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Table A.1.

Description of the mathematical notations for the variables used in the RSM detection map, KLIP FM, and LOCI FM computation.

Symbol Dimension Comments

RSM map
xia θ × θ × TLa Patch of residuals centred on pixel ia
Fia TLa Realisation of a two-state Markov chain representing the state in which the system is for pixel ia
m θ × θ Cropped planetary signal (off-axis PSF)
εs,ia 2θ × θ × TLa Error terms associated with the two regimes
Sia TLa State in which the system is for every pixel ia
ξs, ia 2 × TLa Probability associated with state s for every pixel ia
ηs, ia 2 × TLa Likelihood of being in each state for every pixel ia
pq, s 2 × 2 Transition probabilities between the regimes
μ 1 Mean of the residuals contained in an annulus a, with width equal to θ
σ 1 Standard deviation of the residuals contained in an annulus a, with width equal to θ
β 1 Parameter representing the intensity of the planetary signal in the cube of residual
a 1 Annulus index
La 1 Number of pixels included in the annulus a
T 1 Number of frames in the cube of residuals
ia 1 Index associated with every pixel from every frame in the annulus a (ranges from 1 to TLa)
θ 1 Angular size of the considered planetary signal (set to 1λ/D)

KLIP FM

i Npix Vectorised science image/annulus before speckle subtraction
m Npix Vectorised planetary signal (off-axis PSF) inside the selected annulus
p Npix Vectorised forward-modelled planetary signal inside the selected annulus
x Npix Vectorised processed annulus after speckle subtraction
pia/pj θ × θ Derotated and cropped forward-modelled planetary signal for pixel ia/frame j
R NR × Npix Reference library matrix
VK K × NR Eigenvector matrix of the covariance matrix RRwith VK = [v1,v2,…,vk]
Λ NR × Npix Diagonal matrix with the eigenvalues of the covariance matrix RR with Λ = diag(μ1, μ2, …, μk)
ZK K × NR Karhunen-Loéve image matrix
ΔZK K × NR Perturbation of the Karhunen-Loéve image matrix
M NR × Npix Planet signal component in the reference library R
CMR NR × NR Covariance Matrix between M and R
Npix 1 Number of pixels in the selected annulus of width equal to one FWHM
NR 1 Number of reference frames used for the speckle field computation
K 1 Number of principal components used for the speckle field computation

LOCI FM

ck NR Factors of the linear combination used to model the speckle field
oi Npix Selected annulus in the science image used for the estimation of the linear factors ck
ok Npix Selected annulus in frame k of the reference library used for the estimation of the linear factors ck
rk Npix Selected annulus in frame k of the reference library used for the computation of the speckle field
mi Npix Selected annulus in frame corresponding to the science image in the planetary signal matrix
mk Npix Selected annulus in frame k of the planetary signal
Npix 1 Number of pixels in the annulus with width equal to three FWHM used for the factors estimation
Npix 1 Number of pixels in the annulus with width equal to one FWHM used for speckle field computation
NR 1 Number of reference frames used for the speckle field computation

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