Image Reconstruction Algorithms for 2D Aperture Synthesis Radiometers

Transcription

1 Image Reconstruction Algorithms for 2D Aperture Synthesis Radiometers E. Anterrieu 1 and A. Camps 2 1 Dept. Signal, Image & Instrumentation Laboratoire d Astrophysique de Toulouse-Tarbes Université de Toulouse & CNRS 2 Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya, Campus Nord, D4-016, Barcelona - IEEC CRAE/UPC, camps@tsc.upc.edu 1

2 Outline of the presentation: 1. Introduction: Aperture Synthesis in Radioastronomy and Earth Observation 2. Fundamentals on Aperture Synthesis Techniques 3. Image Reconstruction Algorithms: Ideal Situation 4. Image Reconstruction Algorithms: Real Situation Basic equation governing an interferometric radiometer Pre-processing of Visibility Samples: Cancellation of Sun and Moon effects Computation of differential visibilities Instrument equation in terms of differential visibilities Solution of Instrument Equation in Terms of Differential Visibilities: direct, iterative 5. Regularization of an ill-posed inverse problem Ill-posed problem Tikhonov / Minimum-norm / Band-limited regularization 6. Projection onto Earth grid and interpolation 7. Apodization and adpative strip-processing 8. Conclusions 2

3 1. Introduction: Aperture Synthesis in Radioastronomy and Earth Observation Aperture synthesis radiometers conceived as a way to increase the angular resolution achievable by real aperture antennas in radioastronomy VLA, New Mexico, Socorro ESTAR MIRAS 3

4 2. Fundamentals on Aperture Synthesis Techniques Channel 1 H 1 (f) H 2 (f) Channel 2 Baseline b 1 (t) b 2 (t) Complex Correlator 1 2 bb * 1 2 T ( ξη, ) = 1 1 * V ( u,v ) = b 1( t ) b2 ( t ) k BB GG 2 B * F ( ξη) ( ξη) ( ξη, ) T n1, F n, T B ΩΩ ξ η 2 ph, rec (u,v): ( uv, ) = ( Δx, Δy) λ 0 Director cosines ( ξ, η) = ( sinθ cosϕ,sinθ sinϕ) = antenna spacing normalized to the wavelength set of (u,v) points depend on the antenna separation and the array geometry set of spatial frequencies where the visibility function V(u,v) is sampled Ideal case: - Identical antenna patterns - Negligible spatial decorrelation - No antenna positioning errors 2D Fourier Transform V ( uv, ) = F T( ξ, η) 4

5 3. Image Reconstruction Algorithms: Ideal Situation (i) Antenna Positions Spatial frequencies (u,v) Periodic extension y [wavelengths] v u x [wavelengths] V ( uv, ) = F T( ξ, η) T ( ξη, ) = 2 F ( ) TB (, ) T n ξη, ξη ph rec Ω 1 ξ η 5

6 3. Image Reconstruction Algorithms: Ideal Situation (ii) In SMOS the alias-free FOV can be enlarged since part of the alias images are the cold sky (including the galaxy!) T B image limited by Earth replicas β η ξ Extension of Alias-Free FOV 6

7 4. Image Reconstruction Algorithms: Real Situation (i) Even in the ideal case: - Antenna spacing > λ/ 3 aliasing - Gibbs phenomenon (ringing) near the sharp transitions (mainly alias borders) In the real case: - Antenna patterns are different - Receivers frequency responses are different ( FWF different) - Antenna positioning errors (u,v,w) real different from (u,v,0) ideal IHFFT cannot be used as image reconstruction method More sophisticated algorithms must be devised But it will be good that the second ones tend to IHFFT in ideal conditions and obviously instrumental errors must be calibrated first! 7

