Sparsity-Driven Feature-Enhanced Imaging

Transcription

1 Sparsity-Driven Feature-Enhanced Imaging Müjdat Çetin Faculty of Engineering and Natural Sciences, Sabancõ University, İstanbul, Turkey Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Contributors: W. Clem Karl, Randy Moses, David A. Castañon, Dmitry Malioutov, Kush Varshney, Robin Cleveland, Emmanuel Bossy, Alan S. Willsky, Aaron Lanterman This work is being supported by AFRL under Grant FA IMA, October 17, 2005

2 Waves involved in today s talk ~10 GHz X-band synthetic aperture radar ~200 MHz passive radar ~700 khz ultrasound ~200 Hz acoustic localization

3 Motivating Application: Synthetic Aperture Radar (SAR) Data Collection Image Formation Interpretation, Decision This is a T-72 tank

4 Motivation for our work Some challenges for automatic decision-making from SAR images: Accurate localization of dominant scatterers Limited resolution Clutter and artifact energy Region separability Speckle Object boundaries Low SNR, limited apertures

5 Ingredients of Our Approach Exploit the sparsity of the underlying signals or features Regularized image formation framework Efficient iterative algorithms

6 Outline Some theoretical results on sparse signal recovery SAR and ultrasound imaging Sparse-aperture imaging Passive radar Sparse-array ultrasound Wide-angle SAR

7 Motivation for Sparsity: Sensor Array Signal Processing Goal: Estimate directions of arrival of acoustic sources using a microphone array Data collection setup Underlying sparse spatial spectrum f * Forward Inverse

8 Underdetermined Linear Inverse Problems Basic problem: find an estimate of, where [ ] Underdetermined -- non-uniqueness of solutions Additional information/constraints needed for a unique solution A typical approach is the min-norm solution: What if we know is sparse (i.e. has few non-zero elements)?

9 Prefer the sparsest solution: Sparsity constraints Number of non-zero elements in f Can be viewed as finding a sparse representation of the signal y in an overcomplete dictionary A Intractable combinatorial optimization problem Are there tractable alternatives that might produce the same result? Empirical observation: l 1 -norm-based techniques produce solutions that look sparse l 1 cost function can be optimized by linear programming!

10 l 1 -norm and sparsity a simple example A sparse signal A non-sparse signal Goal: Rigorous characterization of the l 1 sparsity link For these two signals f 1 and f 2 we have A*f 1 =A*f 2 where A is a 16x128 DFT operator

11 l 0 uniqueness conditions Prefer the sparsest solution: Let where When is? Number of non-zero elements in f Thm. 1: where and K(A) is the largest integer such that any set of K(A) columns of A is linearly independent. Unique l 0 solution What can we say about more tractable formulations like l 1?

12 l 1 equivalence conditions Consider the l 1 problem: Can we ever hope to get? Thm. 2 (*) : where is sparse enough! exact solution by l 1 optimization l 1 solution = l 0 solution! Can solve a combinatorial optimization problem by convex optimization! (*) Donoho and Elad obtained a similar result concurrently. [Malioutov, Çetin, and Willsky, ICASSP, May 2004]

13 l p (p 1) equivalence conditions Consider the l p problem: How about? Thm. 3: where l p solution = l 0 solution! Smaller p Smaller p! more non-zero elements tolerated As p!0 we recover the l 0 condition, namely [Malioutov, Çetin, and Willsky, ICASSP, May 2004]

14 Formulation in the presence of noise Noisy observation model: Modify the optimization problem as: where is a parameter controlling noise tolerance The problem can be recast in the form Data fidelity Regularizing sparsity constraint where is a regularization parameter

15 Outline Some theoretical results on sparse signal recovery SAR and ultrasound imaging Sparse-aperture imaging Passive radar Sparse-array ultrasound Wide-angle SAR

16 SAR Ground-plane Geometry Scalar 2-D complex reflectivity field Transmitted chirp signal: Received, demodulated return from circular patch:

17 SAR Observation Model Observations are related to projections of the field: SAR observations are band-limited slices from the 2-D Fourier transform of the reflectivity field: Range profiles: Discrete tomographic SAR observation model: (combining all measurements) Observed data SAR Projection Operator Unknown field Noise

18 Conventional Image Formation Given SAR returns, create an estimate of the reflectivity field f O Support of observed data in the spatial frequency domain N Polar format algorithm: Each pulse gives slice of 2-D Fourier transform of field Polar to rectangular resampling 2-D inverse FFT

19 Feature-Enhanced Imaging Noisy observation model: Perform reconstruction through optimization: How should we choose to enhance features?

