Combining reference-star and angular differential imaging for high-contrast imaging of extended sources

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Volume 688 (August 2024)

A&A, 688 (2024) A185

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Table 1

Notations.

Notation	Definition
Constants

m	Width of an image (i.e., the number of pixels on one side of the square images)
n	Number of frames in the ADI cube
r	Number of frames in the reference library
q	Number of dimensions in the PCA low-rank subspace (aka rank)

Matrices

Y ∈ ℝ^n,m×m	ADI dataset, i.e., a cube of vectorized images
L_r ∈ ℝ^r,m×m	Library of r reference stars
$\bar{S}$ $\[\bar{S}\]$ ∈ ℝ^n,m×m	Estimated speckle fields for each image in Y
$\bar{d}$ $\[\bar{d}\]$ ∈ ℝ^+,m×m	Estimated circumstellar signal (forced to be positive)
U · ∑ · V	Singular value decomposition of a matrix

Operators

∥_∥	Absolute value
ℋ_q(Y)	Operator (ℝ^n,m×m ↦ ℝ^n,m×m) returning the estimated speckle fields via PCA using q principal components.
$H_{q}^{L_{r}} (Y)$ $\[\mathcal{H}_q^{L_r}(Y)\]$	Operator (ℝ^n,m×m ↦ ℝ^n,m×m) returning the estimated speckle fields via PCA using the q principal components of a set of reference L_r
\|_\|₁	l₁-norm, sum of the absolute value of all elements in a matrix
$Q_{Y}$ $\[\mathcal{Q}_Y\]$	Rotation operator (ℝ^m×m ↦ ℝ^n,m×m) creating a cube of n images where each frame contains the same image rotated according to the parallactic angles of the sequence of observations Y. This operator is used to inject simulated disks or planets into the data.
$Q_{Y}^{- 1}$ $\[\mathcal{Q}_Y^{-1}\]$	Median image of a cube after de-rotation (ℝ^n,m×m ↦ ℝ^m×m)
[_;_]	Concatenation of two collections of images

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