I’ve attempted to implement a Sobel Edge detector, like so:

def sobel(img):
    img = cv2.GaussianBlur(img, (5, 5), 0)
    
    filter_x = np.array([
        [-1, 0, 1],
        [-2, 0, 2],
        [-1, 0, 1]
    ])
    
    filter_y = np.array([
        [-1, -2, -1],
        [0, 0, 0],
        [1, 2, 1]
    ])
    
    S_x = cv2.filter2D(img, -1, filter_x)
    S_y = cv2.filter2D(img, -1, filter_y)
    
#     S_x = cv2.convertScaleAbs(S_x)
#     S_y = cv2.convertScaleAbs(S_y)
#     grad = S_x * 0.5 + S_y * 0.5
    grad = np.sqrt(S_x ** 2 + S_y ** 2)
    grad = np.clip(grad, 0, 255)
    return grad.astype(np.uint8)

img_gray = cv2.cvtColor(img, cv2.COLOR_RGB2GRAY)
plt.figure(figsize=(10, 10))
edge = sobel(img_gray)
plt.imshow(edge, cmap='gray')

The outputs I’m getting are very noisy. For example:

This is happening because I’m computing the L2 Magnitude of the gradients in x and y directions. However, when I switch the strategy to use L1 norms (i.e, using `S_x * 0.5 + S_y * 0.5), I get a much less noisier output:

Why is this happening? According to various sources, both of these combinations are used, but I want to understand why the L2 variant is so noisy, and how I can avoid it.

Thank you!