Last updated on Saturday, September 30, 1995 at 5:00 PM.
Some forms of noise are fixed and predictable: for example, a 60 Hz hum on a transmission wire introduced by cross-over from power lines.
In general, though, noise is a random process. As such, it is usually characterized using the language of statistics.
Since the units of measure may be arbitrary, we typically normalize
the standard deviation by comparing it to the mean: 
.
We can consider the picture 
as a random variable from which we sample an
ensemble of images from the space of all possibilities.
This ensemble has a mean (average) image, which we'll denote as
.
As with most stochastic processes, if we sample enough images,
the ensemble mean 
approaches the noise-free
original signal.
So, one way to often eliminate noise is to take a lot of pictures. However, this usually isn't feasible.
If we compare the strength of a signal or image (the mean of
the ensemble) to the variance between individual acquired images
we get a signal-to-noise ratio:
A high signal-to-noise ratio indicates a relatively clean signal
or image; a low signal-to-noise ratio indicates that the noise is
great enough to impair our ability to discern the signal in it.
If 
is the ensemble mean, we can measure the
difference between each sampled image 
and the ensemble mean by
.
We can think of this as a (really big!) vector that measures how each
pixel in the sample differs from the average at that position.
By multiplying
by its transpose, we get a matrix where each ij-th element
is the product of
the difference between pixel i and its average and
the difference between pixel j and its average.
By taking the average of this matrix over all images in the ensemble,
we get a covariance matrix
(Notice that this is the same form that we used in the last lecture
when discussing the Hotelling transform.
Then, we were seeing how each element of a multi-valued pixel correlated
with the others.
Here, we're seeing how each pixel correlates with every other
pixel--a much bigger matrix!)
If the covariance matrix is diagonal, each element of the diagonal is the variance of the corresponding pixel, and there is no correlation between the effect of noise on one pixel and the effect of noise on another pixel. This type of noise is called uncorrelated.
If the covariance matrix has non-zero elements off the diagonal, there exists a correlation between the effect of noise on one pixel and effect on another. Such noise is called correlated.
|
In-class question: what do you think correlated noise looks like? |
One common distribution for the values of a each pixel is determined by the nature of light itself. Light isn't a continuous quantity, but occurs in discrete photons. These photons don't arrive in a steady stream, but sometimes vary over time. Think of it like a flow of cars on a road--sometimes they bunch together, sometimes they spread out, but in general there's an overall average flow.
Discrete arrivals over a period of time are modeled statistically
by a Poisson distribution:
A Poisson distribution is similar to a Normal or Gaussian distribution with the following exceptions/properties:
.
Consider then, what happens to the signal-to-noise ratio for
an image with Poisson image:
In other words, the signal-to-noise ratio of an image with Poisson noise increases with the square root of the mean.
Remember, though-the mean value is such an image represents the average number of photons counted during a particular period of time. Any imaging modality with low photon counts is inherently noisy-you simply can't get around the Poisson noise due to the quantum nature of light.
So, one way to decrease the effects of Poisson noise is to increase the photon count. This is not, however, as easy as it sounds. Increasing photon count may mean
This often isn't feasible when imaging moving objects. Sports photographers, astronomers, and cardiologists all have to deal with this tradeoff.
High-speed photographers may want to use brighter light to illuminate their scene. This is usually harmless, but may be something over which he has no control.
Doctors can get better medical images by increasing the strength of the radiation. Make the X-ray beam stronger, give the patient a little more radiopharmeceutical to drank, etc. The patients may object to this, however, so this sometimes isn't feasible. (You can get really great images from cadavers, though.)
If the problem is not catching enough photons, use a larger photon-catcher. This, however, means a loss of image resolution.
This tradeoff between noise and resolution is an inevitable one in vision. If you want a perfect-resolution image, use an infinitely-small imaging element, but don't hold your breath waiting for a photon to hit this infinitely small hole. If you want a noise-free image, use an infinitely-large imaging element, but don't hope to discern any spatial sructure.
Unlike Poisson noise, which varies with the image intensity, such Gaussian-distributed noise is usually uniform over the image.
Noise 
that is Gaussian-distributed, zero-mean
(doesn't change the average intensity level), uncorrelated, and
additive is called white noise.
Some forms of noise are "colored". "Blue noise", for example, has light low-frequency content and large high-frequency content.
© Bryan S. Morse, 1995