PaperlayerView exact revision on arXiv

Source-linked AI summary

Compressed Sensing in Astronomy

J. Bobin, J-L Starck, R. Ottensamer

arXiv:0802.0131v1astro-ph

TL;DR

Astronomical remote sensing needs effective compression under narrow transmission bands while preserving spatial information. This paper applies compressed sensing with prior-aware recovery and data fusion, reporting improved recovery over standard techniques.

  • Problem

    Astronomical remote sensing faces narrow transmission bands, while spatial resolution remains a crucial instrumental feature.

  • Method

    The paper uses compressed sensing with M<t projected measurements, ProxIT iterative-threshold decoding, and physical priors.

  • Results

    Compressed sensing data fusion produces astounding improvements over standard techniques by exploiting redundant observations.

  • Takeaways & Limitations

    Compressed sensing provides an elegant and effective compression technique for astronomical imaging and remote sensing.

Abstract

from arXiv · show

Recent advances in signal processing have focused on the use of sparse representations in various applications. A new field of interest based on sparsity has recently emerged: compressed sensing. This theory is a new sampling framework that provides an alternative to the well-known Shannon sampling theory. In this paper we investigate how compressed sensing (CS) can provide new insights into astronomical data compression and more generally how it paves the way for new conceptions in astronomical remote sensing. We first give a brief overview of the compressed sensing theory which provides very simple coding process with low computational cost, thus favoring its use for real-time applications often found on board space mission. We introduce a practical and effective recovery algorithm for decoding compressed data. In astronomy, physical prior information is often crucial for devising effective signal processing methods. We particularly point out that a CS-based compression scheme is flexible enough to account for such information. In this context, compressed sensing is a new framework in which data acquisition and data processing are merged. We show also that CS provides a new fantastic way to handle multiple observations of the same field view, allowing us to recover information at very low signal-to-noise ratio, which is impossible with standard compression methods. This CS data fusion concept could lead to an elegant and effective way to solve the problem ESA is faced with, for the transmission to the earth of the data collected by PACS, one of the instruments on board the Herschel spacecraft which will be launched in 2008.

I. AN OVERVIEW OF COMPRESSED SENSING THEORY · A. The gist of compressed sensing

Compressed sensing replaces conventional sampling with a framework that acquires fewer linear measurements and exploits sparse signal structure for recovery. Its effectiveness depends on compressibility in a representation basis and measurement vectors that are non-adaptive and incoherent with that basis.

  • I. AN OVERVIEW OF COMPRESSED SENSING THEORY: Compressed sensing observes a signal x ∈ R^t through only M < t samples obtained by projecting x onto measurement vectors.The measurements are denoted {y_k}_{k=1,…,M}.
  • I. AN OVERVIEW OF COMPRESSED SENSING THEORY: The framework relies on compressible data that admit sparse representations in a basis and on non-adaptive measurement vectors incoherent with that basis.The measurement vectors should not depend on x.
  • A. The gist of compressed sensing: Natural signals often contain structured features that can be efficiently represented using sparse bases such as wavelets, ridgelets, curvelets, and contourlets.The passage identifies contours and textures as examples of structured image content.
  • A. The gist of compressed sensing: A signal satisfies x = Φα, and it is sparse in Φ when most coefficients of α are zero or close to zero while only a few have significant amplitudes.Sparse representations provide efficient approximation with low ℓ2 approximation error.
  • A. The gist of compressed sensing: In the K-sparse case, x is exactly synthesized from K ≪ t coefficients, so only K parameters are needed for perfect reconstruction.This is the extreme case of sparse representation described in the passage.
  • A. The gist of compressed sensing: The paper focuses on K-sparse decompositions in orthonormal bases Φ and uses signal compressibility to devise efficient non-adaptive compression.Extensions to overcomplete dictionaries are described as straightforward based on cited prior work.
  • A. The gist of compressed sensing: Compressed-sensing measurements are simple linear projections, and measurement design is crucial because random ensembles are likely to provide incoherent projections.In practice, measurements can be formed by randomly selecting vectors indexed by Λ from a deterministic ensemble Θ, giving y = Θ_Λx.
  • A. The gist of compressed sensing: Practical measurement vectors are selected at random from a deterministic ensemble Θ, producing measurements according to y = Θ_Λx.The selected vectors are indexed by Λ.

