KL-Projection Post-processing

• Post-processing via KL-projection is a method that projects raw data onto subspaces minimizing sensitivity to nuisance parameters using the Kullback–Leibler divergence.
• It employs mathematical frameworks, including singular value decomposition and information projection, to isolate and reduce dominant error modes in complex imaging systems.
• The technique enhances detection limits in astronomical imaging by integrating robust observables with classical post-processing pipelines and variational control methods.

Post-processing via KL-projection refers to a family of model-based methods for suppressing dominant error modes or extracting statistically optimal observables from raw data by projecting onto subspaces that minimize sensitivity to nuisance parameters, as quantified by the Kullback–Leibler (KL) divergence. In astronomical imaging, particularly in high-contrast exoplanet detection with coronagraphs, KL-projection is applied to isolate astrophysical signals from instrumental aberrations by leveraging knowledge of the system response. In a broader mathematical setting, KL-projection is unified with information projection principles on Banach spaces, where it identifies an optimal shift of a reference Gaussian measure to be closest to a target in KL-divergence, yielding solutions that connect control, variational, and stochastic optimization perspectives.

1. Mathematical Frameworks of KL-projection

In direct imaging of exoplanets with coronagraphs, the observed intensity $I(\Delta E_s)$ at the detector plane is governed by the optical system's response to wavefront errors (WFE) $\Delta E_s$. The nominal starlight intensity for a reference (ideal) wavefront $E_{s0}$ is $I_0 = |C E_{s0}|^2$, where $C$ is the coronagraph operator. When aberrations and astrophysical signals (e.g., planets) are present, the intensity expands as

$$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$

where $E_{p0}$ is the planetary field. The dominant error terms are the linear "speckle-pinning" contribution $2\Re[(C E_{s0})^* (C \Delta E_s)]$ and the quadratic aberration term $|C \Delta E_s|^2$. These contributions can be recast as linear and quadratic forms via transfer matrices $A_l$ and $\Delta E_s$0:

• Linear: $\Delta E_s$1
• Quadratic: $\Delta E_s$2, where $\Delta E_s$3 comprises pairwise products of WFE coefficients. These models underpin downstream KL-projection techniques for robust observable extraction (Xin et al., 2024).

Within the general measure-theoretic setting, KL-projection becomes minimizing divergence between a shifted Gaussian measure $\Delta E_s$4 and a target $\Delta E_s$5 over allowable shifts $\Delta E_s$6 in the Cameron–Martin space $\Delta E_s$7:

$\Delta E_s$8

where $\Delta E_s$9 and $E_{s0}$0 is the Kullback-Leibler divergence. On Banach or Wiener space, this projection is equivalent to minimizing Onsager–Machlup functionals, which encode both the control energy of the shift and the relative entropy cost (Selk et al., 2020).

2. KL-projection via Instrumental Mode Decomposition

To mitigate the impact of dominant aberration modes in coronagraphic data, KL-projection techniques employ singular value decomposition (SVD) of the transfer matrix $E_{s0}$1 (either $E_{s0}$2 or $E_{s0}$3). For a system with $E_{s0}$4 pixels and $E_{s0}$5 WFE basis modes (or $E_{s0}$6 quadratic terms), SVD yields orthonormal pixel-space basis vectors $E_{s0}$7 ordered by speckle sensitivity.

The projection operator $E_{s0}$8 is then defined to remove the $E_{s0}$9 leading modes (highest singular values) most sensitive to WFE:

$I_0 = |C E_{s0}|^2$0

such that, for a measured frame $I_0 = |C E_{s0}|^2$1, the projected data $I_0 = |C E_{s0}|^2$2 yields "robust observables." In pixel space, the equivalent projector $I_0 = |C E_{s0}|^2$3 is

$I_0 = |C E_{s0}|^2$4

which annihilates the subspace of maximal error contamination. This procedure generalizes principal component analysis for application to structured error models with known system response (Xin et al., 2024).

