Lambda Selection

TotalVariationImageFiltering.jl includes two ROF-specific parameter selection tools:

select_lambda_discrepancy
select_lambda_sure

Both operate on N-D arrays (for example, 2D images and 3D volumes).

1) Discrepancy Principle

select_lambda_discrepancy finds lambda such that:

\[\|u_\lambda - f\|_2^2 \approx \text{target\_scale}\cdot N \sigma^2,\]

where N = length(f).

Search strategy in code:

evaluate at lambda_min and lambda_max;
expand upper bracket if needed;
bisection (max_bisect) on the bracket.

Example:

using TotalVariationImageFiltering

selection = TotalVariationImageFiltering.select_lambda_discrepancy(
    noisy,
    TotalVariationImageFiltering.ROFConfig();
    sigma = 0.12,
    lambda_min = 0.0,
    lambda_max = 0.2,
    rtol = 0.05,
)

lambda_hat = selection.lambda
u = selection.u

Returned diagnostics include residual norm, target norm, mismatch, evaluations, and final search bracket.

2) Monte-Carlo SURE

select_lambda_sure minimizes SURE on a provided grid:

\[\mathrm{SURE}(\lambda) = -N\sigma^2 + \|u_\lambda-f\|_2^2 + 2\sigma^2\,\operatorname{div}(u_\lambda(f)).\]

Divergence is estimated with Monte-Carlo finite differences:

\[\operatorname{div}(u_\lambda(f)) \approx \frac{1}{\epsilon} b^\top\!\left(u_\lambda(f+\epsilon b)-u_\lambda(f)\right), \quad b \sim \mathcal{N}(0,I).\]

Example:

using Random
using TotalVariationImageFiltering

selection = TotalVariationImageFiltering.select_lambda_sure(
    noisy,
    TotalVariationImageFiltering.ROFConfig();
    sigma = 0.12,
    lambda_grid = [0.0, 0.03, 0.06, 0.1, 0.16],
    mc_samples = 2,
    rng = MersenneTwister(1),
)

lambda_hat = selection.lambda
u = selection.u

Returned diagnostics include selected SURE, per-grid SURE values, residuals, divergence estimates, epsilon, and total number of solves.

Practical Notes

Both selectors are implemented for the ROF path (ROFConfig).
warm_start=true reuses solver state between lambda evaluations.
Smaller rtol (discrepancy) and larger mc_samples (SURE) increase runtime.

References

V. A. Morozov, Methods for Solving Incorrectly Posed Problems, 1984. DOI:10.1007/978-1-4612-5280-1
Y. Wen and R. H. Chan, "Parameter selection for total-variation based image restoration using discrepancy principle," IEEE TIP 21(4):1770-1781, 2012. DOI:10.1109/TIP.2011.2181401
S. Ramani, T. Blu, M. Unser, "Monte-Carlo SURE: A black-box optimization of regularization parameters for general denoising algorithms," IEEE TIP 17(9):1540-1554, 2008. DOI:10.1109/TIP.2008.2001404
Y. Lin, B. Wohlberg, H. Guo, "UPRE method for total variation parameter selection," Signal Processing 90(8):2546-2551, 2010. DOI:10.1016/j.sigpro.2010.02.025
C.-A. Deledalle et al., "Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection," 2014. HAL:hal-00987295