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Non-negative Matrix Factorization: Robust Extraction of Extended Structures

Bīn Rén, Laurent Pueyo, Guangtun Ben Zhu, John Debes, Gaspard Duchêne

arXiv:1712.10317v2astro-ph.IMastro-ph.EPcs.LG

TL;DR

Recovering extended circumstellar disks from high-contrast images is challenging because existing methods can lose astrophysical flux or struggle with disk morphology. This paper applies vectorized NMF to direct-imaging data and compares it with existing approaches. NMF better recovers disk morphology and faint structure in the reported comparisons.

  • Problem

    Existing methods can reduce astrophysical flux, require forward modeling, or struggle to calibrate extended and irregular disks.

    Source: §1 · p. 2

  • Method

    The paper applies vectorized NMF, decomposing reference data into non-negative components and modeling targets with those components.

    Source: §2.2 · p. 4

  • Results

    NMF outperforms other methods in recovering disk morphology and faint structure, including the faint far side of inclined disks.

    Source: §3.2.1 · p. 8

  • Takeaways & Limitations

    For first-order recovery of disk morphology, NMF subtraction can be corrected with a simple one-dimensional search.

    Source: §1 · p. 3

    KLIP forward modeling assumes prior understanding of the object and is time-consuming for recovering disk surface brightness.

    Source: §2.1 · pp. 3–4

Abstract

from arXiv · show

We apply the vectorized Non-negative Matrix Factorization (NMF) method to post-processing of direct imaging data for exoplanetary systems such as circumstellar disks. NMF is an iterative approach, which first creates a non-orthogonal and non-negative basis of components using given reference images, then models a target with the components. The constructed model is then rescaled with a factor to compensate for the contribution from a disk. We compare NMF with existing methods (classical reference differential imaging method, and the Karhunen-Lo\`eve image projection algorithm) using synthetic circumstellar disks, and demonstrate the superiority of NMF: with no need for prior selection of references, NMF can detect fainter circumstellar disks, better preserve low order disk morphology, and does not require forward modeling. As an application to a well-known disk example, we process the archival Hubble Space Telescope (HST) STIS coronagraphic observations of HD~181327 with different methods and compare them. NMF is able to extract some circumstellar material inside the primary ring for the first time. In the appendix, we mathematically investigate the stability of NMF components during iteration, and the linearity of NMF modeling.

1. INTRODUCTION

High-contrast disk imaging depends on removing stellar PSF and speckle residuals, but existing subtraction methods can distort disk signals or struggle with irregular morphologies. The paper studies vectorized NMF as an alternative, including its modeling approach and comparisons with existing methods.

1. INTRODUCTION

  • Aggressive LOCI PSF subtraction biases photometry and astrometry, whereas KLIP projection onto eigen-images reduces astrophysical-object flux and requires forward modeling to recover it (Marois et al. 2010; Pueyo et al. 2012; Soummer et al. 2012; Choquet et al. 2016, 2017).Source: §1 · p. 2
  • Irregularly shaped disks are difficult to recover through forward modeling because their initial surface-brightness distributions are hard to reconstruct.Accurate morphology can reveal asymmetric structures, disk evolution, and possible perturbing planets.Source: §1 · p. 2
  • Because point-source methods perform poorly on extended disks and full calibration is difficult, the paper investigates NMF for more robust, well-calibrated PSF subtraction.It adopts Zhu’s vectorized method for its parallel computational efficiency.Source: §1 · p. 2
  • NMF constructs non-negative, non-orthogonal components from references and models a target with non-negative component coefficients.Unlike KLIP, NMF does not subtract each image’s mean and keeps entries non-negative.Source: §1 · p. 2
  • For disk morphology, NMF subtraction still needs forward modeling, but the paper shows that first-order correction can use a simple one-dimensional search.The paper compares methods on modeled disks and archival HD 181327 observations, and examines NMF component stability and modeling linearity in appendices.Source: §1 · p. 3

2. METHODS

The section motivates NMF as an alternative to current subtraction methods, whose disk recovery can be biased or require time-consuming forward modeling. It describes constructing non-negative components, modeling targets, and correcting disk over-fitting with a scaling factor.

