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Fast Compressed Sensing SAR Imaging based on Approximated Observation

Jian Fang, Zongben Xu, Bingchen Zhang, Wen Hong, Yirong Wu

arXiv:1302.3120v1cs.IT

TL;DR

Existing exact-observation CS-SAR models are costly for high-dimensional imaging. The paper replaces exact observation with an approximation derived from inverse MF focusing and combines it with sparse regularization. The resulting method is reported to support high-quality, high-resolution sparse imaging with reduced measurements and substantially lower computational and memory costs.

  • Problem

    Exact-observation CS-SAR models have much higher computational complexity and memory cost than MF-based methods, making high-dimensional applications inefficient.

  • Method

    The paper replaces the exact observation matrix with an approximated observation derived from inverse MF procedures and incorporates it into sparse regularization.

  • Results

    The proposed method achieves high-quality, high-resolution imaging with far fewer measurements, while reported reconstruction speedups reach hundreds of times.

  • Takeaways & Limitations

    The approximated-observation framework makes CS-SAR reconstruction compatible with efficient MF-based processing and reduces the computational burden for large-scale imaging.

Abstract

from arXiv · show

In recent years, compressed sensing (CS) has been applied in the field of synthetic aperture radar (SAR) imaging and shows great potential. The existing models are, however, based on application of the sensing matrix acquired by the exact observation functions. As a result, the corresponding reconstruction algorithms are much more time consuming than traditional matched filter (MF) based focusing methods, especially in high resolution and wide swath systems. In this paper, we formulate a new CS-SAR imaging model based on the use of the approximated SAR observation deducted from the inverse of focusing procedures. We incorporate CS and MF within an sparse regularization framework that is then solved by a fast iterative thresholding algorithm. The proposed model forms a new CS-SAR imaging method that can be applied to high-quality and high-resolution imaging under sub-Nyquist rate sampling, while saving the computational cost substantially both in time and memory. Simulations and real SAR data applications support that the proposed method can perform SAR imaging effectively and efficiently under Nyquist rate, especially for large scale applications.

I. INTRODUCTION

CS-SAR offers reduced sampling and improved imaging outcomes, but exact-observation models impose substantially higher computational and memory costs than traditional MF methods. The paper proposes approximated observations derived from inverse MF procedures within sparse regularization to reduce these costs.

  • CS-SAR can relax measurement requirements, reduce sidelobes, and further suppress noise relative to traditional SAR imaging methodologies.
  • Existing CS-SAR models have much higher computational complexity and memory cost than MF-based methods, limiting high-dimensional applications.
  • The proposed framework replaces the exact observation function with approximated observations derived from inverse traditional MF procedures.
  • The approximation is incorporated into a sparse regularization framework solved by an iterative thresholding algorithm, requiring only well-focused imaging for CS reconstruction.
  • The method produces high-quality, high-resolution images with significantly reduced measurements, memory cost O(n), and one-step complexity O(n log n).These orders match traditional SAR imaging methods.

II. CS-SAR MODELS BASED ON EXACT OBSERVATION

This section presents the SAR observation and conventional CS formulation based on an exact observation matrix, then describes sparse recovery through regularization and iterative thresholding. The exact matrix formulation creates substantial computational and memory burdens for large-scale CS-SAR imaging.

  • A. Stripmap Mode SAR Model: Stripmap SAR models the received echoes using azimuth and range variables, a time-variant convolution kernel, and additive noise.
  • A. Stripmap Mode SAR Model: The discretized SAR model represents the scene as x and the observations through an exact matrix H with noise n0.
  • B. Formulation of CS-SAR models: CS samples and compresses the data, seeking recovery of sparse x from fewer measurements than the Nyquist criterion requires.
  • B. Formulation of CS-SAR models: The sparse recovery problem is converted to regularization and solved using an iterative thresholding algorithm with parameter λ controlling regularization.
  • B. Formulation of CS-SAR models: The main computation load is time-domain correlation A^H A x, while the two-dimensional sensing matrix must also be collected and stored, consuming substantial memory.
  • B. Formulation of CS-SAR models: These computational and storage demands hamper effective application of exact-observation CS-SAR models.

