使用低通頻率域濾波器平滑圖像

理想低通濾波器(ILPF)

在以原點為中心的一個圓內無衰減地通過所有頻率，而在這個圓外“截止”所有的頻率的二維低通濾波器。

$$H(u,v)=\{\begin{array}{ll}1,& D(u,v)\le D_0\\ 0,& D(u,v)>D_0\end{array}\text{\hspace{1em}(4.111)}$$

$$D(u,v)=[(u?P/2{)}^2+(v?Q/2{)}^2{]}^{1/2}\text{\hspace{1em}(4.12)}$$

$D_0$是一個正常數，控制圓的大小

def idea_low_pass_filter(source, center, radius=5):"""create idea low pass filter param: source: input, source imageparam: center: input, the center of the filter, where is the lowest value, (0, 0) is top left corner, source.shape[:2] is center of the source imageparam: radius: input, the radius of the lowest value, greater value, bigger blocker out range, if the radius is 0, then allvalue is 0return a [0, 1] value filter"""M, N = source.shape[1], source.shape[0]u = np.arange(M)v = np.arange(N)u, v = np.meshgrid(u, v)D = np.sqrt((u - center[1]//2)**2 + (v - center[0]//2)**2)D0 = radiuskernel = D.copy()kernel[D > D0] = 0kernel[D <= D0] = 1return kernel

def plot_3d(ax, x, y, z):ax.plot_surface(x, y, z, antialiased=True, shade=True)ax.view_init(20, 60), ax.grid(b=False), ax.set_xticks([]), ax.set_yticks([]), ax.set_zticks([])

# 理想低通濾波器 ILPF
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import cmcenter = img_ori.shape
ILPF = idea_low_pass_filter(img_ori, center, radius=50)# 用來繪制3D圖
M, N = img_ori.shape[1], img_ori.shape[0]
u = np.arange(M)
v = np.arange(N)
u, v = np.meshgrid(u, v)fig = plt.figure(figsize=(21, 7))
ax_1 = fig.add_subplot(1, 3, 1, projection='3d')
plot_3d(ax_1, u, v, ILPF)ax_2 = fig.add_subplot(1, 3, 2)
ax_2.imshow(ILPF,'gray'), ax_2.set_xticks([]), ax_2.set_yticks([])h = ILPF[img_ori.shape[0]//2:, img_ori.shape[1]//2]
ax_3 = fig.add_subplot(1, 3, 3)
ax_3.plot(h), ax_3.set_xticks([0, 50]), ax_3.set_yticks([0, 1]), ax_3.set_xlim([0, 320]), ax_3.set_ylim([0, 1.2])plt.tight_layout()
plt.show()

總圖像能量$P_T$是對填充零后圖像的功率譜的各個分量在點$(u,v)$處求和得到。
$$P_T=\sum _{u=0}^{P?1}\sum _{v=0}^{Q?1}P(u,v)\text{\hspace{1em}(4.113)}$$

半徑為$D_0$的圓將包含$a$%的功率
$$a=100[\sum _{u=0}\sum _{v=0}P(u,v)/P_T]\text{\hspace{1em}(4.114)}$$

def mask_ring(img_ori, d=10):ILPF_1 = idea_low_pass_filter(img_ori, img_ori.shape, radius=d)ILPF_2 = idea_low_pass_filter(img_ori, img_ori.shape, radius=d-1)ILPF = ILPF_1 - ILPF_2return ILPF

# 測試模型
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0441(a)(characters_test_pattern).tif', -1)
M, N = img_ori.shape[:2]# 填充
fp = pad_image(img_ori, mode='reflect')radius = [10, 30, 60, 160, 460]
mask = np.zeros(fp.shape)
for i in range(len(radius)):mask += mask_ring(fp, d=radius[i])plt.figure(figsize=(16, 8))
plt.subplot(1, 2, 1), plt.imshow(img_ori, cmap='gray'), plt.xticks([]), plt.yticks([])
plt.subplot(1, 2, 2), plt.imshow(mask, cmap='gray'), plt.xticks([]), plt.yticks([])
plt.tight_layout()
plt.show()

