q3.normalization(f21), (f21.shape[0]//2,f21.shape[1]//2)) )
cv2.imshow('F1_house02',cv2.resize(Q3.normalization(F21) , (F21.shape[0]//2,F21.shape[1]//2)))
cv2.imshow('F2_house02',cv2.resize(Q3.normalization(np.abs(F22)), (F22.shape[0]//2,F22.shape[1]//2)))
cv2.imshow('F3_house02',cv2.resize( Q3.normalization(F23), (F23.shape[0]//2,F23.shape[1]//2)))
f31 = pad(Q3.normalization(f3))
F31 = Q3.dft2D(f31)
F31 = Q3.normalization(np.abs(F31))
f32 = center(f31)
F32 = Q3.dft2D(f32)
F33 = np.log(1 + np.abs(F32))
cv2.imshow('orign_lena_gray_512',cv2.resize(Q3.normalization(f31), (f31.shape[0]//2,f31.shape[1]//2)) )
cv2.imshow('F1_lena_gray_512', cv2.resize(Q3.normalization(F31) , (F31.shape[0]//2,F31.shape[1]//2)))
cv2.imshow('F2_lena_gray_512', cv2.resize(Q3.normalization(np.abs(F32)), (F32.shape[0]//2,F32.shape[1]//2)))
cv2.imshow('F3_lena_gray_512', cv2.resize( Q3.normalization(F33), (F33.shape[0]//2,F33.shape[1]//2)))
f41 = pad(Q3.normalization(f4))
F41 = Q3.dft2D(f41)
F41 = Q3.normalization(np.abs(F41))
f42 = center(f41)
F42 = Q3.dft2D(f42)
F43 = np.log(1 + np.abs(F42))
cv2.imshow('orign_lunar_surface', cv2.resize(Q3.normalization(f41), (f41.shape[1]//2,f41.shape[0]//2)) )
cv2.imshow('F1_lunar_surface', cv2.resize(Q3.normalization(F41) , (F41.shape[1]//2,F41.shape[0]//2)))
cv2.imshow('F2_lunar_surface', cv2.resize(Q3.normalization(np.abs(F42)), (F42.shape[1]//2,F42.shape[0]//2)))
cv2.imshow('F3_lunar_surface',cv2.resize( Q3.normalization(F43), (F43.shape[1]//2,F43.shape[0]//2)))
f51 = pad(Q3.normalization(f5))
F51 = Q3.dft2D(f51)
F51 = Q3.normalization(np.abs(F51))
f52 = center(f51)
F52 = Q3.dft2D(f52)
F53 = np.log(1 + np.abs(F52))
cv2.imshow('orign_Characters_test_pattern', cv2.resize(Q3.normalization(f51), (f51.shape[1]//2,f51.shape[0]//2)) )
cv2.imshow('F1_Characters_test_pattern', cv2.resize(Q3.normalization(F51) , (F51.shape[1]//2,F51.shape[0]//2)))
cv2.imshow('F2_Characters_test_pattern', cv2.resize(Q3.normalization(np.abs(F52)), (F52.shape[1]//2,F52.shape[0]//2)))
cv2.imshow('F3_Characters_test_pattern',cv2.resize( Q3.normalization(F53), (F53.shape[1]//2,F53.shape[0]//2)))
cv2.waitKey(0)
cv2.destroyAllWindows()
结果展示
