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IMAGE SENSORS, PIXEL SIZE, AND POISSON NOISE (continued)

8. COPING WITH POISSON
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Key words:
ENOB | GRAY-LEVEL RESOLUTION | FULL-WELL CAPACITY | SATURATION EXPOSURE | COVERAGE | RADIANT ENERGY | PIXEL AREA | RADIANT EXPOSURE | PIXEL SIZE | ILLUMINATION | SILICON AREA | SPATIAL RESOLUTION | BINNING | TIMING SCHEME | AVERAGING | PHOTON CONTENTS |

Illumination and pixel size for ENOB = 12:

What we would like to get is a clean and noise-free signal with a GRAY-LEVEL RESOLUTION of, say, 12 bits.

Is that possible?
How much FULL-WELL CAPACITY would be needed?
Which PIXEL SIZE at which SATURATION EXPOSURE could fill such a well?

Let   Np = number of photons collected during 1 integration time.

If Np represents saturation exposure, then Np equals FULL-WELL CAPACITY.

Then, as shown in Para. 5, it is clear that
SNR = (1/6) * Np^0.5                   (1)
and
ENOB = ld [(1/6) * Np^0.5]             (2)

Equation (2) shows in general, how ENOB depends on FULL-WELL CAPACITY for 99.7% COVERAGE. Important enough for plotting it as a function graph: Do you need the exact numbers? - Check my spreadsheet for Microsoft Excel (23 kByte)!

If we need
ENOB = 12 ,  then
SNR = 4096                             (3)
is necessary
(because   ld 4096 = 12   or   2^12 = 4096 ).

We use eqation (1) with (3) and get:

4096 = (1/6) * Np^0.5
Np = (6*4096)^2 = 6.04E8              (4)

So,
FULL-WELL CAPACITY should be at least 600 E6 electrons.

As we learnt in para. 7, single photon energy at a wavelength of  650 nm  is
Ep = 30E-20 J

So, 6.04 E8 photons represent a (pixel saturating) RADIANT ENERGY of
Esat = 6.04 E8 * 30E-20 J = 181.2 E-12 J     per Pixel.

At bigger or smaller PIXEL AREAs  Ap , this leads to smaller or bigger necessary RADIANT EXPOSUREs  Hsat  for saturating the sensor array:
Hsat = Esat / Ap = 181.2 E-12 J / Ap    (5)

Assuming that the pixels are square, the PIXEL AREA Ap will be:
Ap  =  (pixel edge)^2                 (6)

Replacing  (6)  for  Ap  in  (5) , we get  Hsat  as a function of pixel edge length. This function is shown in the next figure. The sharp increase of necessary exposure at PIXEL SIZEs below 5 micrometers strikes the eye. You could assume that  ENOB=12  is rendered impractical  --  if not impossible  --  at PIXEL SIZEs below 2 micrometers.

Again I must add a note of caution:
Obeying the limits of figures 8-a and 8-b does not ensure the 12th (i.e. least significant) bit to stay clean and noise-free eternally. As stated in para. 5, in the long run 0.3% of the pixel gray-level values can be deteriorated by Poisson noise.

Well, this was one method of keeping Poisson noise small:

* choose big pixels,
* choose big FULL-WELL CAPACITY, and hence:
* choose big SATURATION EXPOSURE ... and
* use this sensor close to its saturation limit.
This will help. But because of other constraints, in most cases this means
- low sensitivity, and
- low SPATIAL RESOLUTION.

Three golden rules against Poisson noise can be derived:
1) Image ILLUMINATION is the stronger the better;
2) sensor SILICON AREA is the bigger the better;
3) use your image sensor as close to SATURATION EXPOSURE as possible.
Coarse, but effective.

Below, I show two more ways of minimizing Poisson noise, but look out:
On these ways, too, you are burdened with nearly the same drawbacks like above.

BINNING:

With some CCD arrays, you can use special TIMING SCHEMEs that "pour" the charges from two (or more) neighbouring pixels at a time into the charge detector. (And CMOS arrays, too, can be designed in an equivalent way.)

Result is a larger effective number of photons per signal pixel and hence, improved SNR.
At the expense of lower SPATIAL RESOLUTION.

AVERAGING:

Having stored an image in a frame memory (e.g. one pixel grey level per byte), you can add the next subseqent frame (from the same scene) pixel by pixel to the stored image.
Then, divide every pixel grey level sum by two and you get the average out of two images.
With doubled PHOTON CONTENTS per pixel and accordingly improved SNR.

Drawback is reduced scanning speed.

Of course this algorithm is not limited to two frames.
Provided the scene is really still, you may average a hundred or more frames.

Struggling against the laws of physics is always costly.

Continued: 9. CONCLUSION

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