To calculate the chi-square test of a grayscale image with randomly selected 2000 bits in Python, you can follow these steps:

Import necessary libraries: You will need to import the numpy library for numerical computations and the scipy.stats library for the chi-square test.

Read and prepare the image data: Read the grayscale image and convert it into a 1D array of pixel values. You can use libraries like OpenCV or PIL for this step.

Randomly select 2000 bits: Use numpy.random.choice to randomly select 2000 pixel values from the image array.

Calculate the observed frequencies: Count the frequency of each pixel value (0 to 255) in the selected 2000 bits. You can use numpy.bincount for this step.

Calculate the expected frequencies: Calculate the expected frequency of each pixel value under the assumption of uniform distribution. This is simply the total number of bits (2000) divided by the number of possible pixel values (256).

Calculate the chi-square statistic: Use the formula for the chi-square statistic: χ² = Σ [(observed frequency - expected frequency)^2 / expected frequency]. You can use a loop or numpy vector operations to calculate this.

Calculate the p-value: Use the scipy.stats.chi2.sf function to calculate the p-value from the chi-square statistic and the degrees of freedom (255 in this case).

Interpret the result: If the p-value is less than a certain significance level (e.g., 0.05), you can reject the null hypothesis that the pixel values are uniformly distributed.

Here's some sample code to get you started:

import numpy as np

from scipy.stats import chi2_contingency, chi2_sf

# Read and prepare the image data

img = cv2.imread('image.jpg', cv2.IMREAD_GRAYSCALE)

img_array = img.flatten()

# Randomly select 2000 bits

selected_bits = np.random.choice(img_array, size=2000, replace=False)

# Calculate the observed frequencies

observed_freq = np.bincount(selected_bits, minlength=256)

# Calculate the expected frequencies

expected_freq = 2000 / 256

# Calculate the chi-square statistic

chi2 = np.sum((observed_freq - expected_freq) ** 2 / expected_freq)

# Calculate the p-value

p_value = chi2_sf(chi2, df=255)

# Interpret the result

if p_value < 0.05:

print("Reject the null hypothesis: The pixel values are not uniformly distributed.")

else:

print("Fail to reject the null hypothesis: The pixel values are uniformly distributed.")