B. Signal recovery

Signal recovery in compressed sensing is formulated as sparse decomposition in the overcomplete system Ψ = ΘΛΦ. Exact recovery uses convex ℓ1 optimization, while approximate recovery handles nonsparsity and noise through LASSO or Basis Pursuit denoising.

  • a) Exact solutions: The decoding problem recovers x from y = ΘΛx by exploiting the prior representation x = Φα, where α is sparse.This reduces recovery to sparse decomposition in the overcomplete system ΘΛΦ.
  • a) Exact solutions: Compressed-sensing decoding is a sparse decomposition problem in the overcomplete representation Ψ = ΘΛΦ.The particular structure of this representation distinguishes the compressed-sensing framework from generic sparse decomposition.
  • b) Approximate solutions: Practical recovery allows signals to be nonsparse and measurements to contain noise, yielding projected data y = ΘΛx + nΛ.Because ΘΛ is orthonormal, projected noise nΛ remains white and Gaussian with the same properties as n.
  • b) Approximate solutions: The noisy recovery problem is known as LASSO or Basis Pursuit denoising and constrains the noise ℓ2 norm.In the noiseless case, the solution provides an approximation error close to the optimal sparse approximation and stable solutions.
  • b) Approximate solutions: The convex second-order cone program provides an efficient and robust mechanism for approximate signal recovery, supported by fast optimization algorithms.A range of optimization techniques has been developed to solve the problem efficiently.

II. COMPRESSED SENSING IN ASTRONOMY … 2) Coding information :

The paper presents compressed sensing as an alternative framework for astronomical remote sensing, linking low-cost coding, detector design, physical priors, and flexible recovery. It applies these ideas through sensor-level incoherent measurements and non-adaptive noiselet projections suited to sparse astronomical data.

  • II. COMPRESSED SENSING IN ASTRONOMY: Compressed sensing provides alternatives to astronomical instrument design through its way of coding information.The paper focuses on applying compressed sensing to astronomical remote sensing.
  • A. A new way of coding signals: Compressed sensing can improve compression by incorporating physical priors during decoding.The paper identifies physical priors as a way to improve the whole compression process.
  • A. A new way of coding signals: Its low-computational-cost coding is attractive for narrow transmission bands and large data volumes in fast scanning or wide-field sensing.These applications include remote sensing and compressing large amounts of data.
  • 1) Measuring physics :: A detector can directly measure incoherent projections, encoding and compressing the signal in the analog domain.The proposed philosophy applies compressed sensing directly to detector design through an optical system.
  • 1) Measuring physics :: Coded-mask γ-ray imaging already approximates this measurement paradigm, with sparse point-source data and Fourier measurements providing near-optimal incoherent projections.Coded masks are used as aperture masks, and the mask-detector pair is equivalent to selecting Fourier-space projections.
  • 1) Measuring physics :: The compressed sensing camera, inspired by coded masks, provides effective image compression with a single pixel.The paper notes that coded-mask concepts influenced the design of this camera.
  • 1) Measuring physics :: Sparsity-based recovery could enhance coded-aperture decoding beyond iterative maximum-entropy techniques.Coded-aperture decoding is often performed with iterative methods based on maximum entropy.
  • 2) Coding information :: Noiselets offer near-optimal, non-adaptive measurements for astronomical data sparse in wavelet bases, while enabling fast-transform on-board compression.The coding process depends on the sparse representation basis Φ rather than directly on the data x and is efficient for a wide signal class.

B. Practical signal recovery · 1) A practical and effective CS decoding algorithm:

The paper introduces ProxIT, a fast CS recovery algorithm for large-scale inverse problems that uses iterative thresholding and can incorporate physical constraints. Experiments on astronomical images show effective reconstruction from noiselet projections, including at compression rates above ρ = 0.1.