20 Enhance features: Scatterers, edges l p -norm-based Functionals for Feature Enhancement Features are in magnitude: Uniform-phase field Focus data: Feature sparsity Choose: Use of l p -norms leads to sparsity of argument Choice of L selects feature: Point Enhancement Region Enhancement

21 Efficient Solution of the Optimization Problem Cost functional Challenging optimization problem (non-quadratic, non-convex cost, constraint on f, large size) Extension of half-quadratic regularization techniques to complex-valued, random-phase fields Quasi-Newton algorithm with special Hessian approximation [Çetin and Karl, IEEE Trans. Image Proc., April 2001]

22 Point-Enhanced Superresolution Imaging Synthetic scene Original Conventional Point-Enhanced

23 Point-Enhanced Superresolution Imaging MIT Lincoln Laboratory ADTS data Conventional Point-Enhanced

24 Region-Enhanced Imaging MIT Lincoln Laboratory ADTS data Conventional Region-Enhanced

25 Region-Enhanced Imaging MIT Lincoln Laboratory ADTS data Conventional Region-Enhanced

26 Feature-based Evaluation MSTAR Data (tanks, personnel carriers), 216 images Evaluation Criteria for Point-Enhanced Images Target-to-Clutter Ratio 3-dB Main-Lobe Width Peak-Matching Accuracy Average Associated Peak Distance Evaluation Criteria for Region-Enhanced Images Speckle Suppression Segmentation Accuracy Statistical Separability of Regions [Çetin, Karl, and Castañón, IEEE Trans. AES, October 2003]

27 Impact on ATR performance Goal: Evaluate the impact of using point-enhanced or region-enhanced images instead of conventional images for ATR

28 Summary of ATR Experiments (in terms of P cc ) Template-based classifier Likelihood-based classifier Point-feature-based classifier Low-SNR Experiment Conventional 69.85% 87.05% 44.80% Point-Enh % 94.38% 81.43% These preliminary experiments show the robustness of feature-enhanced imaging to loss of SNR or resolution Much more extensive experiments needed: More vehicle classes Confuser vehicles Advanced feature-based ATR system Region-Enh % 99.15% Reduced-Resolution Experiment Conventional Point-Enh., Superresolution [Çetin, Karl, and Castañón, IEEE Trans. AES, October 2003]

29 Ultrasound Imaging Ultrasound test facility Aluminum object to be imaged Basic setup Potential application: Non-destructive evaluation

30 Ultrasound Measured and Theoretical PSF Measured PSF Theoretical PSF 730 khz. Green s function: 300 khz. (Real parts shown) Observation model based on the physical optics approximation:

31 Ultrasound Imaging Results Conventional Proposed 730 khz. (λ = 2 mm) 300 khz. (λ = 5 mm) [Çetin, Karl, and Willsky, Optical Engineering, to appear]

32 Outline Some theoretical results on sparse signal recovery SAR and ultrasound imaging Sparse-aperture imaging Passive radar Sparse-array ultrasound Wide-angle SAR

33 Passive Radar Imaging Data Collection Geometry FM Radio stations Flight paths of the target Receiver VHF TV stations

34 Passive Radar Imaging Fourier sampling pattern Magnitude of PSF Flight path - I Flight path - II

35 Passive Radar Imaging Results Electromagnetic Simulations via FISC Flight path - I Original Conventional Region-Enhanced [Çetin and Lanterman, IEE Proc. Radar, Sonar, and Nav., June 2005]