3. Information Projection and Variational Characterization

In a probabilistic setting, information projection on Banach spaces with Gaussian reference measure $I_0 = |C E_{s0}|^2$5 seeks the member of a feasible set $I_0 = |C E_{s0}|^2$6 (typically shifts of $I_0 = |C E_{s0}|^2$7) closest to a target $I_0 = |C E_{s0}|^2$8 in KL-divergence. If $I_0 = |C E_{s0}|^2$9 has a Radon–Nikodym derivative with respect to $C$0 given by

$C$1

where $C$2 is a cost (potential) function, the KL-projection minimization is equivalent to KL-weighted control and to Onsager–Machlup minimization:

• (a) Information projection: $C$3
• (b) KL-weighted control: $C$4
• (c) Onsager–Machlup minimization: $C$5,

where $C$6. All three formulations coincide in optimizer due to the portmanteau theorem, and the solution can be constructed via Euler–Lagrange ODEs in explicit cases (Selk et al., 2020).

4. Algorithmic Implementation for Robust Observable Extraction

In astronomical data analysis, the KL-projection is implemented as follows (Xin et al., 2024):

1. Select a region of interest and extract corresponding submatrices $C$7 of $C$8.
2. Compute the thin SVD: $C$9.
3. Isolate robust (low-sensitivity) left-singular vectors and construct $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$0.
4. For each raw frame $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$1, form $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$2 and project: $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$3.
5. Aggregate over time for subsequent detection via matched-filter or $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$4 statistics.

The resulting robust observables $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$5 have reduced sensitivity to instrumental error, enabling improved scientific yield. Quantitative gains on Roman HLC simulated data are:

• Linear regime (7 nm RMS WFE, $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$6): $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$728% improvement in flux ratio detection limits after KL-projection (from $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$8 to $$I(\Delta E_s) \approx I_0 + 2\Re[(C E_{s0})^* (C \Delta E_s)] + |C \Delta E_s|^2 + |C E_{p0}|^2,$$9, FPR=0.01, TPR=0.9).
• Quadratic regime (110 nm RMS WFE, $E_{p0}$0): $E_{p0}$1 contrast gain (from $E_{p0}$2 to $E_{p0}$3) (Xin et al., 2024).

5. Integration with Hybrid Post-processing Pipelines

The KL-projection approach is fundamentally model-based, requiring only the instrument operator $E_{p0}$4 and a baseline instrument model. This allows seamless integration as an initial step in data reduction pipelines:

• The KL-projection is first applied to suppress speckle-dominated modes and produce robust observables.
• Classical post-processing techniques such as Angular Differential Imaging (ADI), Reference Differential Imaging with KLIP, Non-negative Matrix Factorization (NMF), and spectral differential methods may then be applied to the robust observables instead of the raw intensities.
• Model-informed observables facilitate the incorporation of reference-star or deformable mirror (DM) modulation data, by extending $E_{p0}$5 or imposing priors on $E_{p0}$6.

Statistically, KL-projection operates as a prior favoring subspaces least sensitive to instrumental noise, analogous to kernel-phase or closure-phase corrections but generalized to full coronagraphic post-processing (Xin et al., 2024).

6. Abstract Information Projection: Banach and Wiener Space Examples

In the abstract framework (Selk et al., 2020), KL-projection on Banach space $E_{p0}$7 with centered Gaussian prior $E_{p0}$8 and Cameron–Martin space $E_{p0}$9 is analyzed for shift measures. For classical Wiener space $2\Re[(C E_{s0})^* (C \Delta E_s)]$0, the optimal shifted measure $2\Re[(C E_{s0})^* (C \Delta E_s)]$1 is determined by minimizing the Onsager–Machlup functional,

$2\Re[(C E_{s0})^* (C \Delta E_s)]$2

subject to specified boundary or regularity conditions. The minimizer is found by solving the associated Euler–Lagrange equation, e.g., for $2\Re[(C E_{s0})^* (C \Delta E_s)]$3, the optimal drift is $2\Re[(C E_{s0})^* (C \Delta E_s)]$4. This formalism reveals the deep equivalence between KL-optimal projection, energy-optimal controls, and variational solution of SDEs under cost constraints.

These mathematical methods underpin KL-projection techniques in practical post-processing, providing a foundation for designing robust, model-informed filters in complex inference and imaging problems (Selk et al., 2020).