Page 318 - Fundamentals of Light Microscopy and Electronic Imaging

SIGNAL-TO-NOISE RATIO 301

obtained by dividing the sum of components contributing to the signal by the square
root of the sums of the variances of the various noise components. If the noises are inde-
2
2
2
pendent of each other, they are added in quadrature (N N N . . .), and the noise
2
1
3
is shown as the square root of this term:
S + S + S K
SN = 1 2 3 .
/
2
2
2
( N + N + N K )
1 2 3
In digital microscope images, the principal signal components are the photoelec-
trons corresponding to the object and underlying background, and bias and thermal sig-
nals; the principal noises are photon noise (see below) and the camera readout noise (for
which there are several independent sources). We will recognize this general form of the
equation in descriptions of practical solutions. Two additional points should be noted:
(1) The statistics of photon counting are based on the number of photons converted to
electrons by the detector, and correspondingly, S/N is always calculated in electrons
(never in terms of analogue-to-digital units, which are the numeric values assigned to
pixels in digital images). The electron equivalent of an ADU is given as ADU gain,
where gain is the number of electrons per digital unit. (2) S/N statistics can be applied
on a per pixel basis for describing equipment performance, or to a group of pixels when
describing an extended object in an image.
Photon Noise

Because the number of photons recorded by the camera over a discrete interval of time
and space is stochastic, the accumulation of photoelectrons is described by a Poisson
distribution. If the photon signal is large, as it is for most microscope images, the prin-
cipal noise is photon noise (also called the shot noise), which is described as the stan-
dard deviation of the signal amplitude in photoelectrons. For a Poisson distribution, the
standard deviation (and therefore the photon noise) is simply the square root of the sig-
nal. Thus, for a background signal of 10,000 electrons, the photon noise 10,000
100 electrons. The fact that the photon noise is the square root of the signal explains why
S/N must be calculated in electrons, not in ADUs. Several additional points are worth
noting:

• An image is considered to be photon limited if the photon noise of the object signal

is greater than the camera read noise. For a CCD camera with 15 e read noise, this

occurs when the corrected photon count from the object reaches 225 e ( 20

ADU/pixel), since at this level the photon noise is 225 or 15 e , the read noise of
the camera. Because microscope images are relatively bright, most images are usu-
ally photon limited, and the following discussion is based on this assumption.
• Under photon-limited conditions, the S/N increases in proportion to the square root
of the exposure time or the square root of the number of averaged frames. Thus,
increasing the exposure time by 4 or averaging 4 like frames increases S/N 2-fold.
The poor S/N of a dim image can be greatly improved by simply increasing the
exposure time. For this reason, images should be acquired so that the brightest fea-
tures in the image approach the saturation level of the camera (see Chapter 14). This
relationship also explains why the S/N of confocal images can sometimes be dra-
matically improved by averaging replicate frames.