注：功率計算不太正確

#功率計算不太正確
# 填充
fp = pad_image(img_ori, mode='constant')
# 中心化
fp_cen = centralized_2d(fp)
# 正變換
fft = np.fft.fft2(fp_cen)
spectrum = spectrum_fft(fft)
# spectrum = np.log(1 + spectrum)
PT = spectrum.sum()ILPF = idea_low_pass_filter(fp, fp.shape, D0=10)fh = fft * ILPF
spectrum = spectrum_fft(fh)
# spectrum = np.log(1 + spectrum)
P = spectrum.sum()
print(f"Power is -> {100*(P / PT)}")

Power is -> 3.269204251436661

def ilpf_test(img_ori, mode='constant', radius=10):M, N = img_ori.shape[:2]# 填充fp = pad_image(img_ori, mode='reflect')# 中心化fp_cen = centralized_2d(fp)# 正變換fft = np.fft.fft2(fp_cen)# 濾波器H = idea_low_pass_filter(fp, center=fp.shape, radius=radius)# 濾波HF = fft * H# 反變換ifft = np.fft.ifft2(HF)# 去中心化gp = centralized_2d(ifft.real)# 還回來與原圖像的大小g = gp[:M, :N]dst = np.uint8(normalize(g) * 255)return dst

# 頻率域濾波過程
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0441(a)(characters_test_pattern).tif', -1)radius = [10, 30, 60, 160, 460]fig = plt.figure(figsize=(15, 10))
for i in range(len(radius)+1):ax = fig.add_subplot(2, 3, i+1)if i == 0:ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])else:img = ilpf_test(img_ori, radius=radius[i-1])ax.imshow(img, 'gray'), ax.set_title("radius = " + str(radius[i-1])), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()

# 頻率域ILPF傳遞函數對應的空間核函數
img_temp = np.zeros([1000, 1000])
ILPF = idea_low_pass_filter(img_temp, img_temp.shape, radius=15)
ifft = np.fft.ifft2(ILPF)
ifft = np.fft.ifftshift(ifft)
space = ifft.real * 1200
space_s = abs(space)
# space_s = np.clip(space, 0, space.max())
space_s = normalize(space_s)hx = space[:, 500]
hx = centralized_2d(hx.reshape(1, -1)).flatten()fig = plt.figure(figsize=(15, 5))
ax_1 = fig.add_subplot(1, 3, 1)
ax_1.imshow(ILPF, 'gray'), ax_1.set_xticks([]), ax_1.set_yticks([])ax_2 = fig.add_subplot(1, 3, 2)
ax_2.imshow(space_s, 'gray'), ax_2.set_xticks([]), ax_2.set_yticks([])ax_3 = fig.add_subplot(1, 3, 3)
ax_3.plot(hx), ax_3.set_xticks([]), ax_3.set_yticks([])plt.tight_layout()
plt.show()

高斯低通濾波器(GLPF)

$$H(u,v)=e^{?D^2(u,v)/2D_0^2}\text{\hspace{1em}(4.116)}$$
$D_0$是截止頻率。當$D(u,v)=D_0$時，GLPF傳遞函數下降到其最大值1.0的0.607。

def gauss_low_pass_filter(source, center, radius=5):"""create gauss low pass filter param: source: input, source imageparam: center: input, the center of the filter, where is the lowest value, (0, 0) is top left corner, source.shape[:2] is center of the source imageparam: radius: input, the radius of the lowest value, greater value, bigger blocker out range, if the radius is 0, then allvalue is 0return a [0, 1] value filter"""    M, N = source.shape[1], source.shape[0]u = np.arange(M)v = np.arange(N)u, v = np.meshgrid(u, v)D = np.sqrt((u - center[1]//2)**2 + (v - center[0]//2)**2)D0 = radiuskernel = np.exp(- (D**2)/(2*D0**2))return kernel

def plot_3d(ax, x, y, z):ax.plot_surface(x, y, z, antialiased=True, shade=True)ax.view_init(20, 60), ax.grid(b=False), ax.set_xticks([]), ax.set_yticks([]), ax.set_zticks([])

