Digital image processing (spatial filtering)
* Spatial filtering *
It is an area to pixel operation. A filter is applied to each pixel, and a product-sum operation is performed to smooth the image (noise removal) and detect edges. For the image, apply the filter at high speed like this video, and input the calculated value obtained from the filter to the case pixel. (Also called convolution)
Smoothing (reducing the unevenness of pixel values)
* Image to use *
Library used
python
import numpy as np
from numpy.lib.stride_tricks import as_strided
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import axes3d
import matplotlib.gridspec as gridspec
import seaborn as sns
import math
import cv2
* Average filter *
The average of the pixel values in the region is taken as the value of the pixel of interest.
python
av_filter = np.array([[ 1/25,1/25,1/25,1/25,1/25,],
[ 1/25,1/25,1/25,1/25,1/25,],
[ 1/25,1/25,1/25,1/25,1/25,],
[ 1/25,1/25,1/25,1/25,1/25,],
[ 1/25,1/25,1/25,1/25,1/25,]], np.float32)
# ddepth = -1, means destination image has depth same as input image
dst1 = cv2.filter2D(image_1, -1, av_filter)
cv2.imwrite('2_av_fil.jpg', dst1)
↓ It is smoothed as a whole.
* Weighted average filter *
Also called * weighted averaging *, a distribution centered on the pixel of interest, with the weight decreasing toward the outside.
python
add_av_fil = np.array([[1, 4, 6, 4, 1],
[4, 16, 24, 16, 4],
[6, 24, 36, 24, 6],
[4, 16, 24, 16, 4],
[1, 4, 6, 4, 1]], np.float32)/256
dst1 = cv2.filter2D(res, -1, add_av_fil)
cv2.imwrite('3_add_av_fil.jpg', dst1)
↓ It is relatively smoothed while preserving the edges.
* Gaussian filter *
Weighted according to Gaussian distribution.
python
def gausian(X, y, siguma):
# exp(-(x2 + y2)/2σ2)
h2 = math.e**(-1*((X**2) + (y**2))/(2*(siguma**2)))
# 2πσ2
h3 = 2*math.pi*(siguma**2)
h = h2/h3
return h
sns.set_style('ticks')
fig = plt.figure(figsize=(10,8))
ax = fig.add_subplot(111, projection='3d')
x = y = np.arange(-3,3,0.1)
X,Y = np.meshgrid(x,y)
ax.plot_surface(X,Y, gausian(X,Y,1),cmap='coolwarm')
plt.show()
plt.savefig("gausian_fig.jpg ")
↓ Such an image
Handmade
python
def gausian_kernel(size):
if size%2==0:
print('kernel size should be odd')
return
sigma = (size-1)/2
# [0,size]→[-sigma, sigma]Shift
x = y = np.arange(0,size) - sigma
X,Y = np.meshgrid(x,y)
print(sigma)
mat = gausian(X,Y,sigma)
#So that the sum is 1
kernel = mat / np.sum(mat)
return mat
g_fil_hand = gausian_kernel(3)
#Filter shape
#[[0.05854983 0.09653235 0.05854983]
#[0.09653235 0.15915494 0.09653235]
#[0.05854983 0.09653235 0.05854983]]
dst1 = cv2.filter2D(res, -1, g_fil_hand)
cv2.imwrite('4_g_hand_fil.jpg', dst1)
by opencv
python
sigma = 2
blur = cv2.GaussianBlur(res, (0, 0), sigmaX = 3, sigmaY = 3)
cv2.imwrite('5_g_cv_fil.jpg', blur)
* Smoothing in a specific direction *
By using the following filters, smoothing is performed only in a specific direction. The picture looks like it's shaken sideways.
python
flat_d_fil = np.zeros([15, 15])
for i in range(15):
flat_d_fil[i, i] = 1/15
dst1 = cv2.filter2D(res, -1, flat_d_fil)
cv2.imwrite('6_shaky_fil.jpg', dst1)
* Derivative filter *
For vertical edges
python
diff_col = np.array([[0, 0, 0],
[1, 0, -1],
[0, 0, 0]])
dst1 = cv2.filter2D(res, -1, 8*diff_col)
cv2.imwrite('7_diff_col.jpg', dst1)
For horizontal edges
python
diff_row = np.array([[0, -1, 0],
[0, 0, 0],
[0, 1, 0]])
dst1 = cv2.filter2D(res, -1, 8*diff_row)
cv2.imwrite('8_diff_row.jpg', dst1)
* Pruwit filter *
Combination of differential filter and average filter Pick up the gradient of the pixel value in the horizontal direction + Smooth the gradient of the pixel value in the vertical direction
python
pulu_fil = np.array([[1, 0, -1],
[1, 0, -1],
[1, 0, -1]])
dst1 = cv2.filter2D(res, -1, 2*pulu_fil)
cv2.imwrite('9_pulu_fil.jpg', dst1)
* Sobel filter *
Combination of differential filter and weighted average filter Achieves a smoother degree of blurring than Pruwit. Pick up the gradient of the pixel value in the horizontal direction + Smooth the gradient of the pixel value in the vertical direction
python
sobel_fil = np.array([[1, 0, -1],
[2, 0, -2],
[1, 0, -1]])
dst1 = cv2.filter2D(res, -1, sobel_fil)
cv2.imwrite('10_sobel_fil.jpg', dst1)
* Second derivative filter *
Feeling that the differential filter is doubled
python
todiff_fil = np.array([[0, 0, 0],
[1, -2, 1],
[0, 0, 0]])
dst1 = cv2.filter2D(res, -1, 2*todiff_fil)
cv2.imwrite('11_to_diff_fil.jpg', dst1)
* Laplacian filter *
It feels like the vertical + horizontal of the second derivative.