8 4. Image Reconstruction Algorithms: Real Situation (ii) Basic equation governing an interferometric radiometer V 12 ( ξη) δ, Tph,rec * u12ξ + v12η + w12 1 ξ η F 1 ( ξη) ( ξη) np, Fnp 2, r12 1 ξ η f0 1 T = % ΩΩ 1 2 ξ + η 1 fore ( j ( u12 + v12 + w12 1 )) exp 2π ξ η ξ η d ξd η Includes: 1. -T rec (Corbella eqn. term) 2. On-ground measured antenna patterns (amplitude and phase: F np, i ξ, η ) 3. Measured fringe-washing function ( ) from correlations at different time lags r% 12 with correlated noise injected 4. Antenna positioning errors (phase center): (u 12, v 12, w 12 ) 5. Antenna voltage pattern frequency-dependence can be minimized with the following weighted average 6. Extension to the polarimetric case including the co- and cross-polar patterns 7. A second integral must be added with -w 12 to account for antenna backlobes ( ) 8

9 4. Image Reconstruction Algorithms: Real Situation (iii) Pre-processing of Visibility Samples: 1.Computation of auxiliary visibilities to extend the AF-FOV to the periodic repetition of Earth aliases : a) Term corresponding to the physical temperature of the receivers (T ph,rec ): VR ( uv, ) Appears as an offset in the visibilities, except in V(0,0), which is measured by the 3 NIRs. V R ξ η ξ η ( ) ˆq u12 + v12 + w12 n ( ) ( ) Δ 1 -T 1 = Fˆ ξ, η F ξ, η ˆ r exp j2 π( u ξ + v η + w 1 ξ η ) dξdη, % ph,rec p * n ΩΩ ξ + η ξ η f0 This term can also be estimated from the Flat Target Response (FTR) when looking at a homogeneous cold sky at T sky, as: b) V (0,0) = T A : weighted average of the 3 NIR measurements V c) Term corresponding to the sky: V sky (u,v), sky V R -Tph,rec = FTR, T -T sky ph,rec L-band noise map of the sky (cosmic + galactic contributions) Δ ( ξη) 1 T ξ η ξ η Bsky, + + = ˆp ξ η ˆq * ξ η ˆ u12 v12 w12 1 π ξ + η + ξ η ξ η ΩΩ F % n1, Fn 2, 12 r exp j2 ( u12 v12 w12 1 ) d d, ξ + η ξ η 1 2 sky 1 f0 ( ) ( ) ( ) 9

10 4. Image Reconstruction Algorithms: Real Situation (iv) d) Term corresponding to the antenna back lobes, since there are two directions (θ,φ) and (π-θ,φ) that are imaged in the same (ξ,η) point: ( ) ( ) ( ( )) ( ξη) ξ + η ξ η = 1 T, back, T rec ˆ ξ η ˆ* ξ η π ξ + η ξ η ξ η ΩΩ ˆ u % 12 v12 w12 1 V,, ξ + η ξ η xp back Fnp Fnp r e j u v w d d f0 Note: uncertainty in measured antenna patterns from back side is large, comparable to its value. e) Term coming from a constant T B over land (TBD) and a modeled T B over the sea so that differential visibilities are zero-mean: V V land sea Δ { ( ) ( ) ' 0, 0 0 ( ) ( ) ( ) ( ),,,,,, 0 0 A Vsky Vs ea Vback TSun dir VSun dir u,v TSun dirv Sun scatt, TMoon dirv Moon dir, TMoon, dirv Moon, scatt ( 0, 0)} land ( 0,0) T = T - 0, 0,0 00 V, land ( ) ˆq * u12ξ + v12η+ w12 1 ξ η ( )% n ( ) Δ 1 1 = Fˆ ξ, η F ξ, η rˆ exp j2 π( u ξ + v η+ w 1 ξ η ) dξdη, p n ΩΩ ξ + η ξ η f land or iced sea 0 ( ξη) ξ η ξ η ( ) ˆq * + + ( ) ˆ u12 v12 w12 1 % n ( 2 ) Δ 1 T, = Fˆ ξ, η F ξ, η r exp j2 π( u ξ + v η+ w 1 ξ η ) dξdη, Bs, ea p n ΩΩ ξ + η ξ η 1 2 se 1 f a 0 Note: this term will be computed once the Sun (++) and Moon (--) contributions are estimated 10