III. APPROXIMATED OBSERVATION

The paper motivates approximated observation by exploiting the efficient, decoupled structure of MF focusing and constructs it through inverse focusing procedures. This approximation is intended to enable CS-SAR processing with MF-like computational complexity.

  • The section constructs an approximated observation operator from the inverse of the Range-Doppler Algorithm and analyzes its relation to the corresponding focusing method.
  • MF focusing can be implemented through decoupled one-dimensional frequency-domain operators with O(n log n) complexity using FFT-type operations.
  • Directly decoupling the exact sensing matrix is impossible because it has intrinsic two-dimensional structure.
  • Because MF imaging M approximately inverts the observation H, M^-1 can approximate H and may inherit a decoupled structure.
  • The proposed method uses the approximated observation to target computational complexity comparable to traditional MF-based methods.

B. How to construct an admissible approximated observation

The paper constructs an approximated observation G as a generalized right inverse of a high-precision imaging procedure M, using low sidelobes to preserve sparse-recovery behavior. For RDA, the inverse operations yield a linear G whose conjugate transpose equals M.

  • A high-precision imaging procedure M can provide an admissible substitute for the exact observation when its inverse is sufficiently accurate.The paper formalizes G as a generalized right inverse of M and calls it an approximated observation.
  • The approximated model reconstructs an image approximation whose sparsity may differ from the original because of phase and sidelobe errors.When reconstructed sidelobes are low enough, the sparsity remains unchanged and the required least sampling rate equals that of the original model.
  • Low sidelobes and strong focusing are criteria for using a focusing method to construct an approximated observation without demanding more samples.The construction is described as flexible and can be obtained from established algorithms, subject to focusing ability.
  • For RDA, G is obtained by inverting the Fourier transforms, phase multiplications, and interpolation-based range-cell migration correction operations.The resulting operator is built from the inverse of each RDA sub-operation.
  • RDA-derived G is linear and satisfies G^H = M, so the approximated observation is the conjugate transpose of the matched-filter imaging operator.Theorem 1 states the equivalent property as G^H = M.

D. Generalization

The approximated-observation principle extends beyond the RDA setting by adapting the focusing procedure to squint and pulse-form conditions. Its implementation remains based on invertible or similarly structured sub-operations.

  • The inverse-based approach retains the speed advantage of matched filtering because its decoupled structure makes the inverse achievable.The authors connect this property to applying approximated rather than exact observations in CS-SAR.
  • High-squint systems can use secondary compression in RDA or derive approximated observations from Chirp-Scaling and ω-k focusing methods.These alternatives are proposed to enhance focusing ability.
  • The general principle is presented as extensible to other algorithms with similar FFT, phase-multiplication, and interpolation sub-operations.The paper does not enumerate all possible extensions.
  • The construction can extend to non-chirp cases by replacing the chirp-specific phase operation, while range convolution can be modeled directly.The paper notes that azimuth modulation is the main SAR-specific difficulty, whereas range convolution has a simple one-dimensional structure.

IV. CS-SAR IMAGING BASED ON APPROXIMATED OBSERVATION

The proposed CS-SAR model replaces exact observation with an approximated observation and combines sparse regularization with matched-filter processing. Its iterative thresholding update alternates residual MF processing and sparsity enforcement.

  • The new CS-SAR imaging method uses an approximated observation within an Lq regularization model solved by a fast iterative thresholding algorithm.The paper formulates the model explicitly around approximated observation and selects q = 1 in implementation.
  • Because the approximated observation operator G is linear, the resulting model can be solved efficiently by iterative thresholding.
  • The method uses separate azimuth and range sampling operators corresponding to the discrete azimuth and continuous range signal structures.The paper notes that the two sampling procedures are usually physically separated.
  • Each iteration simulates compressed data, applies a matched filter to the residual, and thresholds the updated estimate.The thresholding step enforces sparsity by regularizing noise and undersampling ambiguity.
  • The iteration terminates upon convergence or when the maximum iteration count is reached.Algorithm 1 lists SAR echoes, G, M, and the sampling operators as inputs and the recovered image as output.