36 Passive Radar Imaging Results Electromagnetic Simulations via FISC Flight path - II Original Conventional Region-Enhanced [Çetin and Lanterman, IEE Proc. Radar, Sonar, and Nav., June 2005]

37 Sparse-Aperture Ultrasound Imaging Sensor array configuration

38 Sparse-Aperture Ultrasound Imaging Results Conventional Proposed 730 khz. (λ = 2 mm) [Çetin, Bossy, Cleveland, and Karl, ICASSP 2006, submitted]

39 Wide-Angle SAR Imaging Hamming windowed phase history data Two approaches: Subaperture-based approach Sparsity-driven anisotropy characterization (Preliminary)

40 Anisotropy Much reflection from one angle

41 Anisotropy Little reflection from another angle

42 Limitations of Conventional Imaging An isotropic point scatterer coherently imaged over a wide-angle aperture leads to an irregular PSF Isotropic scattering assumption is not usually valid Leads to inaccuracies in relative reflectivities of scatterers with different levels of anisotropy No characterization of aspect dependence Frequency or angle-band omissions cause yet more irregular PSFs, hence more pronounced artifacts Band structure determines the type of artifact Hard to adapt to and interpret the resulting imagery

43 Proposed Subaperture-based Approach Main pieces: Point-enhanced subaperture image reconstruction Composite wide-angle image formation Desirable features: Robustness to bandwidth limitations and band omissions Preservation of limited-persistence anisotropic scatterers Partial characterization of aspect dependence

44 Experimental Setup 2D imaging based on the AFRL Backhoe Data Dome Center frequency: 10 GHz Azimuthal span: 110 Bandwidths: 4 GHz, 2 GHz, 1 GHz, 500 MHz Full, 70%, 30% frequency-band availability 19 subapertures: {0,5,,90 } Subaperture width: 20 Subaperture response shape: Hamming

45 Sample Results Bandwidth = 4 GHz, Full band availability Conventional Proposed [Çetin and Moses, SPIE Defense and Security Symp., April 2005]

46 Sample Results Bandwidth = 1 GHz, Full band availability Conventional Proposed [Çetin and Moses, SPIE Defense and Security Symp., April 2005]

47 Sample Results Bandwidth = 1 GHz, 70% band availability Conventional Proposed [Çetin and Moses, SPIE Defense and Security Symp., April 2005]

48 Sample Results Bandwidth = 1 GHz, 30% band availability Conventional Proposed [Çetin and Moses, SPIE Defense and Security Symp., April 2005]

49 Visualization of Aspect Dependence 70% band availability Bandwidth = 4GHz Bandwidth = 500MHz [Çetin and Moses, SPIE Defense and Security Symp., April 2005]

50 Joint imaging and sparsity-based anisotropy characterization Main pieces: Formulate anisotropy characterization as an overcomplete representation problem Incorporate into point-enhanced imaging Intended desirable features: Characterization of aspect dependence (both direction and extent)! aid to target recognition Level of coherent integration tuned to anisotropy of the scatterer Accurate reflectivity estimation for both isotropic and anisotropic scatterers

51 Preliminary Experiments -- Setup

52 Preliminary Results Color-coded aspect angle [Varshney, Çetin, Fisher, and Willsky, SPIE Def. & Sec. Symp. 2006, submitted]

53 Preliminary Results Samples of estimated scattering functions [Varshney, Çetin, Fisher, and Willsky, SPIE Def. & Sec. Symp. 2006, submitted]

54 Conclusion Theoretical results showing the possibility of finding optimally sparse representations by efficient techniques Framework for sparsity-driven, feature-enhanced, model-based coherent imaging Efficient numerical algorithms for reconstructing images of complex-valued fields Practical outcomes: Preservation of scatterer locations and object boundaries Robustness to noise, data loss, resolution loss Suppression of noise artifacts, sidelobes, speckle Positive impact on automated decision-making Promising approach for sparse-aperture imaging Robustness to the sparsity and shape of the aperture Extension to the case of anisotropic scattering