# 高斯低通濾波器 GLPF
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import cmcenter = img_ori.shapeGLPF_10 = gauss_low_pass_filter(img_ori, center, radius=10)
h_10 = GLPF_10[img_ori.shape[0]//2:, img_ori.shape[1]//2]
GLPF_20 = gauss_low_pass_filter(img_ori, center, radius=20)
h_20 = GLPF_20[img_ori.shape[0]//2:, img_ori.shape[1]//2]
GLPF_40 = gauss_low_pass_filter(img_ori, center, radius=40)
h_40 = GLPF_40[img_ori.shape[0]//2:, img_ori.shape[1]//2]
GLPF_60 = gauss_low_pass_filter(img_ori, center, radius=60)
h_60 = GLPF_60[img_ori.shape[0]//2:, img_ori.shape[1]//2]# 用來繪制3D圖
M, N = img_ori.shape[1], img_ori.shape[0]
u = np.arange(M)
v = np.arange(N)
u, v = np.meshgrid(u, v)fig = plt.figure(figsize=(21, 7))
ax_1 = fig.add_subplot(1, 3, 1, projection='3d')
plot_3d(ax_1, u, v, GLPF_60)ax_2 = fig.add_subplot(1, 3, 2)
ax_2.imshow(GLPF_60,'gray'), ax_2.set_xticks([]), ax_2.set_yticks([])ax_3 = fig.add_subplot(1, 3, 3)
ax_3.plot(h_10, label='$D_0=10$'), ax_3.set_xticks([0, 50]), ax_3.set_yticks([0, 1]), ax_3.set_xlim([0, 320]), ax_3.set_ylim([0, 1.2])
ax_3.plot(h_20, label='$D_0=20$')
ax_3.plot(h_40, label='$D_0=40$')
ax_3.plot(h_60, label='$D_0=60$')
plt.legend(loc='best')
plt.tight_layout()
plt.show()

def glpf_test(img_ori, mode='constant', radius=10):M, N = img_ori.shape[:2]# 填充fp = pad_image(img_ori, mode=mode)# 中心化fp_cen = centralized_2d(fp)# 正變換fft = np.fft.fft2(fp_cen)# 濾波器H = gauss_low_pass_filter(fp, center=fp.shape, radius=radius)# 濾波HF = fft * H# 反變換ifft = np.fft.ifft2(HF)# 去中心化gp = centralized_2d(ifft.real)# 還回來與原圖像的大小g = gp[:M, :N]dst = np.uint8(normalize(g) * 255)return dst

# 高斯低通濾波器在頻率域濾波的使用，這效果要比理想低通濾波器好很多，不會出現振鈴效應
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0441(a)(characters_test_pattern).tif', -1)radius = [10, 30, 60, 160, 460]fig = plt.figure(figsize=(15, 10))
for i in range(len(radius)+1):ax = fig.add_subplot(2, 3, i+1)if i == 0:ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])else:img = glpf_test(img_ori, mode='reflect', radius=radius[i-1])ax.imshow(img, 'gray'), ax.set_title("radius = " + str(radius[i-1])), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()

# 頻率域GLPF傳遞函數對應的空間核函數
img_temp = np.zeros([1000, 1000])
GLPF = gauss_low_pass_filter(img_temp, img_temp.shape, radius=15)
ifft = np.fft.ifft2(GLPF)
ifft = np.fft.ifftshift(ifft)
space = ifft.real * 1200
space_s = abs(space)
space_s = normalize(space_s)hx = space[:, 500]
hx = centralized_2d(hx.reshape(1, -1)).flatten()fig = plt.figure(figsize=(15, 5))
ax_1 = fig.add_subplot(1, 3, 1)
ax_1.imshow(GLPF, 'gray'), ax_1.set_xticks([]), ax_1.set_yticks([])ax_2 = fig.add_subplot(1, 3, 2)
ax_2.imshow(space_s, 'gray'), ax_2.set_xticks([]), ax_2.set_yticks([])ax_3 = fig.add_subplot(1, 3, 3)
ax_3.plot(hx), ax_3.set_xticks([]), ax_3.set_yticks([])plt.tight_layout()
plt.show()

