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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 RR⊤with 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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