How do I use 1D gradients to compute a 2D Sobel in OpenCV with a different vector norm?

I’ve accepted the answer from @fmw42 as correct, but I want to add more context.

The direct answer to my question is that I found the documentation misleading. The "Formulation" section in this link implied to me that the cv2.Sobel() method does the convolutions with the x and y kernels, and then returns the magnitude (L2 norm) of those derivatives.

The mistake that @fmw42 pointed out to me, and which I then verified, was that cv2.Sobel() does not calculate the magnitude for you. It only does the convolution using the Sobel kernels. You can verify by looking at the output of this code:

import cv2
import numpy as np


z_img = cv2.imread(".\some_image.tif", cv2.IMREAD_UNCHANGED)

z_px_rows = z_img.shape[0]
z_px_cols = z_img.shape[1]

kernel_x = np.array([[-1, 0, 1],
                     [-2, 0, 2],
                     [-1, 0, 1]])

kernel_y = np.array([[-1, -2, -1],
                     [ 0,  0,  0],
                     [ 1,  2,  1]])
  
gx = cv2.filter2D(src=z_img, ddepth=-1, kernel=kernel_x)
print(f'Center pixel intensity (gx): {gx[z_px_rows // 2, z_px_cols // 2]}')

native_sobel_x = cv2.Sobel(z_img, cv2.CV_32F, 1, 0, ksize=3)
print(f'Center pixel intensity (native sobel x): {native_sobel_x[z_px_rows // 2, z_px_cols // 2]}')

gy = cv2.filter2D(src=z_img, ddepth=-1, kernel=kernel_y)
print(f'Center pixel intensity (gy): {gy[z_px_rows // 2, z_px_cols // 2]}')

native_sobel_y = cv2.Sobel(z_img, cv2.CV_32F, 0, 1, ksize=3)
print(f'Center pixel intensity (native sobel y): {native_sobel_y[z_px_rows // 2, z_px_cols // 2]}')

# Now take gx and convolve using the y kernel
mixed = cv2.filter2D(src=gx, ddepth=-1, kernel=kernel_y)
print(f'Center pixel intensity (mixed): {mixed[z_px_rows // 2, z_px_cols // 2]}')

# And compare to calling Sobel(dy=1) on the output of Sobel(dx=1)
native_sobel_mixed = cv2.Sobel(native_sobel_x, cv2.CV_32F, 0, 1, ksize=3)
print(f'Center pixel intensity (native sobel mixed): {native_sobel_mixed[z_px_rows // 2, z_px_cols // 2]}')

native_sobel_both = cv2.Sobel(z_img, cv2.CV_32F, 1, 1, ksize=3)
print(f'Center pixel intensity (native sobel): {native_sobel_both[z_px_rows // 2, z_px_cols // 2]}')

This will produce intensities of the center pixel as follows:

What remains a mystery to me is why calling cv2.Sobel(dx=1, dy=1) generates a different result from calling cv2.Sobel(dx=1) on the original image and then calling cv2.Sobel(dy=1) on the result of the first call (the last three rows). That seems like a different question, though.

You question why the mixed filter2D gradient is not the same as the mixed sobel gradient with kernel dx=1 and dy=1. You have a couple of issues. 1) You did not ensure that your filter2D outputs were floats. 2) The answer is to do multiplication (element wise) between the two kernels first, then do the convolution with filter_2D (rather than successive convolutions). Note that it is element-wise multiplication rather than matrix multiplication. To show that, see the following script and results.

