tomographic sar inversion by the generic log- barrier algorithm the second order...
TRANSCRIPT
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CoSeRa Bonn 2013
Tomographic SAR Inversion by The Generic Log-Barrier Algorithm the Second Order Cone
Programming Approach.
Prof. Nazzareno PIERDICCA (University La Sapienza)Prof. Piero CIOTTI (University of L'Aquila)
Dr. Filippo BIONDI (Italian Ministry Of Defence)
University of L'Aquila (Italy)
Electrical Engineering Faculty
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LIST OF ARGUMENTS
● Tomographic SAR analysis by spectral estimators;
● Under-sampled Tomographic SAR data-sets;
● Validation of Compressed Sensing (CS) signal processing for mixed environment by the Second Order Cone Programming (SOCP).
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Multi-Baseline Geometry Acquisition● SAR Tomography;
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● Statistical De-correlation Effects in Repeat-Pass Tom-SAR acquisition Campaign;F. Biondi, P. Ciotti, N. Pierdicca. Tomographic SAR Inversion by the Generic Log-Barrier Algorithm, The Second Order Cone Programming Approach
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Vegetated Environmental
Hybrid Environmental
Coherent Environmental
[1] Yue Huang, Laurent Ferro-Famil, Fabrizio Lombardini: Improved Tomographic SAR Focusing using Automatic Baseline Error Compensation, ESA PolInSAR 2012.
MB-PolInSAR Environments
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Tomographic Problem Solution
● y=A*x
● y= Known Observation Parameter;
● A=Geometric Built, Steering Matrix;
● x=Unknown reflectivity function, (vertical reflectivity profile);
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RangeRange
AzimuthAzimuth
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● A is not directly invertible, P<N
● Direct Method:
● Indirect Method, (Covariance Matrix Estimation);
● This leads to poor results for many practical applications
A−1=A†
Tomographic Problem Solution
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Covariance Matrix Estimation● Tomographic Processing Procedure: Covariance Matrix
Estimation;
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Compressed Sensing
An Optimization Problem seeks to find the best solution so that is true:
min∥ x ∥1 subject to Ax= y
min∥ x ∥1 subject to ∥ A x - y∥2 SOCPSOCP
LPLP
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CVX – Convex Optimization
minimize f 0 x subject to f i x 0
x=x1xN optimization variablef 0 :ℂN
ℂ objective functionf i :ℂ
Nℂ ; i=1m constraint functions
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and y=Ax
Inequality Constrained Problem
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CVX – Convex Optimization
F. Biondi, P. Ciotti, N. Pierdicca. Tomographic SAR Inversion by the Generic Log-Barrier Algorithm, The Second Order Cone Programming Approach
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∃D s.t. y=Ax and f ix 0 ; i=1⋯m.s.t.
f 0 ,⋯, f m :∈ℂnℂ are convex and twice
continuously differentiable
A∈ℂPxN with Rank A=P<N continuously differentiable
x , f x = p is the optimal solition
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CVX – Convex Optimization Duality:
Karush-Kuhn-Tucker KKT Condition:any optimization problem with differentiable objective and
constraint functions for which strong duality obtains,any pair of primal and dual optimal points must satisfy the following
∃*∈ℂ
m , v∈ℂP s.t.
Ax*= y , f i x
*0 ; i=1 ,⋯,m
* ≽ 0
∇ f 0 x*∑
i=1
m
i ∇ f ix*AT v*
=0
i* f ix
*=0 ; i=1⋯m
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CVX – Convex Optimization
Interior Point Methods Applying Newton Methods
Inequality Constrained Problem
Equality Constrained Problem
Use of Log-Barrier Method13 F. Biondi, P. Ciotti, N. Pierdicca. Tomographic SAR Inversion by the Generic Log-Barrier Algorithm, The Second Order
Cone Programming Approach
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● The Log-Barrier Algorithm for SOCP
CVX – Convex Optimization
The best Equality Constrained Problem is:
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minimize f 0x ∑i=1
m
I - f i x subject to y=Ax
where I - :ℝℝ has the following indicator function for the non-positive reals
I -= { 0 ;u0∞ ;u0 }
No Inequality ConstraintBut is not differentiable
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CVX – Convex Optimization● The Log-Barrier Algorithm for SOCP
I - is approximated into: I -
I -=−1/ t log −u
minimize f 0 x∑i=1
m
[−1/t log −f ix ]
subject to y=Ax for the non-positive reals= [−1/ t log −f ix ]
The Log-Barrier is twice differentiable
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CVX – Convex Optimization● The Log-Barrier function is twice differentiable
● Approximation of the Log-Barrier Problem
f 0x x≈x⟨gx∣x⟩12⟨H xx∣x ⟩=q x x
Solution: find x that minimizes q xx
subject to Ax=y is the solutionof the following linear equiation set :
H x A0T
A0 0 x
v =−gx
gx=∇x =∑i=1
m1
−f i x∇ f ix
H x=∇2=
1∑i=1
m1
f ix2 ∇ f i x
T∑
i=1
m1
−f ix 2 ∇
2 f i x
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● We have used a “matrix free” solver such as Conjugate Gradients (C.G.)
● The following two algorithms results has been opposed:
● Min-l1 with bounded residual correlation (Dantzig Selector) LP;
CVX – Convex Optimization
min∥x∥1 subject to ∥A*Ax− y ∥∞
● Min-l1 with quadratic constraints SOCP
min∥x∥1 subject to ∥Ax− y∥2
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CVX – Convex Optimization● Validation of the Log-Barrier algorithm to L1● Point Scatterers
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● Validation of the Log-Barrier algorithmCVX – Convex Optimization
● Point Scatterers
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● Validation of the Log-Barrier algorithmCVX – Convex Optimization
● Point Scatterers
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CVX – Convex Optimization● Distributed Scatterer - Case (a)
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● Distributed Scatterer - Case (b)CVX – Convex Optimization
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CVX – Convex Optimization
● Distributed Scatterer Case (c)
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CVX – Convex Optimization● Realistic PolSARPro Vegetation Environment
P-band L-band
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CVX – Convex Optimization
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CVX – Convex Optimization
Capon Filter: Two layer definitionSOCP: Three layer definition
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Airborne Real Data-Set
● Geometry:
● 6 Tracks;
● Full POL;
● P and L-Band;
● Vertical Baseline 50 ft;
● Height 4000m.
● Mission: TROPISAR-2009 (SETHI ONERA ESA)
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Real data P-bandoptical radar
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CVX – Convex Optimization•Blue line case Tomogram
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CVX – Convex Optimization•Red line case Tomogram
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CVX – Convex Optimization
(b)-
line
(a)-
line
A great resolution improvement is observed
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CVX – Convex Optimization
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optical
radar
CVX – Convex Optimizationradar
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CVX – Convex Optimization
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CVX – Convex Optimization
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CVX – Convex Optimization
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CVX Convex Optimization
●The Resolution is doubled
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Conclusions● Demonstration that Convex Optimization can be
successfully used for SAR Tomography;
● L1 Norm minimization with quadratic constraints where a Log-Barrier algorithm with Newton iteration was implemented;
● Good resolution improvement has been observed.
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Future Work
● Efficient Tomographic reconstruction based on the Total Variational (TV) minimization of estimated image;
● TV may have higher computational speed respect to Classical CS.
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THANKS FOR YOUR ATTENTION