python
lap_fil = np.array([[0, 1, 0],
[1, -4, 1],
[0, 1, 0]])
dst1 = cv2.filter2D(res,ddepth=cv2.CV_16S,kernel=lap_fil)
cv2.imwrite('12_lap_hand_fil.jpg', dst1+100)
plt.plot(ds[400])
plt.savefig("lap_fig_1.jpg ")
with opencv
python
ddepth = cv2.CV_16S
kernel_size = 3
dst = cv2.Laplacian(res, ddepth, ksize=kernel_size)
cv2.imwrite('13_lap_cv_fil.jpg', dst)
plt.plot(dst[400]+100)
plt.savefig("lap_fig_2.jpg ")
* Log filter (Laplacian Gaussian) *
Handmade
python
def Log_fil(X, y, siguma):
# X2+y2-2σ2
h1 = ((X**2) + (y**2) - (2*(siguma**2)))
# exp(-(x2 + y2)/2σ2)
h2 = math.e**(-1*((X**2) + (y**2))/(2*(siguma**2)))
# 2πσ6
h3 = 2*math.pi*(siguma**6)
h = h1*h2/h3
return h
sns.set_style('ticks')
fig = plt.figure(figsize=(10,8))
ax = fig.add_subplot(111, projection='3d')
x = y = np.arange(-3,3,0.1)
X,Y = np.meshgrid(x,y)
ax.plot_surface(X,Y,Log_fil(X,Y,1),cmap='coolwarm')
plt.savefig("log_fig.jpg ")
plt.show()
python
def L_of_g_kernel(size, sigma):
if size%2==0:
print('kernel size should be odd')
return
ad_size = (size-1)/2
# [0,size]→[-sigma, sigma]Shift
x = y = np.arange(0,size) - ad_size
X,Y = np.meshgrid(x,y)
print(sigma)
mat = Log_fil(X,Y,sigma)
print(mat)
#So that the sum is 0
kernel = mat - np.sum(mat)/size**2
return kernel
log_fil = L_of_g_kernel(5, 1)
#Filter shape
#[[ 0.01749015 0.0391927 0.04307856 0.0391927 0.01749015]
# [ 0.0391927 0. -0.09653235 0. 0.0391927 ]
# [ 0.04307856 -0.09653235 -0.31830989 -0.09653235 0.04307856]
# [ 0.0391927 0. -0.09653235 0. 0.0391927 ]
# [ 0.01749015 0.0391927 0.04307856 0.0391927 0.01749015]]
python
dst1 = cv2.filter2D(res,ddepth=cv2.CV_16S,kernel=log_fil)
cv2.imwrite('14_log_fil_hand.jpg', dst1*20+200)
plt.savefig("log_fill_1.jpg ")
plt.plot(dst1[400]*20+200)
by opencv
python
ddepth = cv2.CV_16S
kernel_size = 5
# Apply Gaussian Blur
blur = cv2.GaussianBlur(res,(5, 5), sigmaX = 1, sigmaY = 1)
# Apply Laplacian operator in some higher datatype
dst = cv2.Laplacian(blur, ddepth, ksize=kernel_size)
cv2.imwrite('15_log_fil_cv.jpg', dst+200)
plt.plot(dst[400])
plt.savefig("log_fill_2.jpg ")
Extraction of 0 intersections
python
def Zero_crossing(image, thresh):
z_c_image = np.zeros(image.shape)
# For each pixel, count the number of positive
# and negative pixels in the neighborhood
for i in range(1, image.shape[0] - 1):
for j in range(1, image.shape[1] - 1):
negative_count = 0
positive_count = 0
neighbour = [image[i+1, j-1],image[i+1, j],image[i+1, j+1],image[i, j-1],image[i, j+1],image[i-1, j-1],image[i-1, j],image[i-1, j+1]]
d = max(neighbour)
e = min(neighbour)
for h in neighbour:
if h>0:
positive_count += 1
elif h<0:
negative_count += 1
# If both negative and positive values exist in
# the pixel neighborhood, then that pixel is a
# potential zero crossing
z_c = ((negative_count > 0) and (positive_count > 0) and (d-e > thresh))
# Change the pixel value with the maximum neighborhood
# difference with the pixel
if z_c:
z_c_image[i, j] = 200
return z_c_image
dst2 = Zero_crossing(dst1, 10)
cv2.imwrite('16_log_fil_zero.jpg', dst2)
* Detection by Canny *
1, smoothing
Smoothing with Gaussian filter
2, Processing with vertical and horizontal differential filters
You are using the sobel filter.
