*********** Square Rooter ************

[MYPIXEL at aol.com]

Posted to CCD-world:
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Michael Hoffberg,

Your question below reminds me of an optimum CCD encoding technique we tried
that involves nonlinear encoding. . .  the following discussion on Square
Rooters may help answer your question .. . . . 

Jim Janesick

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Starting from basics . . . .

The number of bits required to encode the dynamic range of a CCD is given by:

BITs =  FW / K(e-/DN)
where FW is the full well (e-). 

To assure there is no quantizing noise we set the camera gain to K(e-/DN) =
CCD Read Noise (a rule of thumb).
Substituting in the above equation yields . . .

BITs = FW / R(e-)
where R(e-) is the read noise. 

i.e.,  the number of bits required is simply equal to the dynamic range of the
CCD.
                                 

Example  1.

         Find the number of bits required to encode a CCD that has a full well
capacity of 100,000 e- and read noise of 3 e-.

	Solution:

	From Equation above:

		Bits = 100,000 / 3

		Bits = 33,333

	A 16-bit (216 = 65,536 bits) ADC would be sufficient to cover the specified
dynamic range and remain CCD noise limited. 

                       


SQUARE ROOTING 

The discussion above shows that CCD read noise can be properly encoded when
K(e-/DN) = R(e-). The same arguments also apply when encoding shot noise
efficiently. In that shot noise increases with signal (by the square-root of
signal) a linear ADC encodes shot noise with unnecessary precision. For
efficient processing the camera gain, K(e-/DN), should increase with signal
as,

K(e-/DN) = (S(e-) + R^2)^1/2

This encoding scheme  is referred to as a "square-rooter" and encodes read
noise and shot noise equally. The number of bits required for a square-rooter
is equal to,

Bits required = Signal ^1/2 / K(e-DN)
                                

Example 2

	Find the number of bits required to encode both linearily and nonlinearily
with a square-rooter for the CCD described in Example 1.

	Solution:

	From Example 1 linear encoding requires 33,333 bits or a 16-bit converter.
Square-rooting only requires 100 bits which can be provided by an 8-bit ADC.
Note that the gain, K(e-DN) at the read noise level is the same for both
encoders (i.e.,  10 e-/DN)). It is interesting to note that the Photon
Transfer response is a straight line of constant noise (1 DN rms) when square
rooting is applied.
                       
Unfortunately nonlinear encoding such as a square-rooter is difficult to
implement in practice. Logarithmic amplifiers do not exhibit a perfect
response leading to camera linearity problems when CCD data is mapped back
into the linear domain. Also, nonlinear circuits used are inherently slow and
show instabilities in offset level. The latter characteristic is important
because an offset change implies a gain change which must be taken into
account when reducing CCD data. A compromise in making the square-rooter
useful for scientific applications is to encode the CCD linearily and employ a
digital square-root look-up table to compress bits. For example, the output of
a 12-bit converter can mapped into 8 bits without loss of information
(employed on the Cassini mission to Saturn).

*************jj




In a message dated 98-02-04 12:16:29 EST, HOFFBEMG at sterlingdi.com writes:

<< Subj:	 logarithmic amplifier question
 Date:	98-02-04 12:16:29 EST
 From:	HOFFBEMG at sterlingdi.com (Michael Hoffberg)
 Reply-to:	CCD-world at cfht.hawaii.edu
 To:	ccd-world at cfht.hawaii.edu
 
 Posted to CCD-world:
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 The noise characteristics of a CCD camera should follow poisson statistics,
that
 is the noise should be proportional to the square root of the signal.  
 
 We are characterizing a Kodak DC 210 camera and have observed that the noise
is
 loosly inversly proportional to the signal.  We suspect that Kodak is using a
 log amplifier and I would like to know how this would impact the noise
 characteristics of the camera output.
 
 MIKE
  
 
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