Databases Reference
In-Depth Information
2
d
Notice we have used the symbol
σ
for the mean squared error, which implies that the variance
of the distortion sequence
d
is equal to the second moment, or that the distortion
sequence is zero mean. This would happen if the error introduced during source coding is
additive. This idea that the error incurred during compression is additive is a common (though
often unstated) assumption. If we are interested in the size of the error relative to the signal,
we can find the ratio of the average squared value of the source output and the mse. This is
called the
signal-to-noise ratio
(SNR):
(
x
n
,
y
n
)
x
=
σ
SNR
(5)
2
d
σ
2
x
2
where
d
is the mse.
The SNR is often measured on a logarithmic scale, and the units of measurement are
decibels
(abbreviated to dB):
σ
is the average squared value of the source output, or signal, and
σ
x
10 log
10
σ
SNR
(
dB
)
=
(6)
2
d
σ
Sometimes we are more interested in the size of the error relative to the peak value of the
signal
x
peak
than in the size of the error relative to the average squared value of the signal. This
ratio is called the
peak-signal-to-noise-ratio
(PSNR) and is given by
x
peak
σ
PSNR
(
dB
)
=
10 log
10
(7)
2
d
Another difference distortion measure that is used quite often, although not as often as the
mse, is the average of the absolute difference, or
N
1
N
1
|
y
n
|
d
1
=
x
n
−
(8)
n
=
This measure seems especially useful for evaluating image compression algorithms.
In some applications, the distortion is not perceptible as long as it is below some threshold.
In these situations, we might be interested in the maximum value of the error magnitude,
d
∞
=
|
x
n
−
y
n
|
(9)
max
n
We have looked at two approaches to measuring the fidelity of a reconstruction. The first
method involving humans may provide a very accurate measure of perceptible fidelity, but it
is not practical or useful in mathematical design approaches. The second is mathematically
tractable, but it usually does not provide a very accurate indication of the perceptible fidelity
of the reconstruction. A middle ground is to find a mathematical model for human percep-
tion, transform both the source output and the reconstruction to this perceptual space, and
then measure the difference in the perceptual space. For example, suppose we could find a
transformation
that represented the actions performed by the human visual system (HVS)
on the light intensity impinging on the retina before it is “perceived” by the cortex. We could
V