2.1. Limitation of Current Methods

  • Classical RDI does not significantly recover the synthetic disk, while KLIP loses flux and radial-profile details and requires forward modeling to reveal the true distribution.The RDI profile's agreement with the model is not statistically significant because of large azimuthal variations.Source: §2.1 · p. 3, Figure 1 · p. 4
  • KLIP forward modeling assumes prior understanding of the object and can require time-consuming iterative recovery of its likely surface-brightness distribution.Source: §2.1 · pp. 3–4

2.2. Non-negative Matrix Factorization (NMF)

  • NMF decomposes reference images into non-negative components and models targets with them, aiming to address forward-modeling difficulties for extended disks with uncertain morphology.The method flattens images and uses vectorized NMF for computational tractability.Source: §2.2 · p. 4

2.2.1. Component Construction

  • NMF constructs components sequentially, retaining earlier components while initializing each new component randomly; this ranks components and supports linear target modeling.A pixelwise variance weighting function accounts for heteroskedastic astronomical data.Source: §2.2.1 · p. 5

2.2.2. Target Modeling (“Projection”)

  • With sequentially constructed components, target modeling linearly separates disk and stellar signals, enabling a first-order scaling-factor correction.The paper presents component stability as the basis for this separation and linearity.Source: §2.2.2 · p. 5, §2.2.3 · p. 5

2.2.3. Disk Retrieval via “Forward Modelling”

  • The Best Factor Finding procedure uses a scaling factor to minimize background noise; for tested synthetic disks, residual variance tracks the distance between the reduced disk and true model as the factor varies.BFF focuses on the first component because higher-order components usually have little influence in the simulations.Source: §2.2.3 · p. 6, Figure 3 · p. 6

3. COMPARISON AND APPLICATION

The comparisons examine noise removal and disk-signal capture before testing NMF against classical subtraction and KLIP on synthetic disks and HD 181327. NMF preserves faint disk structure and radial profiles in the synthetic tests, while the HD 181327 results reveal higher inner-region SNR and recoverable debris structure.

3.1.1. Noise Removal

  • KLIP’s fractional residual variance plateaus over many components, while NMF eventually reaches a lower value and continues capturing quasi-static noise.The FRV comparison uses 81 STIS images of HD 38393.Source: §3.1.1 · p. 7, Figure 4 · p. 8

3.1.2. Disk Signal Capture

  • KLIP captures more disk signal through direct projection, causing over-subtraction; NMF’s sparse, smaller coefficients reduce over-fitting and its model is rescaled using a factor.The factor is obtained through the BFF procedure.Source: §3.1.2 · p. 7

3.2. Synthetic Disk Models

  • The synthetic tests vary disk inclination and brightness, spanning disk-to-star flux ratios of approximately 10^-4 to 10^-6.The models are reduced using classical RDI, KLIP, and NMF.Source: §3.2 · pp. 7–8

3.2.2. Radial Profile

  • NMF recovers the synthetic face-on disk’s radial profile without bias and with small uncertainties, unlike classical subtraction’s large uncertainties and KLIP’s profile distortion.The paper expects the smaller uncertainties to help detect fainter structures, especially for low-inclination disks.Source: §3.2.2 · p. 9, Figure 11 · p. 11

3.2.1. Morphology

  • NMF preserves disk morphology and flux better than KLIP at 20× dimming, and can marginally recover morphology when all methods struggle at 50× dimming.Across the synthetic comparisons, NMF also recovers faint far-side structures better than KLIP.Source: §3.2.1 · p. 8, Figure 8 · p. 10, Figure 9 · p. 10
  • NMF achieves lower or similar χ2 levels to KLIP and performs better than classical subtraction in the synthetic disk comparisons.The authors attribute slightly larger NMF χ2 values than KLIP’s in some cases to KLIP fitting random noise.Source: §3.2.1 · pp. 8–9, Figure 10 · p. 11

3.3. Application to HST-STIS Observations: HD 181327

  • For HD 181327, NMF recovers the faint northwest debris and yields higher SNR inside the primary ring than the other methods.The inner region is non-zero at significance levels better than 1σ.Source: Figure 12 · p. 12, §3.3 · pp. 12–13
  • For HD 181327, NMF and classical azimuthal profiles mostly agree within 1σ, while KLIP is inconsistent; the authors do not claim which better represents the disk signal.The authors note that extended debris may bias the BFF scaling.Source: Figure 12 · p. 12, §3.3 · p. 13

4. SUMMARY

The paper presents NMF as a way to detect faint circumstellar-disk signals while retaining their morphology, addressing limits of current methods that hinder detailed morphological study. It reports advantages over current methods and notes that reference images remain necessary.

4. SUMMARY

  • NMF supersedes current methods in retrieving disk morphology and photometry on synthetic faint disks, and is also applied to HD 181327.The authors report extracting circumstellar material inside the primary ring for the first time in that example.Source: §4 · p. 13
  • The authors report that NMF minimizes over-subtraction and thereby circumvents forward-modeling attempts.Source: §4 · p. 13
  • NMF does not require prior reference selection, although reference images remain necessary and selection is preferred for detailed analyses.Without selection, reference-star spectral types should be evenly sampled.Source: §4 · p. 13
  • NMF can extract disk signals while retaining morphology, supporting study of detailed structures, especially in faint disks.The paper connects well-constrained morphology with studying planetary-system formation, evolution, and planet-disk interaction.Source: §4 · pp. 13–14
  • The paper uses space-based coronagraphic observations for their imaging stability and states that NMF can capture varying stellar PSF and speckles in ground-based exposures.Source: §4 · p. 14

A. LIST OF SYMBOLS

This section lists symbols used in the paper and defines element-wise matrix multiplication.