Input:

import cv2
import numpy as np


z_img = cv2.imread("lena_gray.png")

z_px_rows = z_img.shape[0]
z_px_cols = z_img.shape[1]

kernel_x = np.array([[-1, 0, 1],
                     [-2, 0, 2],
                     [-1, 0, 1]])

kernel_y = np.array([[-1, -2, -1],
                     [ 0,  0,  0],
                     [ 1,  2,  1]])
# matrix multiply            
kernel_xy1 = np.matmul(kernel_x,kernel_y)
kernel_xy2 = np.matmul(kernel_x,kernel_y)
# element-wise multiply
kernel_xy3 = cv2.multiply(kernel_x,kernel_y)
kernel_xy4 = cv2.multiply(kernel_y,kernel_x)

print('')
print("kernel_xy1:n", kernel_xy1)
print('')
print("kernel_xy2:n", kernel_xy1)
print('')
print("kernel_xy3:n", kernel_xy1)
print('')
print("kernel_xy4:n", kernel_xy1)
print('')
  
gx = cv2.filter2D(src=z_img, ddepth=cv2.CV_32F, kernel=kernel_x)
print(f'Center pixel intensity (gx): {gx[z_px_rows // 2, z_px_cols // 2]}')

native_sobel_x = cv2.Sobel(z_img, cv2.CV_32F, 1, 0, ksize=3)
print(f'Center pixel intensity (native sobel x): {native_sobel_x[z_px_rows // 2, z_px_cols // 2]}')

gy = cv2.filter2D(src=z_img, ddepth=cv2.CV_32F, kernel=kernel_y)
print(f'Center pixel intensity (gy): {gy[z_px_rows // 2, z_px_cols // 2]}')

native_sobel_y = cv2.Sobel(z_img, cv2.CV_32F, 0, 1, ksize=3)
print(f'Center pixel intensity (native sobel y): {native_sobel_y[z_px_rows // 2, z_px_cols // 2]}')

mixed1 = cv2.filter2D(src=z_img, ddepth=cv2.CV_32F, kernel=kernel_xy1)
print(f'Center pixel intensity (mixed1 - matrix multiply): {mixed1[z_px_rows // 2, z_px_cols // 2]}')

mixed2 = cv2.filter2D(src=z_img, ddepth=cv2.CV_32F, kernel=kernel_xy2)
print(f'Center pixel intensity (mixed2 - matrix multiply): {mixed2[z_px_rows // 2, z_px_cols // 2]}')

mixed3 = cv2.filter2D(src=z_img, ddepth=cv2.CV_32F, kernel=kernel_xy3)
print(f'Center pixel intensity (mixed3 - element-wise multiply): {mixed3[z_px_rows // 2, z_px_cols // 2]}')

mixed4 = cv2.filter2D(src=z_img, ddepth=cv2.CV_32F, kernel=kernel_xy4)
print(f'Center pixel intensity (mixed4 - element-wise multiply): {mixed4[z_px_rows // 2, z_px_cols // 2]}')

native_sobel_xy = cv2.Sobel(z_img, cv2.CV_32F, 1, 1, ksize=3)
print(f'Center pixel intensity (native sobel xy): {native_sobel_xy[z_px_rows // 2, z_px_cols // 2]}')

Results:

kernel_xy1:
 [[2 4 2]
 [4 8 4]
 [2 4 2]]

kernel_xy2:
 [[2 4 2]
 [4 8 4]
 [2 4 2]]

kernel_xy3:
 [[2 4 2]
 [4 8 4]
 [2 4 2]]

kernel_xy4:
 [[2 4 2]
 [4 8 4]
 [2 4 2]]

Center pixel intensity (gx): [6. 6. 6.]
Center pixel intensity (native sobel x): [6. 6. 6.]
Center pixel intensity (gy): [-184. -184. -184.]
Center pixel intensity (native sobel y): [-184. -184. -184.]
Center pixel intensity (mixed1 - matrix multiply): [2580. 2580. 2580.]
Center pixel intensity (mixed2 - matrix multiply): [2580. 2580. 2580.]
Center pixel intensity (mixed3 - element-wise multiply): [-28. -28. -28.]
Center pixel intensity (mixed4 - element-wise multiply): [-28. -28. -28.]
Center pixel intensity (native sobel xy): [-28. -28. -28.]