Label regions with unique combinations of values in two numpy arrays?

I have two labelled 2D numpy arrays a and b with identical shapes. I would like to re-label the array b by something similar to a GIS geometric union of the two arrays, such that cells with unique combination of values in array a and b are assigned new unique IDs:

I'm not concerned with the specific numbering of the regions in the output, so long as the values are all unique. I have attached sample arrays and desired outputs below: my real datasets are much larger, with both arrays having integer labels which range from "1" to "200000". So far I've experimented with concatenating the array IDs to form unique combinations of values, but ideally I would like to output a simple set of new IDs in the form of 1, 2, 3..., etc.

import numpy as np
import matplotlib.pyplot as plt

# Example labelled arrays a and b
input_a = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                    [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
                    [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
                    [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
                    [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
                    [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
                    [0, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 0],
                    [0, 0, 3, 3, 3, 3, 2, 2, 2, 2, 0, 0],
                    [0, 0, 3, 3, 3, 3, 2, 2, 2, 2, 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]])

input_b = np.array([[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, 1, 1, 1, 3, 3, 3, 3, 3, 0, 0],
                    [0, 0, 1, 1, 1, 3, 3, 3, 3, 3, 0, 0],
                    [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0],
                    [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0],
                    [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0],
                    [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 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]])

# Plot inputs
plt.imshow(input_a, cmap="spectral", interpolation='nearest')
plt.imshow(input_b, cmap="spectral", interpolation='nearest')

# Desired output, union of a and b
output = np.array([[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, 1, 1, 1, 2, 3, 3, 3, 3, 0, 0],
                   [0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 0, 0],
                   [0, 0, 1, 1, 1, 4, 7, 7, 7, 7, 0, 0],
                   [0, 0, 5, 5, 5, 6, 7, 7, 7, 7, 0, 0],
                   [0, 0, 5, 5, 5, 6, 7, 7, 7, 7, 0, 0],
                   [0, 0, 5, 5, 5, 6, 7, 7, 7, 7, 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]])

# Plot desired output
plt.imshow(output, cmap="spectral", interpolation='nearest')

Solution

If I understood the circumstances correctly, you are looking to have unique pairings from a and b. So, 1 from a and 1 from b would have one unique tag in the output; 1 from a and 3 from b would have another unique tag in the output. Also looking at the desired output in the question, it seems that there is an additional conditional situation here that if b is zero, the output is to be zero as well irrespective of the unique pairings.

The following implementation tries to solve all of that -

c = a*(b.max()+1) + b
c[b==0] = 0
_,idx = np.unique(c,return_inverse= True)
out = idx.reshape(b.shape)

Sample run -

In [21]: a
Out[21]: 
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
       [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
       [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
       [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
       [0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0],
       [0, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 0],
       [0, 0, 3, 3, 3, 3, 2, 2, 2, 2, 0, 0],
       [0, 0, 3, 3, 3, 3, 2, 2, 2, 2, 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]])

In [22]: b
Out[22]: 
array([[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, 1, 1, 1, 3, 3, 3, 3, 3, 0, 0],
       [0, 0, 1, 1, 1, 3, 3, 3, 3, 3, 0, 0],
       [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0],
       [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0],
       [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0],
       [0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 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]])

In [23]: out
Out[23]: 
array([[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, 1, 1, 1, 3, 5, 5, 5, 5, 0, 0],
       [0, 0, 1, 1, 1, 3, 5, 5, 5, 5, 0, 0],
       [0, 0, 1, 1, 1, 2, 4, 4, 4, 4, 0, 0],
       [0, 0, 6, 6, 6, 7, 4, 4, 4, 4, 0, 0],
       [0, 0, 6, 6, 6, 7, 4, 4, 4, 4, 0, 0],
       [0, 0, 6, 6, 6, 7, 4, 4, 4, 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, 0, 0, 0, 0, 0, 0, 0]])

Sample plot -

# Plot inputs
plt.figure()                                                    
plt.imshow(a, cmap="spectral", interpolation='nearest')
plt.figure() 
plt.imshow(b, cmap="spectral", interpolation='nearest')

# Plot output
plt.figure()
plt.imshow(out, cmap="spectral", interpolation='nearest')