Adding poisson noise to an image

Question

I have some images that I need to add incremental amounts of Poisson noise to in order to more thoroughly analyze them. I know you can do this in MATLAB, but how do you go a

Answer 1

From the item 1.4.4 - "Gaussian Approximation of the Poisson Distribution" of Chapter 1 of this book:

For large mean values, the Poisson distribution is well approximated by a Gaussian distribution with mean and variance equal to the mean of the Poisson random variable:

P(μ) ≈ N (μ,μ)

Then, we can generate Poisson noise from a normal distribution N (0,1), scale its standard deviation by the square root of μ and add it to the image which is the μ value:

# Image size
M, N = 1000, 1000

# Generate synthetic image
image = np.tile(np.arange(0,N,dtype='float64'),(M,1)) * 20

# -- sqrt(mu) * normal(0,1) --
poisson_noise = np.sqrt(image) * np.random.normal(0, 1, image.shape)

# Add the noise to the mu values
noisy_image = image + poisson_noise

plt.figure(figsize=(10,10))
plt.subplot(2,2,1)
plt.title('Image')
plt.imshow(image,'gray')
plt.subplot(2,2,2)
plt.title('Noisy image noise')
plt.imshow(noisy_image,'gray')  
plt.subplot(2,2,3)
plt.title('Image profile')
plt.plot(image[0,:])
plt.subplot(2,2,4)
plt.title('Noisy image profile')
plt.plot(noisy_image[0,:])

print("Synthetic image mean: {}".format(image[:,1].mean()))
print("Synthetic image variance: {}".format(image[:,1].var()))    
print("Noisy image mean: {}".format(noisy_image[:,1].mean()))
print("Noisy image variance: {}".format(noisy_image[:,1].var()))

As Poisson noise is signal-dependent, as we increase the underlying signal the noise variance also increases, as we can see in this row profiles:

Output for statistics in a single column:

Synthetic image mean: 20.0

Synthetic image variance: 0.0

Noisy image mean: 19.931120555821597

Noisy image variance: 19.39456713877459

Further references: [1][2]