  • B. Practical signal recovery: The decoding step is more complex than coding because it solves the inverse problem, requiring a fast and accurate algorithm for large-scale situations.The decoding framework remains flexible enough to account for physical priors.
  • 1) A practical and effective CS decoding algorithm:: ProxIT solves the CS recovery inverse problem with a projected Landweber iteration based on proximal optimization and thresholded coefficient updates.The method uses a relaxation descent-direction matrix and soft-thresholding within its iterative scheme.
  • 1) A practical and effective CS decoding algorithm:: The threshold γ decreases iteratively from γ(0) = ∥ΦTΘTy♯∥∞ toward γmin, with γmin = 0 without noise and potentially noise-dependent otherwise.In the experiments, γmin = 3σn bounds noise coefficients with overwhelming probability.
  • 1) A practical and effective CS decoding algorithm:: Replacing soft-thresholding with hard thresholding provides better recovery performances and is used in the forthcoming experiments.The algorithm panel initializes x(0) to zero, updates projections and coefficients, reconstructs x, and decreases the threshold.
  • 1) A practical and effective CS decoding algorithm:: For a 512 × 512 HST image, ProxIT reconstructs pointwise structures and diffuse features from 0.2 ∗t random noiselet projections.The reconstruction is described as visually close to the original, including the gravitational arc.
  • 1) A practical and effective CS decoding algorithm:: ProxIT provides reasonable solutions for compression rate higher than ρ = 0.1 when the relative number of noiselet projections varies from 0.05 to 0.9.The experiment evaluates recovery SNR as ρ changes.
  • Comparison with other methods:: Compared with computationally demanding linear-programming and high-cost greedy methods, ProxIT is attractive for its simplicity, implicit fast transforms, and additional constraints such as positivity.The comparison section identifies computational cost as a limitation of alternative methods.

Accounting for physical priors:

Compressed sensing decoding can incorporate physical and instrumental priors through an observation map F, including positivity, convolution, detector response, and image shifts. The ProxIT algorithm handles linear F straightforwardly, whereas nonlinear F makes recovery more difficult; overall, this connects data acquisition with data processing.

  • Accounting for physical priors:: Positivity is a simple physical prior for astronomical photon-intensity data and can be incorporated during decoding.Astronomical data x are often photon intensity, for which positiveness is a useful assumption.
  • Accounting for physical priors:: Compressed sensing decoding can account for physical priors through an observation map F, including generating models, convolution, detector response, and image shifts.The map F models physical or instrumental priors, while Section III uses image shifts.
  • Accounting for physical priors:: The ProxIT algorithm extends straightforwardly to solve the decoding problem when F is linear, including convolution operators.For linear F, F(x) = Fx and F is a t × t matrix.
  • Accounting for physical priors:: When F is nonlinear, decoding becomes far more difficult and depends on the specific expression of F.The paper notes that special nonlinear models have been studied and later considers bijective possibly nonlinear maps F.
  • Accounting for physical priors:: Compressed sensing combines simple, low-cost coding with decoding that accounts for physical priors, bridging data acquisition and data processing.These properties are presented as the section’s concluding advantages of the compression scheme.

C. Compressed sensing versus Standard compression techniques

The section identifies advantages of compressed-sensing-based compression over standard techniques, including the JPEG compression standard. It also considers compressed sensing as a conventional compression method.

  • C. Compressed sensing versus Standard compression techniques: Compressed-sensing-based compression is presented as advantageous compared with standard compression techniques.The passage introduces these advantages without specifying them.
  • C. Compressed sensing versus Standard compression techniques: JPEG is cited as a celebrated standard compression benchmark for comparison.The passage specifically refers to the JPEG compression standard.
  • C. Compressed sensing versus Standard compression techniques: The section considers compressed sensing when used as a conventional compression method.No further conditions or results are provided in the passage.

1) Computational complexity:

CS-based compression has low computational complexity because its projections require no further encoding, leaving projection computation as the main cost. With noiselets, this cost scales as O(t), below JPEG’s O(t log(t)) burden, and optical implementation can accelerate it further.

  • Computational complexity: CS projections require no further encoding, unlike classical compression methods such as JPEG or JPEG2000.
  • Computational complexity: The only computational cost of CS-based compression is computing the projections.
  • Computational complexity: O(t) noiselet projections impose lower CPU load than JPEG’s O(t log(t)) computational burden.
  • Computational complexity: Optical projection systems can make CS-based compression even faster.