B. Parameter Setting

The paper sets the iterative and regularization parameters through convergence and sparsity considerations, then compares approximate- and exact-observation computational costs and memory use. The approximate model avoids the large sensing-matrix burden of exact CS-SAR.

  • Parameter Setting: The step parameter µ controls convergence, and adaptive step selection is used to accelerate the algorithm.The adaptive strategy is described as satisfying the required condition and providing an additional acceleration advantage.
  • Parameter Setting: The regularization parameter λ balances reconstruction precision against solution sparsity and can be set optimally when scene sparsity k is known or bounded.A prior upper estimate of target-scene sparsity can guide the setting.
  • Computation Cost: The approximate-observation method has per-step computational cost Ca = O(In log^2 n), combining inverse MF, MF, and O(n) thresholding.The notation uses I for iteration count and n for the image size.
  • Computation Cost: The exact-observation method has cost Ce = O(Iuns), while the cost ratio rC depends linearly on radar-signal time-bandwidth product u.Large u values used for SNR improvement therefore correspond to high computational cost for time-domain reconstruction.
  • Computation Cost: The required iteration counts are difficult to compare analytically, but no obvious practical difference is observed.

D. A Summary

The proposed approximated-observation CS-SAR model combines sparse regularization with matched-filter operations to retain CS imaging benefits while substantially reducing computational and memory costs.

  • Model and compatibility: The model uses approximated observation derived from inverse focusing procedures to combine compressed sensing and matched filtering.The resulting procedure adds thresholding to operations similar to traditional matched-filter SAR imaging.
  • Computational efficiency: Approximated observation reduces the method to 1-D operations, significantly lowering computational complexity compared with exact-observation CS-SAR.The decoupled structure also enables substantial memory savings.
  • Model and compatibility: The method requires little modification of existing SAR imaging algorithms, simplifying the combination of matched filtering and compressed sensing.It is therefore presented as compatible with traditional matched-filter procedures.
  • Scope: The proposed CS-SAR method is intended for efficient high-dimensional SAR applications.The paper states that simulations and applications will support this benefit.

V. SIMULATIONS AND APPLICATIONS

Simulations compare CSRDA, CSEO, and RDA on reconstruction ability, quality, and cost, followed by applications to real RADARSAT-1 data using compressed sampling.

  • Experimental design: CSRDA uses approximated observation G from inverse RDA, whereas CSEO uses exact observation H.The experiments compare both CS-SAR methods with traditional RDA.
  • Real-data validation: Real-data applications use RADARSAT-1 imagery to further assess CSRDA alongside traditional RDA.The paper reports that these applications support the proposed method’s effectiveness and efficiency.
  • Experimental design: The simulations evaluate reconstruction ability, quality, and cost under varied sampling rates using nine simulated point targets.The target scene is generated from exact time-domain slant-range data and compressed measurements.
  • Reconstruction ability: With full samples, RDA, CSEO, and CSRDA recover the scene, but sparse-regularized methods avoid RDA’s serious side lobes.At reduced sampling, the CS-SAR methods preserve scene recovery while reducing side lobes.
  • Reconstruction ability: At 0.65% sampling, CSRDA and CSEO both reconstruct the image; CSRDA fails at 0.55% while CSEO fails at a lower sampling rate.The results support reconstruction with far fewer samples than Nyquist requires, while approximation requires somewhat more samples.

2) RQ comparison:

CSRDA improves reconstructed resolution and side-lobe suppression while substantially reducing computational cost, though its approximation can require additional measurements or trade reconstruction quality.

