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Lab work on detection and estimation theory, SiCoM 3A 2013

This lab worked is based on a paper published in 2002 whose reference isO.Michel, A.Ferrari : A brief introduction to detection problems and il lustra-tion, in [Astronomy with high contrast imaging, EAS publication series, 2002.

1 Studying a toy model for direct detection

of extra-solar planet

1.1 Problem statement and notations

A set of N elementary contiguous photo-detectors is considered. All the de-tectors are supposed to be perfectly identical to each other and independentfrom each other (the response to a photon or a photonic flux does affect thecell receiving the photons, but not its neighbors). Such effects as diffusion andblooming are not considered here. The effect of the coronographic setup and theeventual adaptive optic are assumed to be perfect. For sake of conciseness in thenotations and without loss of generality, a one-dimensional image is consideredin the sequel. Hence, the toy problem under consideration resumes to testingthe following hypothesis:

H0 : n = 1 . . . N , xn Poiss(f) (1)

H1 :

n = n1, . . . , n1 + r 1, xn Poiss(f + hA(n (n1 +

r12 )))

otherwise: xn Poiss(f)(2)

where f is the background noise intensity, (h, n1) are the brightness and positionof the planet and A(n) is the optical instrument response function (the PSF isusually considered an Airy function) of width r1.

We will consider herein a telescope of 100m2 operating at a wavelength of10m and in a bandwith of 1m. According to these values, every second anaverage of 102 photons will be received from the planet and 108 photon will berecieved from the star. We assume that the coronograph reduces the star fluxby a factor 104, that the optical instrument response function spreads over 10pixels, typical values for the zodiacal and exo-zodiacal background noise, andquantum efficiency of the CCD. These asumptions lead to typical optimisticvalues ofh = 10T, f = 104T and r = 10 pixels, where T stands for the exposuretime.

1.2 Detection algorithm with known parameters

In the sequel, both the position n1 of the planet and the flux h received from itare assumed to be known.

1xn Poiss(f) ifP(xn = k) = exp(f)fk

k!)

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Let X1:N be the N-dimensional joined observation of the N pixels, X1:Ndef=

[x1, . . . , xN]T.

1. Express the likelihood function of X1:N under H0 and the alternativehypothesis H1.

2. Express the Neyman Pearson statistics for the problem.

3. Assuming hf

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Here, the unknown parameters n1 and h are estimated by maximizing thelog-likelihood (n1, h).

1. Prove that h may be obtained by solving

0 = 1 +n1+r1n=n1

xnA(n (n1 +r12 ))

f + hA(n (n1 +r12 ))

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2. Compute 2(n1, h)/2h and show that it is always strictly negative if at

least one variable xn on [n1, n1 + r 1] is non-zero.

As the first derivative of(n1, h) possesses r negative singularities, the equa-tion (n1, h)/h = 0 has at most one positive solution. For a given n1, maxi-mizing the likelihood with respect to h resumes to identify the unique positiveroot of the r-th order polynomial equation:

(h) = P(h) +

(r1)/2n=(r1)/2

xn1+n+(r1)/2Qn(h) (6)

where polynomials P(h) and Qn(h) are defined as:

P(h) = 1

(r1)/2n=(r1)/2

(f + hA(n)) (7)

Qn(h) = A(n)

(r1)/2q=(r1)/2,q=n

(f + hA(q)) (8)

P(h) and Qn(h) do not depend on n1 or the data and can therefore be computedonly once. Note that if the larger root of this polynomial equation is negative,necessarily the value h = 0 maximizes (n1, h) over [0,].

Computation of the roots of (h) is not be a problem here, the values of rbeing r = 10. The estimation of h can also be performed for the model derivedin the previous section. The main drawback of this solution is that it leads tothe computation of the roots of a polynom with order 2r and no direct techniquefor the proper root selection.

TO DO Implement the detection scheme that can now be summarized asfollows

1. For n1 {1, . . . , N r}: compute h(n1) from

h(n1) = max{0, root((h))

2. The ML of n1, n1, is the global maximum ofLP( X1:N; h(n1)) for n1 {1, . . . , N r}.

3. The ML estimate of h ish = h(n1).3

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4. The GLRT is:

LP( X

n1:n1+r1

;h)H0H1

Report the result obtained on a single plot showing the estimated positionsn1 on the x-axis, and thee stimated h on the y-axis. The simulations generatedin the first part of the lab work may be used for this question. Comment yourresults.

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