11 4. Image Reconstruction Algorithms: Real Situation (v) Cancellation of Sun and Moon effects: ˆ B R, -1 - Make a raw T B image using an IHFFT: T = F V (u,v) V ( u v) -The Sun is such a bright source that it completely masks the rest of the image. - Estimate the image of a normalized (T B =1 K) Sun and Moon if visible: ˆ -1 T = F V (u,v) Sun / Moon, dir / ref Sun / Moon, dir / ref - Estimate T B from the Sun as: ˆ T ( ξ,, η, ) T TSun, dir = ˆ T ξ, η B Sun dir Sun dir background ( ) Sun, dir Sun, dir Sun, dir - Compute the contributions to the measured visibilities from the Sun: Number of appearances Most probable value around Sun pixel [Kelvin] V ( u,v ) = T V ( u,v) Sun / Moon, dir / ref Sun / Moon, dir / ref Sun / Moon, dir / ref -T B Moon is much smaller and cannot be easily seen in the raw T B image. T BMoon ~250 K 11

12 Computation of differential visibilities: ( u, v ) back (, v) Vs ky ( u, v) sea (, ) land land (, ) ( ) ( u,v ) T V ( u,v ) T V ( u,v ) ΔV ( u,v) = V (u,v) V V u V u v T V u v R ' T V u,v T V Sun, dir Sun, dir Sun, dir Sun, scatt Moon, dir Moon, dir Moon, dir Moon, scatt, Note: V R (0,0) = 0 and V (0,0) = T A (without the T rec term) Instrument equation in terms of differential visibilities: Δ 1 Δ T B ( ξη, ) ξ + η = ˆp ( ξ η) ˆq * ( ξ η) ˆ u12 v12 V F ( π 12ξ + 12η ) ξ η n Fn r j u v d d ξ η f0 Δ ΩΩ,, % exp 2 ( ), ξ + η Earth Where: T ( ξ, η ) = ˆ ΔT ( ξη, ) + T ( ξ, η ) + T land ( ξ η ) B B sea land, The equation is discretized dec W ( u, v) ΔV ( u, v) = G ΔT ( ξ, η ) Notes: Windowing (W(u,v)) can also be performed in the (ξ,η) domain after image reconstruction Redundant baselines (quasi linear dependent equations) can be weighted-averaged prior to construction of G improves condition number 12

13 If redundant baselines are averaged G-matrix rows correspond to the discretization of the integral equation, sampled at the (ξ,η) points of the (u,v) reciprocal grid (slide #5) Reduces to IHFFT in the ideal case no error amplification (cond. number = 1). Redundant visibilities (same (u,v) point) are averaged to reduce noise the corresponding equations are averaged as well. Solution of Instrument Equation in Terms of Differential Visibilities: Direct: G-matrix pseudo-inverse (used in ESTAR) dec H 1 H ( ξη, ) ( ) (, ) (, ) Δ T = G G G W u v ΔV u v Iterative methods (coded in SEPS can run in standard PC): Extended-CLEAN (UPC) Stabilization + conjugate gradient (UPC): (, ) Δ (, ) = Δ (, ) H H dec G W u v V u v G G T ξ η OK for small arrays Calibration + Sun/Moon cancellation + Differential visibilities + G-matrix formulation + Image reconstruction SMOS L1 Processor 13

14 Simulation examples Original TB FFT retrieved TB G-matrix retrieved TB after visibility decomposition Error: Original TB - FFT retrieved TB Original TB - G-matrix retrieved TB after visibility decomposition [from IGARSS 2006 & TGRS, Jan 2008] 14

15 5. Regularization of an ill-posed inverse problem What is an inverse problem? T G T = V direct problem inverse problem T = G -1 V V What is an ill-posed problem? J. HADAMARD (1902), R. COURANT (1962): A problem satisfying the requirements of existence, uniqueness, and continuity is said to be well-posed. How to regularize an ill-posed inverse problem? J. HADAMARD (1902), R. COURANT (1962): The general principle of regularization is to introduce a priori information in order to compensate for the lack/loss of information in the imaging process. 15