2) Decoupling: · 3) Data Fusion: · III. EXPERIMENT : THE HERSCHEL PROJECT

The paper separates CS decompression from compression, enabling future improvements through better sparse dictionaries. Its data-fusion experiments show that CS preserves and recovers faint sources from multiple noisy observations, motivating a CS-based compression scheme for Herschel/PACS.

  • 2) Decoupling:: CS allows the decompression step to change while retaining the same compressed data.This decoupling means future processing can improve decompressed quality without reacquiring or recompressing the data.
  • 2) Decoupling:: Decompression quality depends on data sparsity in the chosen basis, so a better dictionary could improve reconstruction.The paper identifies sparsity in basis Φ as the relevant quality factor.
  • 3) Data Fusion:: Astronomical raster scans contain redundant information that standard compression techniques cannot account for effectively.This redundancy motivates combining observations during decompression.
  • 3) Data Fusion:: The contrast arises because CS compression is linear, transmitting projections that retain information from the original source.Unlike nonlinear standard methods, CS compressed data always contain a piece of information belonging to x⋆ under the stated argument.
  • 3) Data Fusion:: CS increases source SNR as the number of observations grows and uses observation redundancy during decompression to overcome compression-related SNR loss.Standard compression can remove noisy high frequencies that cannot be recovered by repeated observation.
  • III. EXPERIMENT : THE HERSCHEL PROJECT: Herschel/PACS faces a compression challenge because conventional low-cost techniques cannot achieve a satisfactory compression rate for its detector data.PACS uses two bolometer arrays read out at 40 Hz while imaging in the far-infrared and sub-millimeter range.
  • III. EXPERIMENT : THE HERSCHEL PROJECT: The paper proposes a CS-based compression scheme for Herschel/PACS data as an elegant and effective solution to this compression dilemma.The proposal directly targets the transmission constraints associated with the PACS instrument.

A. The Herschel dilemma · B. Compressed sensing for the Herschel data

Herschel must compress data in real time despite limited transmission bandwidth, high readout noise, and low onboard CPU power. The proposed compressed-sensing scheme jointly models shifted image observations and uses nonlinear, positivity-constrained recovery to preserve sensitivity and spatial resolution.

  • A. The Herschel dilemma: Averaging 6 consecutive images is computationally cheap and increases SNR by a factor of √6 without losing spatial resolution for pointed observations.This strategy is described as near-optimal for pointed observations.
  • A. The Herschel dilemma: For wide-area fast scans, shifts of approximately λ = 1 pixel combined with a PSF FWHM of δ ≃3 pixels make averaging 6 images degrade spatial resolution.The resulting increase in the equivalent point spread function along the scanning direction can be catastrophic for some scientific programs.
  • A. The Herschel dilemma: An effective compression scheme must balance spatial resolution and sensitivity, reducing resolution loss while lowering noise to improve signal detection.Averaging is less optimal for noise reduction when useful signal is spread across shifted consecutive images.
  • B. Compressed sensing for the Herschel data: Compressed sensing projects each signal onto a subspace defined by selected columns of Θ, then decodes the original image x⋆ from the compressed observations.The image stack is represented as shifted versions of x⋆ plus noise, with x⋆ = x0 and λ0 = 0 in practice.
  • B. Compressed sensing for the Herschel data: The proposed decoder adapts the ProxIT algorithm and incorporates astronomical positivity by projecting each iteration onto the cone of vectors with positive entries.The positivity-based formulation assumes the shifting operator Tλi is invertible.

C. A toy-example · 1) Detection performances:

The toy experiment evaluates compressed sensing against averaging six shifted images for detecting point sources in noisy astronomical data. CS achieves similar or slightly better detection performance while providing better spatial resolution, especially at low intensity.

