How to set first order image moments m01 and m10 to zero to achieve translation invariance with Zernike moments

I am struggling to understand the methodology in the paper 'Invariant image recognition by Zernike moments' to achieve translational invariance of Zernike moments.

It is stated "Translation invariancy is achieved by transforming the image into a new one whose first order moments, m01 and m10, are both equal to zero. This is done by transforming the original f(x, y) image into another one which is f(x+ X, y + J ), where X and J are the centroid location of the original image computed from" raw moments M10, M01 and M00. "In other words, the origin is moved to the centroid before moment calculation".

I have plotted a 3x3 square of 1's in an array of zeros.

[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[1. 1. 1. 0. 0. 0. 0. 0. 0.]
[1. 1. 1. 0. 0. 0. 0. 0. 0.]
[1. 1. 1. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]

I calculate the moments: M00 = 9.0, M10 = 9.0, M01 = 27.0. Hence centroid (x,y) = (1.0,3.0)

I transform the image by translating the shape +1 in the x and + 3 in the y using the values from the centroid of the first image.
I calculate the moments of the transformed image: M00 = 9.0, M10 = 36.0, M01 = 36.0. Hence (x,y) = (4.0,4.0)

[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 1. 1. 1. 0. 0. 0.]
[0. 0. 0. 1. 1. 1. 0. 0. 0.]
[0. 0. 0. 1. 1. 1. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0. 0.]

The resulting M10 and M01 are equal, but not equal to zero, and the object is now in the centre of the array. From my understanding the only way to achieve M10 or M01 equalling to zero is to have no pixels in the image. As M10 = sum of the x coordinates where f(x,y) = 1, and M01 = sum of the y coordinates where f(x,y) = 1.

If I apply this process to each image in a set, have I normalised correctly despite not equalling zero? If not, what am I misunderstanding here?

I have included my crude code in python to illustrate below.

import numpy as np
import cv2

image = np.zeros([9,9])

image[2,0] = 1
image[3,0] = 1
image[4,0] = 1
image[2,1] = 1
image[3,1] = 1
image[4,1] = 1
image[2,2] = 1
image[3,2] = 1
image[4,2] = 1

print(image)

M = cv2.moments(image)
m00 = M['m00']
m10 = M['m10']
m01 = M['m01']
x = m10 / m00
y = m01 / m00
print(M)
print(f"centroid is {x},{y}")

image_2 = np.zeros([9,9])

image_2[3,3] = 1
image_2[4,3] = 1
image_2[5,3] = 1
image_2[3,4] = 1
image_2[4,4] = 1
image_2[5,4] = 1
image_2[3,5] = 1
image_2[4,5] = 1
image_2[5,5] = 1

print(image_2)
M = cv2.moments(image_2)
m00 = M['m00']
m10 = M['m10']
m01 = M['m01']
x = m10 / m00
y = m01 / m00
print(M)
print(f"centroid is {x},{y}")

After answer from Cris, realised that the A00 and A11 zernike moments do not need to be included as they are by definition zero after adjusting the coordinate scale such that the centroid is 0,0. No image processing or adjustments to the image are required. Code below calculates zernike moments from a contour up to order 8, the first 2 moments A00 and A11 are removed from the list storing the moments. By computing the Euclidean distance between sets of moments between two identical shapes which have been translated their image moments are the same.

z_Moments = mahotas.features.zernike_moments(thresh, cv2.minEnclosingCircle(cnts)[1], degree=8)
z_Moments = z_Moments[2:]

I'm not sure if I've ever read the paper you refer to, but it might have unclear language.

The idea is to change the coordinate system of the image to have its origin at the centroid of the object. That is, you change the coordinate system such that M01 and M10 become 0. You don't change the image itself, only the coordinate system used in the calculations of the moments.

When computing higher order moments in this translated coordinate system, we will compute the central moments, which are translation invariant. You can accomplish the same thing by modifying the computation of the moments such that you get central moments. This Wikipedia page has a fairly clear explanation for how to compute central moments.

When computing Zernike moments in this translated coordinate system, the computed moments will be independent of the location of the object in the image, i.e. they are translation invariant.