# Coherent Speckle: an extreme class of noise

Most image-processing contexts deal with additive noise, in which the signal received has the form of the signal sent plus a (usually comparatively small) addition. Coherent speckle, by contrast, is a form of multiplicative noise. A signal corrupted with coherent speckle has the form of the source signal multiplied by a random variate from the negative exponential distribution.

Another way to look at this is to say that, with additive noise, the signal sent is the expected value of the signal received (seen as a random variate) whereas, with coherent speckle, the signal sent appears as the standard deviation of the (complex zero-mean Gaussian) signal received. This complex signal's amplitude is then negative exponentially distributed with mean and standard deviation equal to the source signal.

In an image corrupted with coherent speckle there are no regions of even approximately constant signal (unless the signal is truly zero on a region), even when the original image is truly uniform. To obtain a region on which it is possible to estimate the mean signal it is, of course, necessary to have some idea of a region (or segment) on which the source signal was essentially constant. Thus it is necessary to segment such images before estimating their visual properties: however, segmentation depends on having some estimate of the source signal. The resulting bootstrap problem makes the segmentation of images corrupted with coherent speckle uncommonly difficult.

## Why Coherent Speckle ?

Coherent speckle arises from the summation of signals from scatterers illuminated with a coherent signal (sonar, radar or a laser) when the resolution possible with the image formation system is comparable with (or coarser than) the wavelength of the illumination, yet fine enough that there are only a modest number of scatterers within the reolution of any given scatterer. All the illumination, being coherent, approaches the scatterers in phase: the reflected signals from objects in a single resolution cell travel various distances differing by up to twice the cell-size. The cell-size not being small compared to the wavelength, this variation in distance produces arbitrary phase differences. This makes both destructive and constructive interference possible, along with every intermediate. The resulting signal then depends on the details of where in the cell the scatterers lie: at least where it is practical to treat the positions of scatterers as being random and uncorrelated, this leads to a summed signal whose form is the sum of a complex random walk. For large numbers of scatterers, this has the form of a zero-mean complex Gaussian variate whose standard deviation is the total scattering cross-section of the scatterers.

So this form of speckle is called coherent because it arises from the summation of reflected signals from coherent illumination – i.e. the illuminating radiation's waves are all in phase with one another.

Written by Eddy.
This page is an adjunct of my curriculum vitae.