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]

Answer 2

You could use skimage.util.random_noise:

from skimage.util import random_noise

noisy = random_noise(img, mode="poisson")

Answer 3

Actually the answer of Paul doesnt make sense.

Poisson noise is signal dependent! And using those commands, provided by him, the noise later added to the image is not signal dependent.

To make it signal dependent you shold pass the image to the NumPy's poisson function:

filename = 'myimage.png'
img = (scipy.misc.imread(filename)).astype(float)

noise_mask = numpy.random.poisson(img)

noisy_img = img + noise_mask

Answer 4

The answer of Helder is correct. I just want to add the fact that Poisson noise is not additive and you can not add it as Gaussian noise.

Depend on what you want to achieve, here is some suggestions:

• Simulate a low-light noisy image (if PEAK = 1, it will be really noisy)

  import numpy as np
  image = read_image("YOUR_IMAGE")  # need a rescale to be more realistic
  noisy = np.random.poisson(image / 255.0 * PEAK) / PEAK * 255  # noisy image
  

• Add a noise layer on top of the clean image

  import numpy as np
  image = read_image("YOUR_IMAGE") 
  noisemap = create_noisemap() 
  noisy = image + np.random.poisson(noisemap)  
  

Then you can crop the result to 0 - 255 if you like (I use PIL so I use 255 instead of 1).

Answer 5

If numpy/scipy are available to you, then the following should help. I recommend that you cast the arrays to float for intermediate computations then cast back to uint8 for output/display purposes. As poisson noise is all >=0, you will need to decide how you want to handle overflow of your arrays as you cast back to uint8. You could scale or truncate depending on what your goals were.

filename = 'myimage.png'
imagea = (scipy.misc.imread(filename)).astype(float)

poissonNoise = numpy.random.poisson(imagea).astype(float)

noisyImage = imagea + poissonNoise

#here care must be taken to re cast the result to uint8 if needed or scale to 0-1 etc...