  • RQ comparison: CSRDA reconstructs a narrower main lobe and lower side lobes than RDA in the point-target quality comparison.The comparison evaluates side lobes using PSLR and spatial resolution using IRW.
  • RQ comparison: 15 samples versus 8 samples: RDA and CSRDA main-lobe widths are reported in both range and azimuth directions.CSRDA’s width is smaller in the reported simulation.
  • RQ comparison: -13.32 dB versus -21.3 dB and -22.7 dB: CSRDA lowers PSLR relative to RDA in azimuth and range.The reported CSRDA PSLR values are -21.3 dB in azimuth and -22.7 dB in range.
  • RQ comparison: CSRDA separates three point targets that overlap and remain inseparable under RDA.This provides a second simulation-based demonstration of enhanced resolution.
  • RC comparison: CSRDA is reported as much faster than CSEO because matched-filter computation costs O(n log^2 n), while CSEO scales with pulse time-bandwidth product and scene size.For u = 10^6, the paper expects acceleration of more than thousands of times.
  • Trade-off: The method trades reconstruction quality against complexity and may require additional measurements because of the observation approximation.The paper characterizes this as a trade-off between reconstruction complexity and quality.

B. Application

On RADARSAT-1 data, CSRDA reconstructs scenes from reduced samples with suppressed side lobes and improved resolution, while using far less time and memory than CSEO.

  • English Bay application: At 20% sampling, CSRDA and CSEO recover the English Bay image, whereas RDA fails with visible ambiguities.RDA reconstructs with full samples but exhibits strong side lobes.
  • English Bay application: At 5% sampling, both CSRDA and CSEO recover the targets, with CSEO showing slightly higher precision than CSRDA.The comparison uses the RADARSAT-1 English Bay scene.
  • Computational cost: 1 minute versus about 9 hours: CSRDA and CSEO reconstruction times at 20% sampling differ substantially.The result is reported alongside CSRDA’s computational-cost advantage.
  • Computational cost: 16Gb versus 100Mb: CSEO stores a sensing matrix occupying about 16Gb, while CSRDA uses about 100Mb.The memory comparison accompanies the 20%-sampling reconstruction-time comparison.
  • Large-scale application: For a 2048 × 2500-sample RADARSAT-1 scene, CSRDA improves resolution and reduces side lobes compared with RDA.CSEO is not compared in this experiment because its memory cost exceeds the computer’s capability.
  • Conclusion: The applications support effective and efficient approximated-observation CS-SAR imaging for high-dimensional SAR applications.The paper presents this as an improvement over exact-observation CS-SAR imaging.

VI. CONCLUSION

The paper proposes a frequency-domain CS-SAR imaging method that replaces exact observation matrices with approximated observations derived from inverse matched-filter procedures. The resulting framework combines sparse regularization with fast iterative thresholding to reduce computational complexity while supporting sparse-scene reconstruction and feature-enhanced imaging.

  • Method: The method replaces the exact observation matrix with an approximated-observation operator constructed by inverting a traditional matched-filter imaging procedure.This construction generates SAR raw data while retaining a connection to existing matched-filter methods.
  • Method: The proposed CS-SAR model combines compressed sensing and matched filtering within a sparse-regularization framework compatible with existing SAR imaging methods.Only limited modification of current SAR imaging technologies is required.
  • Method: An iterative thresholding algorithm provides a fast solution to the new CS-SAR model and forms a low-complexity imaging method.The approximation is used to reduce the computational burden of solving the model.
  • Results: The experiments support reconstruction of sparse scenes with far fewer measurements than Nyquist requires, together with feature-enhanced, high-quality imaging.The reported applications include simulations and real SAR imaging tasks.
  • Results: The method is reported to provide a reconstruction speedup of hundreds of times compared with exact-observation CS-SAR methods.The paper identifies high-dimensional SAR imaging as a particularly suitable application.
  • Limitations: The main trade-off is that approximated observations require more samples for scene reconstruction, and the effect and selection of the approximation remain under study.The paper also notes multiple possible realizations of the approximated observation without establishing a selection criterion.
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