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My first question :

  1. I want to know why an image is noisy ?.
  2. why the ultrasound, SAR and Laser... images are noisy by a multiplicative noise not an additive noise?

Another question :

So the dominant noise in ultrasound image is the Speckle noise, but the confusion in my mind is :

The speckle noise is of multiplicative nature, the mathematical writing is $$f(x,y)=u(x,u)\eta(x,y)\tag{1}$$ where $f$ is the observed image, $\eta$ the noise and $u$ the dispeckle (recovered) image.

But in some paper (as this) the perturbed images with the speckle noise can write as : $$f(x,y)=u(x,y)+\sqrt{u(x,y)}\eta(x,y)\tag{2}$$ and $\eta$ is with zero-mean Gaussian variable, the first constraint it must $u>0$ and then the model (2) is characteris of the Gaussian distribution, So this is the case of additive noise

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  • $\begingroup$ Ye. I also too have curious about that. $\endgroup$ – gmotree Jul 17 '15 at 8:46
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There are plenty of sources of noise in an imaging system. In the domain of sonar systems (which I work in) there is a big problem of reverberation, which is a fancy way of saying "sound from something that we aren't interested in. When using a sonar at sea, you get reverb from the sea surface, sea bed and the water volume. Taking volume reverb as a simple example, this is the water itself (or particles in it) returning sound to the receiver. The best way to relate to this is like driving your car on a foggy night. The sea bed and surface can absorb or reflect sound energy, depending on their composition (e.g. sea state). It's worth noting that this will bugger up your sonar image.

Another overlooked source of noise is in the signal processing itself. Your sensor is not going to be perfect and will quite happily distort both the amplitude and phase of the phenomena its trying to measure. In acoustics, you'll find microphones and hydrophones will be "tuned" for a certain bandwidth, and are quite useless outside of this frequency range. Typically your analogue to digit convertor will have some form of quantisation error with the values its spits out. It simply doesn't have the infinite precision required to represent measurements of the real world. Before we move off the subject, we should be aware that any electronics experiences interference from its environment (e.g. induction, radiation, poor design). Add all this together and you have an estimate of what's called your "noise floor" and as a designer, you want to both minimise this by design and use other techniques to boost the signal.

Once in the digital world, your sample values are pretty much safe from additional noise. But wait there's more. In acoustics you'll want to electrically steer your array and spatially filter the incoming signal (look in different directions). This is also known as beam forming. The basic principle of beam forming is that you apply delays to the signals you receive and then sum them all together. Using the awesome power of constructive and destructive interference, incoherent signals cancel out, and coherent signals emphasised. This isn't perfect either because the degree of spatial attenuation varies depending on a number of factors (e.g. array length and element spacing). It's worth pointing out that this aspect of processing features quite a bit of arithmetic on data types that have limited precision. Usually it's not noticeable but its there, trust me.

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There are a lot of factors, the answer can be very ambiguous, I'm going to tell you an example: In a simple digital camera you have inside a matrix of sensor to transform the viewed space in a digital sample of it, the sensors are capable to receive red, green, blue and in some cases white light, one pixel is made by a combination of this sensors and the light beams don't arrive in the same proportion to the sensors, actually a normal distribution of the arriving beam is supposed on each sensor. This can be appreciated in the video of a non moving scene, if you look meticulously you will see how some pixels change its color in a noisy way.

As this simple effect is present in a simple camera can you imagine what are the problems in other types of sensors. In ultrasound the image is acquired using the same principle, there is a matrix of sensors to collect some information, this information is the reflected sound which is affected by the tissue of the observed region, the sounds of the observed region and a lot of things.

These are simple examples that could you to understand why images are noisy. If you are interested in more information, take a look in imaging extraction Technics, but always keep in mind that there are a lot of things that could interfere in the extraction process and think which can be these artifacts.

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In Radar systems you have both additive noise and speckle. The additive noise is generally the additive thermal noise. In the case of additive noise, you can improve the SNR by simply increasing the transmitted signal power - whether or not you can practically do this for your system is another matter.

In the case of multiplicative noise increasing the transmitted signal power will not improve the SNR because the noise increases in proportion to the transmitted signal power. See for example the reverberation problem in Sonar mentioned in another post.

In Radar, speckle is caused by the presence of lots of individual scatterers within a resolution cell (resolution due the pulse bandwidth and aperture length in Synthetic Aperture systems). Consider viewing a field of grass at 1m resolution - due the presence of a lot of individual blades of grass - the results add incoherently and produce what is known as speckle. If we increase the transmitted power, the grass will simply reflect more of that energy back to the receiver and they will still add incoherently, so you will not see any improvement in Signal to Noise Ratio.

In the example you give, the noise power is proportional to the sqrt of the signal, so the variance of the noise (assuming noise with zero mean) is proportional to the signal level - but the variance is the measure of noise power. Lets say the power in the signal $u=1$ and the variance of $\eta=1$ i.e. $E(\eta^2)=1$, so the SNR $= 1/1$.

If we scale the input signal by $A$ then, under the model you give, the signal power is $A^2$ but the noise power is $E(A\eta^2)=AE(\eta^2)=A$. Thus the new SNR = $A^2/A=A$ so there is some improvement in the SNR. But in the case of additive thermal noise the improvement would have been $A^2$ since the power of the additive thermal noise is unaffected by the signal power i.e. they are independent.

So your example is kind of a hybrid between the two extremes. For your reference another example of multiplicative noise, is phase noise in clocks for local oscillators, ADC etc.

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  • $\begingroup$ Thank you for this good answer. Can you give me the link of the post of the reverberation problem in Sonar and a reference of the phase noise in clocks for local oscillators, ADC ... $\endgroup$ – Achaire Aug 2 '15 at 1:48
  • $\begingroup$ @Achaire For Sonar you can refer to Neilson's "Sonar Signal Processing", or Burdic's "Underwater Acoustic System Analysis". Online you can read Principles of Radar and Sonar Signal Processing For phase noise - check the white papers by various ADC manufacturers: Analog Devices had some good ones, Linear Technology. $\endgroup$ – David Aug 10 '15 at 14:45
  • $\begingroup$ @Achaire Another online reference for Reverberation $\endgroup$ – David Aug 10 '15 at 14:51

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