Any idea to optimise this algorithm?

Question

Let's imagine I have a binary 40*40 matrix. In this matrix, values can be either ones or zeros.

I need to parse the whole matrix, for any value == 1, apply the following :

If the following condition is met, keep the value = 1, else modify the value back to 0:

Condition: in a square of N*N (centered on the currently evaluated value), I can count at least M values == 1.

N and M are parameters that can be set for the algorithms, naturally N<20 (in this case) and M<(N*N - 1).

The obvious algorithm is to iterate over the whole image and then each time a value == 1. Perform another search around this value. It would make an O^3 complex algorithm. Any idea to make this more efficient?

Edit: Some code to make this more easy to understand.

Let's create a randomly initialized 40*40 matrix of 1s and 0s. 5% (arbitrarily chosen) of the values are 1s, 95% are 0s.

import matplotlib.pyplot as plt
import numpy as np

%matplotlib inline

def display_array(image):
    image_display_ready = image * 255

    print(image_display_ready)

    plt.imshow(image_display_ready, cmap='gray')
    plt.show()

image = np.zeros([40,40])

for _ in range(80):  # 5% of the pixels are == 1
    i, j = np.random.randint(40, size=2)
    image[i, j] = 1

# Image displayed on left below before preprocessing    
display_array(image)   

def cleaning_algorithm_v1(src_image, FAT, LR, FLAG, verbose=False):
    """
    FAT  = 4 # False Alarm Threshold (number of surrounding 1s pixel values)
    LR   = 8 # Lookup Range +/- i/j value
    FLAG = 2 # 1s pixels kept as 1s after processing are flag with this value.
    """

    rslt_img = np.copy(src_image)

    for i in range(rslt_img.shape[0]):
        for j in  range(rslt_img.shape[1]):

            if rslt_img[i, j] >= 1:

                surrounding_abnormal_pixels = list()

                lower_i = max(i - LR, 0)
                upper_i = min(i + LR + 1, rslt_img.shape[0])

                lower_j = max(j - LR, 0)
                upper_j = min(j + LR + 1, rslt_img.shape[1])

                abnormal_pixel_count = 0

                for i_k in range(lower_i, upper_i):
                    for j_k in range(lower_j, upper_j):

                        if i_k == i and j_k == j:
                            continue

                        pixel_value = rslt_img[i_k, j_k]

                        if pixel_value >= 1:
                            abnormal_pixel_count += 1

                if abnormal_pixel_count >= FAT:
                    rslt_img[i, j] = FLAG

    rslt_img[rslt_img != FLAG] = 0
    rslt_img[rslt_img == FLAG] = 1

    return rslt_img

# Image displayed on right below after preprocessing  
display_array(cleaning_algorithm_v1(image, FAT=10, LR=6, FLAG=2))

Which gives the following:

Answer 1

What about using convolution?

Your kernel would be an NxN window of 1's. In this case the kernel is separable so you can process the convolution as 2 1-D convolutions. You could do something like:

import numpy as np
from scipy.ndimage.filters import convolve1d
from time import time

mat = np.random.random_integers(0, 1, (40, 40))

N = 5
M = 15

window = np.ones((N, ), dtype=np.int)

start = time()

interm = convolve1d(mat, window, axis=0)
result = convolve1d(interm, window, axis=1)

[rows, cols] = np.where(result >= M)
result[:, :] = 0
result[(rows, cols)] = 1

end = time()
print "{} seconds".format(end - start)
0.00155591964722 seconds

Not sure how the complexity compares, but convolution is pretty well optimised in a variety of python deep learning libraries.