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EECS 101: HOMEWORK #2 solution

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1. Consider a noiseless CCD camera with perfect quantum efficiency (q(λ) = 1) of 501 × 501
exposed potential wells with an integration time of 1 millisecond. Assume that a spot of light
of 109 photons per second is focused on the single potential well at the center of the array. All
other potential wells receive no light. Assume that the pixel clock operates at 7.5 MHz.
a) Describe in detail the 501 × 501 image (electrons/potential well) that will be generated by
the CCD if interline transfer is used.
b) Describe in detail the 501 × 501 image (electrons/potential well) that will be generated by
the CCD if frame transfer is used.
2. Consider an amplified measurement C in electrons for a collection site (potential well) 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
1 electron, NP is the photon noise source, and A is the constant amplifier gain. (Hint: read
problem 4 before working on this problem)
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 100?
3. Consider an imaging system using a lens with focal length 4cm having an image plane 6cm
behind the lens. Assume the lens diameter is 2cm.
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 500 ×
500 square potential wells (collection sites). Supose the point in part a) images without blur to
the center of a potential well. How far can we move the point in focus towards the lens before
the image extends to more than one potential well.
4. 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. If we assume that the three noise sources are
independent, 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
2
σ
2
A + σ
2
Q (4)
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 compute µb and σb
2
D for each of four images provided by your TA.
Plot σb
2
D versus µb for these four points. Estimate A and σb
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 in lab.
You are required to demonstrate your program to your TA during lab.