Description
1. Consider an amplified measurement C in electrons for a collection site (pixel) in a cooled
CCD camera described by
C = (S + NA + NP )A
where S is the signal in electrons, NA is a zero-mean amplifier noise source with a variance of 2
electrons, NP is a zero-mean photon noise source with a variance of S electrons, A is the constant
amplifier gain, and NA and NP are independent.
a) Write an expression for the variance of the measurement C.
b) Define the signal-to-noise of the quantity C as its mean divided by its standard deviation.
What is the signal-to-noise for the measurement C?
c) For what minimum value of S will the signal-to-noise exceed 50?
2. Consider an imaging system using a lens with focal length 4cm having an image plane 6cm
behind the lens. Assume the lens diameter is 1cm.
a) How far in front of the lens on the optical axis of the system must we place a point to get an
image without blur?
b) Suppose the image plane has an active area of 2cm × 2cm which is partitioned into 1000 ×
1000 square pixels. Supose the point in part a) images without blur to the center of a pixel.
How far can we move the point in focus towards the lens before the image of the point extends
to more than one pixel?
3. Assume that we have a CCD camera system that is cooled so that noise due to dark current
is negligible. Then digitized pixel values will be given by
D = (S + NA + NP )A + NQ (1)
where S is the signal in electrons, NA is the zero-mean amplifier noise source with variance σ
2
A,
NP is the zero-mean photon noise with variance S, A is the gain of the amplifier, and NQ is the
zero-mean quantization noise with variance σ
2
Q. Assume that the noise sources are independent.
a) Show that for a constant signal level S the expected value of D is
µ = SA (2)
and the variance of D is given by
σ
2
D = Aµ + σ
2
C (3)
where
σ
2
C = A
2σ
2
A + σ
2
Q (4)
b) Write a program that reads a digital image I(x, y) of size N ×N where N = 100 and estimates
µ by
µb =
1
N2
X
1≤x≤N
X
1≤y≤N
I(x, y) (5)
and σ
2
D by
σb
2
D =
1
N2 − 1
X
1≤x≤N
X
1≤y≤N
(I(x, y) − µb)
2
(6)
Run your program to find µb and σb
2
D for each of four images provided by your TA.
c) Plot σb
2
D versus µb for these four points. Estimate A and σ
2
C using a least squares fit of the
line given by (3) to your data. A program that shows how to read an image will be provided.
Demonstrate your program as described by your TA.
