When working with binary images, it can be useful to add Gaussian noise to simulate real-world conditions or test the robustness of algorithms. In Python, there are several ways to accomplish this task, depending on whether you know the noise variance or the signal-to-noise ratio (SNR).

Option 1: Adding Gaussian noise with known variance

If you know the noise variance, you can use the NumPy library to generate Gaussian noise and add it to the binary image. Here’s how you can do it:

import numpy as np

def add_gaussian_noise(image, variance):
    noise = np.random.normal(0, np.sqrt(variance), image.shape)
    noisy_image = image + noise
    return noisy_image

# Example usage
binary_image = ...  # Your binary image
variance = ...  # Known noise variance
noisy_image = add_gaussian_noise(binary_image, variance)

This code snippet defines a function add_gaussian_noise that takes the binary image and the noise variance as inputs. It generates Gaussian noise using np.random.normal with a mean of 0 and a standard deviation of the square root of the variance. The noise is then added to the binary image, resulting in a noisy image.

Option 2: Adding Gaussian noise with known SNR

If you know the signal-to-noise ratio (SNR) instead of the noise variance, you can calculate the variance from the SNR and then use the same approach as in option 1. Here’s an example:

import numpy as np

def add_gaussian_noise_snr(image, snr):
    signal_power = np.mean(image ** 2)
    noise_power = signal_power / snr
    variance = noise_power / 2
    noisy_image = add_gaussian_noise(image, variance)
    return noisy_image

# Example usage
binary_image = ...  # Your binary image
snr = ...  # Known SNR
noisy_image = add_gaussian_noise_snr(binary_image, snr)

This code snippet defines a function add_gaussian_noise_snr that takes the binary image and the SNR as inputs. It first calculates the signal power as the mean squared value of the image. Then, it calculates the noise power by dividing the signal power by the SNR. Finally, it calculates the variance by dividing the noise power by 2 and calls the add_gaussian_noise function from option 1 to add the noise to the binary image.

Option 3: Comparing the options

Both options 1 and 2 provide ways to add Gaussian noise to a binary image, but they differ in the input parameter required. Option 1 requires the noise variance, while option 2 requires the SNR. If you already know the noise variance, option 1 is simpler and more straightforward. However, if you only have the SNR, option 2 allows you to calculate the variance and add the noise accordingly.

In conclusion, the better option depends on the available information. If you have the noise variance, option 1 is recommended. If you only have the SNR, option 2 provides a convenient way to calculate the variance and add the noise to the binary image.