Noise Distributions

  

Noise present in  a scientific image can be described depending of its distribution. Thus, it is important to figure out how to define the type of noise in order to elaborate a strategy to denoise the images...

1- Impulse noise (or Salt and Pepper noise)

An impulse noise is characterized by extreme values - most of the time due to high-energy particles hitting the camera recording the image - and is rather easy to 

Create a new 8-bit image of 256x256 pixels entitled test with a background gray level equal to 128.0. Then, add the impulse noise (Process > Noise > Salt and Pepper).

Fig. 1: Salt and pepper image.

If you calculate the histogram, three peaks are available corresponding respectively to pixel values of 0 (the "pepper"), 128 (image background) and 255 (the "salt"). The fact that you only get extreme values (0 and 255 in a 8-bit image) is the signature of an impulse noise.

Note: We assume that we have an additive noise meaning that the pixels corresponding to the noise are added to the sample image.

2- Noise with a Gaussian Distribution

Gaussian noise is one of the most common noise found in scientific images. This type of noise is characterized by a probability density function (PDF) corresponding to a normal distribution.

 

Fig.2: Formula of the PDF p_G of a random variable z (source Wikipedia).

2-1- Gaussian Noise

In ImageJ, there is a dedicated function for adding a Gaussian noise onto an image. Go to Process > Noise > Add noise. If you want to modify the standard deviation, use instead Process > Noise > Add specified noise....

Fig. 3: Gaussian noises added to the 8-bit test image with a 128.0 background with their corresponding histograms. Standard deviations of A) Default (25). B) 10 and C) 100.

The aspects of these noisy images are very different from the salt-and-pepper and it is easy to imagine that removing such a noise covering the whole dynamic range of the image will be more difficult than removing the extreme values of the impulse noise.

2-2- DIY Gaussian Noise

It is easy to generate your own Gaussian Noise. Let's do it!!

Create a 32-bit image with a black background and fill it with Process > Math > Macro... according to the following formula ...

v= v + sqrt(-2 * log(random()))*cos((2*PI) * random()) 

 ... then, compute the histogram. Et voilà! you have generated a bell curve with a mean of 0.0 and a standard deviation of 1.0 (Fig. 4).

Fig. 4: Histogram of the DIY Gaussian noise.

 

Note: The algorithm used to generate the gaussian noise is based on the Box-Muller transform [Wikipedia]. A discussion about the various possible implementations is here [Link].

If you want a Gaussian noise with a different standard deviation, just multiply this noisy 32-bit image by 25.0 (for comparison to the default Add Noise in ImageJ) in Process > Math > Multiply > 25.0, then add this noise to a 32-bit test image (with a background of 128.0) with the Image calculator (Process > Image Calculator...).

3- Other distributions

TODO 

Now, we know what the noise looks like, it is time to try to denoise these images...

Thank you for reading.

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3. Other crazybiocomputing posts

Further readings are available in ...

• Image Enhancement Series  [Link]
• Image Processing TOC [Link]
• Wikipedia about Kernel [Link]