A. LIST OF SYMBOLS

D 1 × Npix Flattened image of the astrophysical signal (i.e., no stellar information). ˆD T −ˆfTNMF 1 × Npix Reduced best image of the astrophysical signal (D), obtained from BFF procedure.

This glossary defines notation for images, NMF components and coefficients, variances, and the scaling factor used in the paper.

D 1 × Npix Flattened image of the astrophysical signal

B. NMF WITH WEIGHTING FUNCTION

This section gives weighted update rules for constructing NMF components and modeling a target, with the reference and variance inputs defined in the accompanying prose.

B. NMF WITH WEIGHTING FUNCTION

  • The paper adopts weighted update rules for constructing NMF components from the reference cube.R is the reference cube, V its variance matrix, H the component matrix, and W the coefficient matrix.Source: B. NMF WITH WEIGHTING FUNCTION · p. 16
  • For HST-STIS images, the reference variance matrix comes from the square of the ERR extension; for exposures with simulated disks, V = D + ERR^2.Source: B. NMF WITH WEIGHTING FUNCTION · p. 16
  • The weighted target-modeling rule uses target T, its variance v, the constructed components H, and target coefficients ω.Source: B. NMF WITH WEIGHTING FUNCTION · p. 16

C. CONSTRUCTION AND STABILITY OF THE COMPONENT BASIS

Sequential construction stabilizes previously built NMF components when reference pixels have sufficiently high signal-to-noise ratios, supporting the paper’s scaling-factor rationale.

C. CONSTRUCTION AND STABILITY OF THE COMPONENT BASIS

  • The scaling-factor rationale depends on sequentially constructed components remaining stable, alongside disk capture resembling the stellar PSF and first-order target-model linearity.The passage identifies stability as one of three premises for using the scaling factor.Source: C. CONSTRUCTION AND STABILITY OF THE… · p. 16
  • The authors consider both direct random construction with a fixed component count and sequential construction that initializes each larger basis from the preceding one.The sequential procedure starts with one component and proceeds to two, and so on.Source: C. CONSTRUCTION AND STABILITY OF THE… · p. 16
  • With sufficiently large reference-pixel values and high signal-to-noise ratios, sequential updates have little impact on already constructed components.The stability result concerns components built sequentially as additional components are introduced.Source: C. CONSTRUCTION AND STABILITY OF THE… · pp. 17–18

D. TARGET MODELING

The target-modeling analysis shows that, under sequential construction and component stability, NMF modeling is linear to first order and can separate stellar and disk signals.

D. TARGET MODELING

  • NMF coefficients have smaller absolute values than normal-projection coefficients because NMF components are non-negative, which the authors give as evidence of lower susceptibility to over-fitting.The passage attributes the inequality to the denominator in the update rule being greater than one.Source: D. TARGET MODELING · p. 18
  • Sequentially created, iteration-stable NMF components make target modeling linear to the first order.The theorem states component stability as a condition for this result.Source: D. TARGET MODELING · p. 19
  • To the first order, the target NMF model equals the sum of the stellar-signal and disk-signal NMF models, plus a higher-order term.The paper identifies this as linear separation of stellar PSF and speckle signals from circumstellar disk signal.Source: D. TARGET MODELING · p. 20

E. THE BEST FACTOR FINDING (BFF) PROCEDURE

The BFF procedure selects a model scaling factor by minimizing residual noise in identified diskless pixels, complementing a classical approach focused on diffraction spikes.

E. THE BEST FACTOR FINDING (BFF) PROCEDURE

  • BFF identifies background pixels iteratively by removing values with large deviations from the median until the background-pixel count stops changing.The standard deviation is then calculated over the remaining pixels.Source: E. THE BEST FACTOR FINDING (BFF)… · p. 20
  • BFF chooses the scaling factor that minimizes the standard deviation of pixels identified as diskless in the scaled reduced image.It tests factors from 0 to 1 and calculates the standard deviation after background identification converges.Source: E. THE BEST FACTOR FINDING (BFF)… · pp. 20–21
  • Unlike the classical scaling method’s focus on residual noise along major diffraction spikes, BFF evaluates the entire field of view and is more able to minimize overall PSF-template differences when spikes are unstable.The passage particularly notes unstable diffraction spikes in ground-based observations.Source: E. THE BEST FACTOR FINDING (BFF)… · p. 21
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