Spatial Frequency Transfer Tuning in Optical Phase Retrieval Systems
Phase retrieval accuracy hinges on balancing low-frequency noise amplification against high-frequency contrast decay through tuned transfer filters.
Regularization Parameter
Meaning
Inverse problems in image reconstruction suffer from ill-posedness, where small measurement noises lead to large reconstruction errors. The regularization parameter is a positive scalar that balances the fidelity to the measured data against a prior assumption about the solution. It controls the strength of the regularization term relative to the data misfit term in the objective function.
This parameter is adjusted to prevent noise amplification while preserving the true features of the reconstructed object.
Error Balancing
Choosing the regularization parameter represents a trade-off between bias and variance. If the parameter is too large, the reconstruction is overly smoothed, which introduces bias and removes fine details. Conversely, if the parameter is too small, the solution is dominated by high-frequency noise, which increases variance.
Thus, the regularization parameter must be selected carefully to balance these two competing sources of error.
Optimal Selection
Methods like the L-curve or generalized cross-validation are used to find the optimal parameter value. The L-curve method plots the norm of the regularized solution against the norm of the residual for different parameter values, and the optimal parameter corresponds to the corner of the resulting L-shaped curve. This corner represents the point where further decrease in the residual requires a massive increase in the solution norm.
However, this calculation is computationally expensive for large-scale imaging systems. For this reason, practitioners often rely on heuristic estimates or automated parameter tuning sweeps during the calibration phase of the instrument.
Numerical Stability
In iterative reconstruction algorithms, the regularization parameter is often decreased gradually as the iterations progress. This cooling schedule stabilizes the early iterations and prevents the algorithm from getting stuck in poor local minima. As the algorithm converges, the parameter is kept at its target value to ensure the final solution remains physical.