  • C. A toy-example: The experiment uses a 128×128 image containing 49 uniformly scattered point sources with randomly Gaussian-distributed amplitudes.The assumed instrument FWHM is δ = 3 pixels, shifts are λ_i = i pixels, and white Gaussian noise models instrumental noise.
  • 1) Detection performances:: Detection compares MO6, the mean of six images, with compressed sensing across intensities from f = 700 to f = 140000.This range corresponds to SNR values from −13.2 to 33dB, using the rate of detected point sources as the evaluation measure.
  • 1) Detection performances:: Both methods use the same wavelet-based preprocessing and 5σ_M hard-thresholding to avoid false detections.The procedure applies an “a trous” B-spline wavelet transform, estimates σ_M with MAD at each scale, and reconstructs the image.
  • 1) Detection performances:: At total source intensity f = 3500, MO6 and CS provide similar detection performances, while CS has better spatial resolution.The comparison uses a noisy six-image input and a noiselet-based CS reconstruction recovered with the ProxIT algorithm.
  • 1) Detection performances:: At low intensity, CS provides slightly better detection results than MO6.The paper attributes this partly to spreading caused by averaging shifted images, despite MO6 being theoretically near-optimal for low shifts.
  • 1) Detection performances:: Overall, the experiment shows that CS can provide similar or better detection performances than MO6 while preserving spatial resolution.Spatial resolution is identified as a crucial instrumental feature, and averaging shifted images is described as clearly affecting it.

2) Resolution: · D. Realistic data · 1) The data:

The CS-based method mitigates spatial-resolution loss caused by jointly compressing six observations, outperforming MO6 especially at high source intensity. For realistic Herschel/PACS data, the model incorporates a slowly varying, nearly fixed flat-field component that may be known or estimated within ProxIT.

  • 2) Resolution:: The simulated signal consists of a couple of point sources, with the worst case occurring when they are aligned along the scan direction.The original datum is denoted x⋆.
  • 2) Resolution:: Resolution is evaluated using the Rayleigh criterion, identifying the minimum separation at which two point sources become resolved.The limit is measured for intensities from f = 100 to f = 2000.
  • 2) Resolution:: At high intensity, CS provides a resolution gain equal to a third of the instrumental FWHM, or 1 pixel, compared with MO6.At low intensity, the CS-based resolution gain slightly decreases.
  • 2) Resolution:: Joint compression of 6 observations causes resolution loss, which the CS-based reconstruction mitigates relative to the input datum.The simulated experiment compares the original signal with MO6 and CS-based reconstructions.
  • D. Realistic data: Real Herschel/PACS data are more complex than the preceding simulations.The realistic-data experiment uses pointwise sources added to real calibration measurements.
  • 1) The data:: The original datum x⋆ is contaminated by a slowly varying flat-field component cf.This component is almost fixed across a short sequence of 6 consecutive images.
  • 1) The data:: When cf is known, ProxIT can be updated accordingly; when unknown, cf can be estimated within the algorithm.The next section examines the CS-based resolution gain for realistic Herschel/PACS data.

2) Resolution: · IV. CONCLUSION · .2 0.3 0.4 0.5 0.6 0.7 Relative number of noiselet projections

The CS-based compression scheme achieves improved spatial resolution, with gains reaching 30% of the instrument PSF’s FWHM across a wide range of signal intensities. The paper concludes that CS offers computationally cheap coding, flexible prior-informed recovery, and strong data-fusion benefits for astronomical remote sensing.

  • 2) Resolution:: The experiment illustrates that CS-based compression reliably handles real-world astronomical data.The tested PACS input includes approximately Gaussian noise, and reconstruction uses noiselet-based CS projections with ProxIT.
  • IV. CONCLUSION: CS provides an easy and computationally cheap coding scheme for on-board astronomical remote sensing.This computational profile favors applications such as space-mission remote sensing.
  • IV. CONCLUSION: The ProxIT recovery algorithm uses iterative thresholding to provide efficient approximate decoding solutions with only a few parameters.The algorithm can be adapted to incorporate physical priors, improving recovery results.
  • IV. CONCLUSION: CS data fusion exploits redundancy across repeated observations of the same sky area to produce substantial recovery improvements over standard techniques.The conclusion characterizes CS’s advantage in multiple scanning observations as huge and its recovery results as astounding.
  • IV. CONCLUSION: Numerical experiments support CS-based compression as an effective solution to ESA’s astronomical remote-sensing compression problem for missions such as Herschel.The authors identify more realistic Herschel experiments with additional physical information as the next step.
Loading 0802.0131v1…