16 Ill-posed problem min V - G T 2 T eigenvalues of G * G G * G T = G * V T r = G + V with G + = (G * G) -1 G * G * G is singular 16

17 Tikhonov regularization min V - G T 2 + μ T 2 T,μ eigenvalues of G * G + μi (G * G + μi) T = G * V T r = G + μ V with G + μ = (G * G + μi) -1 G * μ is the LAGRANGE regularization parameter 17

18 Minimum-norm regularization min T 2 T { V = G T singular values of G Σ i 1 T r = G + 1 T V with Σ T G + = V i U i G = σ i U i V i i 1 σ i (SVD) T r = G + m V with Σ 1 T G + = V i U i i m Σ T m σ G = σ i U i V i i m i m (TSVD) m plays the role of a regularization parameter 18

19 Band-limited regularization min V - G T 2 T { (I PH ) T = 0 singular values of J J * J ^ T H = J * V with J = G U * Z T r = U * Z J + V with J + = (J * J) -1 J * P H plays the role of a regularization parameter 19

20 Examples Observed scenes: 20

21 Examples Retrieved maps: 21

22 6. Projection onto Earth grid and interpolation Retrieved maps are reconstructed at antenna level but they should be projected onto Earth surface for scientific purpose. As a consequence, an interpolation or re-gridding procedure should be performed FOV in antenna frame FOV at ground level 22

23 6. Projection onto Earth grid and interpolation Retrieved maps are reconstructed at antenna level but they should be projected onto Earth surface for scientific purpose. As a consequence, an interpolation or re-gridding procedure should be performed FOV in antenna frame FOV at ground level 23

24 6. Projection onto Earth grid and interpolation Retrieved maps are reconstructed at antenna level but they should be projected onto Earth surface for scientific purpose. As a consequence, an interpolation or re-gridding procedure should be performed FOV in antenna frame FOV at ground level without introducing artifacts! 24

25 6. Projection onto Earth grid and interpolation Resampling of a band-limited image with a bi-linear interpolation is translation dependent, this not the case of the Fourier approach: Fourier interpolation bi-linear interpolation 25

26 6. Projection onto Earth grid and interpolation Resampling of a band-limited image with a bi-linear interpolation is translation dependent, this not the case of the Fourier approach: Fourier interpolation bi-linear interpolation 26

27 7. Apodization and strip-adaptive processing Gibbs oscillations Owing to the finite extent of the experimental aperture and to the associated sharp frequency cut-off, Gibbs oscillations should be expected: d 27

28 7. Apodization and strip-adaptive processing Gibbs oscillations Owing to the finite extent of the experimental aperture and to the associated sharp frequency cut-off, Gibbs oscillations should be expected: λ o 2 3 d 28

29 7. Apodization and strip-adaptive processing Gibbs oscillations Owing to the finite extent of the experimental aperture and to the associated sharp frequency cut-off, Gibbs oscillations should be expected: 29

30 7. Apodization and strip-adaptive processing Gibbs oscillations Owing to the finite extent of the experimental aperture and to the associated sharp frequency cut-off, Gibbs oscillations should be expected: These effects can be attenuated by using an appropriate apodization window. 30

31 7. Apodization and strip-adpative processing Projection onto Earth Geometrical effect: - constant angular resolution at antenna level - variable spatial resolution at ground level 31

32 7. Apodization and strip-adaptive processing Projection onto Earth Strip adaptive processing with multiple apodization windows: - variable angular resolution at antenna level - constant spatial resolution at ground level 32

33 8. Conclusions Image reconstruction algorithm for SMOS presented, including: Foreign sources removal (sky, Sun, Moon) and visibility decomposition (Corbella term, backlobe, land + iced sea & sea) reduces reconstruction errors and bias G-matrix formulation & inversion Projection onto Earth grid Apodization and strip-adaptive processing 33