Noise:PHOTON NOISE, THERMAL NOISE, KTC NOISE, QUANTIZATION NOISE

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...Image Processing Fundamentals

where Ne is the number of even chain codes, No the number of odd chain codes,

and Nc the number of corners. The specific formulas are given in Table 7.

Coefficients

Formula

Reference

Pixel count

[18]

Freeman

[11]

Kulpa

0.9481

0.9481 · 2

[20]

Corner count

0.980

1.406

0.091

[21]

Table 7: Length estimation formulas based on chain code counts (Ne, No, Nc)

5.2.3 Conclusions on sampling

If one is interested in image processing, one should choose a sampling density

based upon classical signal theory, that is, the Nyquist sampling theory. If one is

interested in image analysis, one should choose a sampling density based upon the

desired measurement accuracy ( bias) and precision (CV). In a case of uncertainty,

one should choose the higher of the two sampling densities (frequencies).

Noise

Images acquired through modern sensors may be contaminated by a variety of

noise sources. By noise we refer to stochastic variations as opposed to deterministic

distortions such as shading or lack of focus. We will assume for this section that

we are dealing with images formed from light using modern electro-optics. In

particular we will assume the use of modern, charge-coupled device (CCD)

cameras where photons produce electrons that are commonly referred to as

photoelectrons. Nevertheless, most of the observations we shall make about noise

and its various sources hold equally well for other imaging modalities.

While modern technology has made it possible to reduce the noise levels associated

with various electro-optical devices to almost negligible levels, one noise source can

never be eliminated and thus forms the limiting case when all other noise sources

are "eliminated".

6.1 P HOTON NOISE

When the physical signal that we observe is based upon light, then the quantum

nature of light plays a significant role. A single photon at λ = 500 nm carries an

energy of E = hν = hc/λ = 3.97 × 1019 Joules. Modern CCD cameras are

sensitive enough to be able to count individual photons. (Camera sensitivity will be

discussed in Section 7.2.) The noise problem arises from the fundamentally

...Image Processing Fundamentals

statistical nature of photon production. We cannot assume that, in a given pixel for

two consecutive but independent observation intervals of length T, the same

number of photons will be counted. Photon production is governed by the laws of

quantum physics which restrict us to talking about an average number of photons

within a given observation window. The probability distribution for p photons in an

observation window of length T seconds is known to be Poisson:

( ρT ) p e-ρT

P( p | ρ , T) =

(62)

where ρ is the rate or intensity parameter measured in photons per second. It is

critical to understand that even if there were no other noise sources in the imaging

chain, the statistical fluctuations associated with photon counting over a finite time

interval T would still lead to a finite signal-to-noise ratio (SNR). If we use the

appropriate formula for the SNR (eq. (41)), then due to the fact that the average

value and the standard deviation are given by:

average = ρT

(63)

Poisson process

σ = ρT

we have for the SNR:

SNR = 10 log10 (ρT ) dB

(64)

Photon noise

The three traditional assumptions about the relationship between signal and noise

do not hold for photon noise:

· photon noise is not independent of the signal;

· photon noise is not Gaussian, and;

· photon noise is not additive.

For very bright signals, where ρT exceeds 105, the noise fluctuations due to photon

statistics can be ignored if the sensor has a sufficiently high saturation level. This

will be discussed further in Section 7.3 and, in particular, eq. (73).

6.2 THERMAL NOISE

An additional, stochastic source of electrons in a CCD well is thermal energy.

Electrons can be freed from the CCD material itself through thermal vibration and

then, trapped in the CCD well, be indistinguishable from "true" photoelectrons. By

cooling the CCD chip it is possible to reduce significantly the number of "thermal

electrons" that give rise to thermal noise or dark current. As the integration time T

increases, the number of thermal electrons increases. The probability distribution of

...Image Processing Fundamentals

thermal electrons is also a Poisson process where the rate parameter is an

increasing function of temperature. There are alternative techniques (to cooling) for

suppressing dark current and these usually involve estimating the average dark

current for the given integration time and then subtracting this value from the CCD

pixel values before the A/D converter. While this does reduce the dark current

average, it does not reduce the dark current standard deviation and it also reduces

the possible dynamic range of the signal.

6.3 ON-CHIP ELECTRONIC NOISE

This noise originates in the process of reading the signal from the sensor, in this

case through the field effect transistor (FET) of a CCD chip. The general form of

the power spectral density of readout noise is:

ω -β

ω < ω min

β> 0



Snn (ω ) ∝  k

ω min < ω < ω max

(65)

Readout noise

 ωα

ω > ω max

α >0



where α and β are constants and ω is the (radial) frequency at which the signal is

transferred from the CCD chip to the "outside world." At very low readout rates (ω

< ωmin) the noise has a 1/ƒ character. Readout noise can be reduced to manageable

levels by appropriate readout rates and proper electronics. At very low signal levels

(see eq. (64)), however, readout noise can still become a significant component in

the overall SNR [22].

6.4 KTC NOISE

Noise associated with the gate capacitor of an FET is termed KTC noise and can be

non-negligible. The output RMS value of this noise voltage is given by:

σ KTC =

(66)

KTC noise (voltage)

where C is the FET gate switch capacitance, k is Boltzmann's constant, and T is the

absolute temperature of the CCD chip measured in K. Using the relationships

Q = C · V = Ne - · e- , the output RMS value of the KTC noise expressed in terms

of the number of photoelectrons ( Ne - ) is given by:

kTC

σ Ne =

(67)

KTC noise (electrons)

where e is the electron charge. For C = 0.5 pF and T = 233 K this gives

Ne - = 252 electrons. This value is a "one time" noise per pixel that occurs during

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