Stochastic Optimization Methods for Parametric Level Set Reconstructions in 2D through-the-Wall Radar Imaging
Abstract
:1. Introduction
2. Materials and Methods
2.1. The Shape-Based Inverse Problem
2.1.1. The Forward Problem
2.1.2. Level Set Representation of Shapes
2.1.3. Calculation of Descent Directions
2.1.4. Specific Choice of Basis Functions
2.1.5. Shape Evolution Schemes
2.2. Stochastic Optimization Algorithms
2.2.1. A Line Search Scheme for Stochastic Shape Optimization
Algorithm 1: Shape-based line search scheme. |
Let and be the coefficient vector at iteration t and the new update direction |
Compute the level set function associated with (Equation (15)) |
Compute the permittivity profile associated with (Equation (8)) |
Initialize the line search parameter , where |
Define the parameters such that and |
Define the interval |
while () do |
% trial coefficient vector |
Compute the corresponding level set function (Equation (15)) |
Compute the corresponding permittivity profile (Equation (8)) |
Count the number of pixels where |
if () then |
end if |
if () then |
end if |
end while |
return |
2.2.2. A Stochastic Gradient Descent Algorithm
2.2.3. The Adam Algorithm
Algorithm 2: SGD algorithm. |
Initialize the parameter vector |
Initialize the level set function |
while not converged do |
Randomly extract a data subset from the dataset |
Compute the gradient of the cost wrt : |
Update the coefficient vector: |
where is chosen according to the line search Algorithm 1 |
end while |
Compute the final level set and the corresponding permittivity profile |
Algorithm 3: Adam algorithm. |
Initialize the parameter vector |
Initialize the level set function |
Define the exponential decay rates for the moment estimates: |
Initialize the first moment vector |
Initialize the second moment vector |
while not converged do |
Randomly extract a data subset from the dataset |
Compute the gradient of the cost wrt : |
Update the first moment vector |
Update the second moment vector , where denotes the elementwise square of |
Rescale the first moment vector |
Rescale the second moment vector |
Update the coefficient vector: , where |
and is chosen according to the line search Algorithm |
end while |
Compute the final level set and the corresponding permittivity profile |
2.2.4. The Online BFGS Algorithm
Algorithm 4: oBFGS algorithm. |
Initialize the parameter vector |
Initialize the level set function |
Initialize the parameters: , , |
Initialize the inverse Hessian approximation of the cost: |
while not converged do |
Randomly extract a data subset from the dataset |
Update the coefficient vector: , where |
with chosen according to the line search Algorithm |
if (t=0) then |
end if |
end while |
Compute the final level set and the corresponding permittivity profile |
3. Results and Discussion
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Incorvaia, G.; Dorn, O. Stochastic Optimization Methods for Parametric Level Set Reconstructions in 2D through-the-Wall Radar Imaging. Electronics 2020, 9, 2055. https://doi.org/10.3390/electronics9122055
Incorvaia G, Dorn O. Stochastic Optimization Methods for Parametric Level Set Reconstructions in 2D through-the-Wall Radar Imaging. Electronics. 2020; 9(12):2055. https://doi.org/10.3390/electronics9122055
Chicago/Turabian StyleIncorvaia, Gabriele, and Oliver Dorn. 2020. "Stochastic Optimization Methods for Parametric Level Set Reconstructions in 2D through-the-Wall Radar Imaging" Electronics 9, no. 12: 2055. https://doi.org/10.3390/electronics9122055
APA StyleIncorvaia, G., & Dorn, O. (2020). Stochastic Optimization Methods for Parametric Level Set Reconstructions in 2D through-the-Wall Radar Imaging. Electronics, 9(12), 2055. https://doi.org/10.3390/electronics9122055