Salt-and-Pepper Noise

A Salt-and-Pepper Noise is an image data noise (on image data) that results in image degradation caused by errors in the data transmission.

• AKA: Impulse Noise, Data Drop-Out Noise, Intensity Spikes, Speckle.
  • …
• Counter-Example(s):
  • Gaussian Noise.
  • Uniform Noise.
  • Quantization Noise.
  • Photon Counting Noise.
  • Photographic Grain Noise.
• See: Noisy Data, Contraharmonic Mean, Image Noise, Pixel, Noise Reduction, Median Filter, Mathematical Morphology.

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References

2017a

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Salt-and-pepper_noise Retrieved:2017-12-3.
  • Salt-and-pepper noise is a form of noise sometimes seen on images. It is also known as impulse noise. This noise can be caused by sharp and sudden disturbances in the image signal. It presents itself as sparsely occurring white and black pixels. An effective noise reduction method for this type of noise is a median filter or a morphological filter. For reducing either salt noise or pepper noise, but not both, a contraharmonic mean filter can be effective.

2017b

• (Fisher et al., 2017) ⇒ R. Fisher, S. Perkins, A. Walker and E. Wolfart (2017). https://homepages.inf.ed.ac.uk/rbf/HIPR2/noise.htm#3 Retrieved:2017-12-3.
  • QUOTE: Another common form of noise is data drop-out noise (commonly referred to as intensity spikes, speckle or salt and pepper noise). Here, the noise is caused by errors in the data transmission. The corrupted pixels are either set to the maximum value (which looks like snow in the image) or have single bits flipped over. In some cases, single pixels are set alternatively to zero or to the maximum value, giving the image a `salt and pepper' like appearance. Unaffected pixels always remain unchanged. The noise is usually quantified by the percentage of pixels which are corrupted.

2010

• (Bovik, 2010) ⇒ Bovik, A. C. (2010). "4.3 Salt and Pepper Noise". In: Handbook of image and video processing. pp 403 Academic press. ISBN: 9780080533612
  • QUOTE: Salt and pepper noise refers to a wide variety of processes that result in the same basic image degradation: only a few pixels are noisy, but they are very noisy. The effect is similar to sprinkling white and black dots - salt and pepper - on the image.

    One example where salt and pepper noise arises is in transmitting images over noisy digital links. Let each pixel be quantized to B bits in the usual fashion. The value of the pixel can be written as [math]\displaystyle{ X=\sum_{i=0}^{B-1}b_i2^i }[/math]. Assume the channel is a binary symmetric one with a crossover probability of [math]\displaystyle{ \epsilon }[/math]. Then each bit is flipped with probability [math]\displaystyle{ \epsilon }[/math] Call the received value, Y. Then, assuming the bit flips are independent,

    [math]\displaystyle{ Pr(|X—Y|=2^i)=\epsilon(1-\epsilon)^{B-1} \quad(33) }[/math]

    for [math]\displaystyle{ i=0,1,\cdots,B-1 }[/math]. The MSE due to the most significant bit is [math]\displaystyle{ \epsilon4^{B-1} }[/math] compared to [math]\displaystyle{ \epsilon(4^{B-1}-1)/3 }[/math] for all the other bits combined. In other words, the contribution to the MSE from the most significant bit is approximately 3 times that of all the other bits. The pixels whose most significant bits are changed will likely appear as black or white dots.

     Salt and pepper noise is an example of (very) heavy-tailed noise. A simple model is the following: Let [math]\displaystyle{ f(x,y) }[/math] be the original image and [math]\displaystyle{ q(x,y) }[/math] be the image after it has been altered by salt and pepper noise.

    [math]\displaystyle{ Pr(q=f)=1-\alpha \quad (34) }[/math]; [math]\displaystyle{ Pr(q = MAX)= \alpha/2 \quad (35) }[/math]; [math]\displaystyle{ Pr(q = MIN) = \alpha/2\quad (36) }[/math]

    where MAX and MIN and the maximum and minimum image values, respectively. For 8 hit images, MIN=0 and MAX=255. The idea is that with probability [math]\displaystyle{ 1-\alpha }[/math] or the pixels are unaltered; with probability [math]\displaystyle{ \alpha }[/math] the pixels are changed to the largest or smallest values. The altered pixels look like black and white dots sprinkled over the image(...)

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