3, Calculation of gradient amount and gradient direction
4, Threshold processing
Only a certain range is detected in order to remove false positives.
python
class Canny_hand:
def __init__(self, image, sigma, h_thresh, l_thresh):
self.image = image.astype(np.float32)#Data type specification
self.h, self.w = image.shape
self.sigma = sigma
self.h_thresh = h_thresh
self.l_thresh = l_thresh
self.sobelx = np.array([[-1, 0, 1],
[-2, 0, 2],
[-1, 0, 1]])
self.sobely = np.array([[-1, -2, -1],
[0, 0, 0],
[1, 2, 1]])
def do(self,):
blur = cv2.GaussianBlur(self.image,(0, 0), sigmaX = self.sigma, sigmaY = self.sigma)
Gx = cv2.filter2D(blur, -1, self.sobelx)
Gy = cv2.filter2D(blur, -1, self.sobely)
G_value = np.sqrt(np.square(Gy) + np.square(Gx))
G_angle = np.arctan2(Gy, Gx)
G_angle_set = self.set_angle(G_angle)
G_max_group = self.extract_max(G_value, G_angle_set)
result = self.thresh_hold(G_max_group, min=self.l_thresh, max=self.h_thresh)
return result
def set_angle(self, G_angle):
pai = math.pi
G_angle[np.where((G_angle >= -pai/8) & (G_angle < pai/8))] = 0
G_angle[np.where((G_angle >= pai/8) & (G_angle < 3*pai/8))] = 45
G_angle[np.where((G_angle >= 3*pai/8) & (G_angle < 5*pai/8))] = 90
G_angle[np.where((G_angle >= 5*pai/8) & (G_angle < 7*pai/8))] = 135
G_angle[np.where((G_angle >= 7*pai/8) & (G_angle < -7*pai/8))] = 180
G_angle[np.where((G_angle >= -7*pai/8) & (G_angle < -5*pai/8))] = 225
G_angle[np.where((G_angle >= -5*pai/8) & (G_angle < -3*pai/8))] = 270
G_angle[np.where((G_angle >= -3*pai/8) & (G_angle < -pai/8))] = 315
return G_angle
def extract_max(self, G_value, G_angle_set):
result = G_value.copy()
for y in range(1, self.h - 1):
for x in range(1, self.w - 1):
if G_angle_set[y][x] == 0:
if (G_value[y][x] < G_value[y][x+1]):
result[y][x] = 0
elif G_angle_set[y][x] == 45:
if (G_value[y][x] < G_value[y+1][x+1]):
result[y][x] = 0
elif G_angle_set[y][x] == 90:
if (G_value[y][x] < G_value[y+1][x]):
result[y][x] = 0
elif G_angle_set[y][x] == 135:
if (G_value[y][x] < G_value[y+1][x-1]):
result[y][x] = 0
elif G_angle_set[y][x] == 180:
if (G_value[y][x] < G_value[y][x-1]):
result[y][x] = 0
elif G_angle_set[y][x] == 225:
if (G_value[y][x] < G_value[y-1][x-1]):
result[y][x] = 0
elif G_angle_set[y][x] == 270:
if (G_value[y][x] < G_value[y-1][x]):
result[y][x] = 0
elif G_angle_set[y][x] == 315:
if (G_value[y][x] < G_value[y-1][x+1]):
result[y][x] = 0
return result
def thresh_hold(self, in_put, min=75, max=150, d=1):
for y in range(0, self.h):
for x in range(0, self.w):
if in_put[y][x] >= max:
in_put[y][x] = 255
elif in_put[y][x] < min:
in_put[y][x] = 0
else:
if np.max(in_put[y-d:y+d+1, x-d:x+d+1]) >= max:
in_put[y][x] = 255
else:
in_put[y][x] = 0
return in_put
canny = Canny_hand(res, 0.3,100, 200 )
can_do = canny.do()
cv2.imwrite('17_canny_hand.jpg', can_do)
by opencv
python
edges = cv2.Canny(res,0.3, 100, 200)
cv2.imwrite('17_canny.jpg', dst2)
* Sharpening *
python
# Apply Gaussian Blur
blur = cv2.GaussianBlur(res,(0, 0), sigmaX = 3, sigmaY = 3)
cv2.imwrite('shap_blur.jpg', blur)
diff = res - blur
cv2.imwrite('18_shape_diff.jpg', diff)
shapen = res + diff*3
cv2.imwrite('19_shape_shapen.jpg', shapen)
* Local averaging *
It can be averaged while leaving a relatively edge. As an example, the following 9 patterns are calculated. Select the one with the least dispersion.