# 不使用傳統方法
import cv2
import numpy as np
import matplotlib.pyplot as pltimg_ic = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0429(a)(blown_ic).tif', 0) #直接讀為灰度圖像plt.figure(figsize=(15, 12))
plt.subplot(221),plt.imshow(img_ic,'gray'),plt.title('origial'), plt.xticks([]), plt.yticks([])#--------------------------------
fft = np.fft.fft2(img_ic)
fft_shift = np.fft.fftshift(fft)
amp_img = np.abs(np.log(1 + np.abs(fft_shift)))
plt.subplot(222),plt.imshow(amp_img,'gray'),plt.title('IC FFT'), plt.xticks([]), plt.yticks([])#--------------------------------
glpf = gauss_low_pass_filter(img_ic, img_ic.shape, radius=20)
plt.subplot(223),plt.imshow(glpf,'gray'),plt.title('mask'), plt.xticks([]), plt.yticks([])#--------------------------------
f1shift = fft_shift * glpf
f2shift = np.fft.ifftshift(f1shift) #對新的進行逆變換
img_new = np.fft.ifft2(f2shift)#出來的是復數，無法顯示
img_new = np.abs(img_new)#調整大小范圍便于顯示
img_new = (img_new-np.amin(img_new))/(np.amax(img_new)-np.amin(img_new))
plt.subplot(224),plt.imshow(img_new,'gray'),plt.title('GLPF'), plt.xticks([]), plt.yticks([])plt.tight_layout()
plt.show()

巴特沃斯低通濾波器

$$H(u,v)=\frac{1}{1+[D(u,v)/D_0{]}^{2n}}\text{\hspace{1em}(4.117)}$$
$$D(u,v)=[(u?M/2{)}^2+(v?N/2{)}^2{]}^{1/2}$$

• 特點
  • 較高的$n$值來控制這個BLPF函數可逼近ILPF的特性
  • 較低的$n$值來控制這個BLPF函數可逼近GLPF的特性，同時提供從低頻到高頻的平滑過渡。
  • 可用BLPF以小得多的振鈴效應來逼近ILPF函數的清晰度

def butterworth_low_pass_filter(img, center, radius=5, n=1):"""create butterworth low pass filter param: source: input, source imageparam: center: input, the center of the filter, where is the lowest value, (0, 0) is top left corner, source.shape[:2] is center of the source imageparam: radius: input, the radius of the lowest value, greater value, bigger blocker out range, if the radius is 0, then allvalue is 0param: n: input, float, the order of the filter, if n is small, then the BLPF will be close to GLPF, and more smooth from lowfrequency to high freqency.if n is large, will close to ILPFreturn a [0, 1] value filter"""  epsilon = 1e-8M, N = img.shape[1], img.shape[0]u = np.arange(M)v = np.arange(N)u, v = np.meshgrid(u, v)D = np.sqrt((u - center[1]//2)**2 + (v - center[0]//2)**2)D0 = radiuskernel = (1 / (1 + (D / (D0 + epsilon))**(2*n)))return kernel

def plot_3d(ax, x, y, z):ax.plot_surface(x, y, z, antialiased=True, shade=True)ax.view_init(20, 60), ax.grid(b=False), ax.set_xticks([]), ax.set_yticks([]), ax.set_zticks([])

# 巴特沃斯低通濾波器 BLPF
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import cmcenter = img_ori.shapeBLPF_60_1 = butterworth_low_pass_filter(img_ori, center, radius=60, n=1)
h_1 = BLPF_60_1[img_ori.shape[0]//2:, img_ori.shape[1]//2]
BLPF_60_2 = butterworth_low_pass_filter(img_ori, center, radius=60, n=2)
h_2 = BLPF_60_2[img_ori.shape[0]//2:, img_ori.shape[1]//2]
BLPF_60_3 = butterworth_low_pass_filter(img_ori, center, radius=60, n=3)
h_3 = BLPF_60_3[img_ori.shape[0]//2:, img_ori.shape[1]//2]
BLPF_60_4 = butterworth_low_pass_filter(img_ori, center, radius=60, n=4)
h_4 = BLPF_60_4[img_ori.shape[0]//2:, img_ori.shape[1]//2]# 用來繪制3D圖
M, N = img_ori.shape[1], img_ori.shape[0]
u = np.arange(M)
v = np.arange(N)
u, v = np.meshgrid(u, v)fig = plt.figure(figsize=(21, 7))
ax_1 = fig.add_subplot(1, 3, 1, projection='3d')
plot_3d(ax_1, u, v, BLPF_60_1)ax_2 = fig.add_subplot(1, 3, 2)
ax_2.imshow(BLPF_60_1,'gray'), ax_2.set_xticks([]), ax_2.set_yticks([])ax_3 = fig.add_subplot(1, 3, 3)
ax_3.plot(h_1, label='$n=1$'), ax_3.set_xticks([0, 50]), ax_3.set_yticks([0, 1]), ax_3.set_xlim([0, 320]), ax_3.set_ylim([0, 1.2])
ax_3.plot(h_2, label='$n=2$')
ax_3.plot(h_3, label='$n=3$')
ax_3.plot(h_4, label='$n=4$')
plt.legend(loc='best')
plt.tight_layout()
plt.show()

def blpf_test(img_ori, mode='constant', radius=10, n=1):M, N = img_ori.shape[:2]# 填充fp = pad_image(img_ori, mode=mode)# 中心化fp_cen = centralized_2d(fp)# 正變換fft = np.fft.fft2(fp_cen)# 濾波器H = butterworth_low_pass_filter(fp, center=fp.shape, radius=radius, n=n)# 濾波HF = fft * H# 反變換ifft = np.fft.ifft2(HF)# 去中心化gp = centralized_2d(ifft.real)# 還回來與原圖像的大小g = gp[:M, :N]dst = np.uint8(normalize(g) * 255)return dst

# 巴特沃斯低通濾波器在頻率域濾波的使用
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0441(a)(characters_test_pattern).tif', -1)radius = [10, 30, 60, 160, 460]fig = plt.figure(figsize=(15, 10))
for i in range(len(radius)+1):ax = fig.add_subplot(2, 3, i+1)if i == 0:ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])else:img = blpf_test(img_ori, mode='reflect', radius=radius[i-1], n=2.25)ax.imshow(img, 'gray'), ax.set_title("radius = " + str(radius[i-1])), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()

空間域的一階巴特沃斯沒有振鈴效應。在2階和3階濾波器中，振鈴效應通常難以察覺，但更高階濾波器中的振鈴效應很明顯。

# 頻率域GLPF傳遞函數對應的空間核函數
img_temp = np.zeros([1000, 1000])
BLPF = butterworth_low_pass_filter(img_temp, img_temp.shape, radius=15, n=25)
ifft = np.fft.ifft2(BLPF)
ifft = np.fft.ifftshift(ifft)
space = ifft.real * 1200
space_s = abs(space)
space_s = normalize(space_s)hx = space[:, 500]
hx = centralized_2d(hx.reshape(1, -1)).flatten()fig = plt.figure(figsize=(15, 5))
ax_1 = fig.add_subplot(1, 3, 1)
ax_1.imshow(GLPF, 'gray'), ax_1.set_xticks([]), ax_1.set_yticks([])ax_2 = fig.add_subplot(1, 3, 2)
ax_2.imshow(space_s, 'gray'), ax_2.set_xticks([]), ax_2.set_yticks([])ax_3 = fig.add_subplot(1, 3, 3)
ax_3.plot(hx), ax_3.set_xticks([]), ax_3.set_yticks([])plt.tight_layout()
plt.show()

def frequen2spatial(filter):ifft = np.fft.ifft2(filter)ifft = np.fft.ifftshift(ifft)spatial = ifft.real * 1200spatial_s = abs(spatial)spatial_s = normalize(spatial_s)return spatial, spatial_s