python
def vari(data):
count = len(data)
av = sum(data)/count
void = []
total = 0
for d in data:
diff = (d - av)**2
total += diff
pvariance = total/count
return pvariance
def selective_ave(image):
av_image = np.zeros(image.shape)
for i in range(2, image.shape[0] - 2):
for j in range(2, image.shape[1] - 2):
center_neighbour = [image[i+1, j-1],image[i+1, j],image[i+1, j+1],image[i, j-1],
image[i, j+1],image[i-1, j-1],image[i-1, j],image[i-1, j+1]]
l_u_neighbour = [image[i-2, j-2],image[i-2, j-1],image[i-1, j-2],image[i-1, j-1],
image[i-1, j],image[i, j-1],image[i, j]]
up_neighbour = [image[i-2, j-1],image[i-2, j],image[i-2, j+1],
image[i-1, j-1],image[i-1, j],image[i-1, j+1],image[i, j]]
r_u_neighbour = [image[i+2, j+2],image[i+2, j+1],image[i+1, j+2],image[i+1, j+1],
image[i+1, j],image[i, j+1],image[i, j]]
right_neighbour = [image[i-1, j+2],image[i, j+2],image[i+1, j+2],
image[i-1, j+1],image[i, j+1],image[i+1, j+1],image[i, j]]
left_neighbour = [image[i-1, j-2],image[i, j-2],image[i+1, j-2],
image[i-1, j-1],image[i, j-1],image[i+1, j-1],image[i, j]]
l_d_neighbour = [image[i+2, j-2],image[i+2, j-1],image[i+1, j-2],
image[i+1, j-1],image[i+1, j],image[i, j-1],image[i, j]]
down_neighbour = [image[i+2, j-1],image[i+2, j],image[i+2, j+1],
image[i+1, j-1],image[i+1, j],image[i+1, j+1],image[i, j]]
r_d_neighbour = [image[i+2, j+2],image[i+2, j+1],image[i+1, j+2],
image[i+1, j+1],image[i+1, j],image[i, j+1],image[i, j]]
nb_list = [center_neighbour, l_u_neighbour, up_neighbour,
r_u_neighbour, right_neighbour, left_neighbour,
l_d_neighbour, down_neighbour, r_d_neighbour]
number = 0
vari_max = 0
for nb in nb_list:
pre_vari = vari(nb)
if vari_max < pre_vari:
vari_max = pre_vari
max_index = number
number += 1
result = statistics.mean(nb_list[max_index])
av_image[i, j] = result
return av_image
select_av = selective_ave(res)
cv2.imwrite('20_select_av.jpg', select_av)
* Bilateral filter (filter using two Gaussian distributions) *
Implementation of smoothing with a Gaussian filter. Weights are distributed according to the distance to the point of interest (the farther the distance, the lighter) Up to this point, it is the same as a normal Gaussian filter. Here, further, the weighting of the Gaussian distribution is adopted for the difference in the pixel value between the pixels. That is, (the larger the difference (the farther the pixel value), the lighter the weight to be smoothed, so the edge portion (the larger the difference in pixel value) is less likely to be smoothed. As a result, the edges are preserved.
python
bi = cv2.bilateralFilter(res,15, 20, 20)
cv2.imwrite('21_bilateral_fil.jpg', bi)
* Non-local mean filter *
Whereas the bilateral filter weights the difference in pixel values, Here, the difference in similarity with other small areas (surrounding image areas) is weighted. That is, if there are many similar images in the periphery, the smoothing becomes heavy, and if they are not similar, the smoothing becomes light.
python
dst = cv2.fastNlMeansDenoising(res, None, 10, 7, 21)
cv2.imwrite('22_non_l_m_fil.jpg', dst)
* Median filter *
At the time of filtering, the median value of the area of interest is output. The median is (M × N + 1) / 2, counting from the pixel with the smallest pixel value for an image of M × N pixels. The value of the second pixel. This filter is effective for spike-like noise such as sprinkled with sesame seeds.
python
median = cv2.medianBlur(noicy_image, 3)
cv2.imwrite('24_median_fil.jpg', median)
Before
After
Referenced site http://optie.hatenablog.com/entry/2018/03/21/185647 https://theailearner.com/2019/05/25/laplacian-of-gaussian-log/ https://algorithm.joho.info/programming/python/opencv-canny-edge-detecte-py/
that's all
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