# 頻率域BLPF傳遞函數對應的空間核函數
img_temp = np.zeros([1000, 1000])
n = [1, 2, 5, 20]fig = plt.figure(figsize=(20, 10))for i in range(len(n)):# 這是顯示空間域的核ax = fig.add_subplot(2, 4, i+1)BLPF = butterworth_low_pass_filter(img_temp, img_temp.shape, radius=15, n=n[i])spatial, spatial_s = frequen2spatial(BLPF)ax.imshow(spatial_s, 'gray'), ax.set_xticks([]), ax.set_yticks([])# 這里顯示是對應的空間域核水平掃描線的灰度分布ax = fig.add_subplot(2, 4, i+5)hx = spatial[:, 500]hx = centralized_2d(hx.reshape(1, -1)).flatten()ax.plot(hx, label=f'n = {n[i]}'), ax.set_xticks([]), ax.set_yticks([])ax.legend(loc='best', fontsize=14)
plt.tight_layout()
plt.show()

低通濾波的例子

出下圖，我們可以清晰看到不同的截止頻率的核對圖像的平滑效果。我們杺選擇合適的核，以平滑圖像，再加上其它圖像處理技術，以達到想要的效果。
如下文字處理的例子，我們可以利用$D_0=20$，來平滑圖像，再經過閾值處理，可以得到文字的蒙板。

# 高斯低通濾波器在印刷和出版業的應用
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0419(a)(text_gaps_of_1_and_2_pixels).tif', -1)radius = [10, 30, 60, 90, 120]fig = plt.figure(figsize=(17, 10))
for i in range(len(radius)+1):ax = fig.add_subplot(2, 3, i+1)if i == 0:ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])else:img = glpf_test(img_ori, mode='reflect', radius=radius[i-1])ax.imshow(img, 'gray'), ax.set_title("radius = " + str(radius[i-1])), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()

# 高斯低通濾波器在印刷和出版業的應用
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0419(a)(text_gaps_of_1_and_2_pixels).tif', -1)radius = [10, 30, 60, 90, 120]fig = plt.figure(figsize=(17, 10))
ax = fig.add_subplot(1, 3, 1)
ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])
ax = fig.add_subplot(1, 3, 2)
img = glpf_test(img_ori, mode='reflect', radius=20)
ax.imshow(img, 'gray'), ax.set_title("radius = " + str(20)), ax.set_xticks([]), ax.set_yticks([])
ax = fig.add_subplot(1, 3, 3)
ret, img_thred = cv2.threshold(img, 0, 255, cv2.THRESH_OTSU + cv2.THRESH_BINARY_INV)
ax.imshow(img_thred, 'gray'), ax.set_title("Thred"), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()

# 高斯低通濾波器在印刷和出版業的應用，平滑后的圖像看上去更柔和、更美觀
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0427(a)(woman).tif', -1)radius = [10, 50, 80, 130, 150]fig = plt.figure(figsize=(17, 10))
for i in range(len(radius)+1):ax = fig.add_subplot(2, 3, i+1)if i == 0:ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])else:img = glpf_test(img_ori, mode='reflect', radius=radius[i-1])ax.imshow(img, 'gray'), ax.set_title("radius = " + str(radius[i-1])), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()

# 高斯低通濾波器在衛星圖像的應用，這里對圖像的濾波的目的是尺可能模糊更多的細節，而保留可識別的大特征。
img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0451(a)(satellite_original).tif', -1)radius = [10, 20, 50, 80, 100]fig = plt.figure(figsize=(17, 10))
for i in range(len(radius)+1):ax = fig.add_subplot(2, 3, i+1)if i == 0:ax.imshow(img_ori, 'gray'), ax.set_title('Original'), ax.set_xticks([]), ax.set_yticks([])else:img = glpf_test(img_ori, mode='reflect', radius=radius[i-1])ax.imshow(img, 'gray'), ax.set_title("radius = " + str(radius[i-1])), ax.set_xticks([]), ax.set_yticks([])
plt.tight_layout()
plt.show()