Recursive Threshold Algorithm

• Recursive Threshold Algorithm is an iterative method using dynamic thresholds to control update rules and decision-making in complex systems.
• It leverages coupled state and threshold sequences to partition domains and optimize processes in applications such as image denoising and stochastic control.
• Practical implementations span image processing, rare-event simulation, optimal stopping, dynamic systems, and secure secret sharing, offering both theoretical and computational benefits.

The recursive threshold algorithm refers to a broad family of algorithmic paradigms and concrete methods in which thresholds are applied recursively, often controlling update or switching rules, partitioning, or stopping criteria. Several distinct mathematical frameworks deploy recursive threshold mechanisms, including but not limited to: image denoising via non-linear filters, rare-event probability estimation for Lipschitz functions, optimal stopping in stochastic processes, piecewise-recursive dynamic systems with adaptive thresholds, and recursive thresholding in secret sharing. These methods are unified chiefly by their exploitation of threshold-based rules embedded within a recursive or iterative computational structure.

1. Mathematical Foundations and General Framework

A canonical recursive threshold system is defined in terms of coupled state and threshold sequences. For example, in the dynamic systems context, sequences $\{a_n\}$ and $\{c_n\}$ are updated iteratively: $$a_{n+1} = \begin{cases} f(a_n), & a_n \leq c_n, \ g(a_n), & a_n > c_n, \end{cases} \qquad c_{n+1} = h(a_n, c_n),$$ where $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are regime-dynamics and $h:\mathbb{R}^2\rightarrow\mathbb{R}$ is the threshold update map. The boundary convention (e.g., $a_n = c_n$ uses regime 1) and the initial condition $(a_0, c_0)$ fully determine the trajectory (Valenti, 25 Jul 2025).

The recursive threshold principle also underpins recursive dyadic partitioning algorithms: subregions are split further only if a test value (computed at the subregion's center or via a local rule) is within a threshold of a target (e.g., in rare-event simulation) (Bernard et al., 2021). In stochastic optimal stopping, threshold-type policies with recursively defined value functions (via dynamic programming) emerge as optimal in a broad class of models (Long et al., 2017).

2. Recursive Threshold Algorithm in Image Denoising

The recursive threshold median filter (RT-MF) formalism for salt-and-pepper denoising is a prototypical example. Let $A(x, y)$ be the observed noisy grayscale image, and initialize $C^{(0)} = A$. Each recursive step consists of a median filter $B^{(k)} = \text{MF}(C^{(k-1)}, W)$, followed by a pixel-wise threshold rule: $$C^{(k)}(x, y) = \begin{cases} A(x, y), & |A(x, y) - B^{(k)}(x, y)| < T^{(k)}, \ B^{(k)}(x, y), & |A(x, y) - B^{(k)}(x, y)| \geq T^{(k)}. \end{cases}$$ The threshold sequence $T^{(k)}$ can be fixed or adaptive, and recursion proceeds until a convergence criterion or maximum iteration $K$ is reached. This scheme can incorporate three-layer denoising autoencoders in place of the median filter (Boriskov et al., 15 Nov 2025).

Several scalable variants exist:

• Two-Median-Filter (2MF): Independent recursive threshold MF runs with distinct window sizes, fused by a final threshold, enhance detail reconstruction.
• MFs–AE: Output images from multiple MF passes (with varying thresholds) are aggregated via an autoencoder to balance local detail and global scene structure (Boriskov et al., 15 Nov 2025).

Performance is quantified by the Structural Similarity Index (SSIM) on both restored images and sample-entropy maps ($\text{SSIM}_{\text{Img}}$, $\text{SSIM}_{\text{Map}}$), with the latter providing sensitivity to blurring and local detail preservation.

Scheme	Robustness to Noise	Computation	Application
RT-MF	Excellent (up to 60% S&P)	Very Low	Edge/IoT
AE	Poor (fails ≥30% S&P)	High	Post-decoded HR
2MF, MFs–AE	Moderate-High	Moderate-High	HR, detail fusion

3. Recursive Threshold Algorithm for Probability Estimation

In rare-event quantification, the recursive threshold algorithm leverages Lipschitz continuity to minimize expensive function calls. The domain $[0,1]^d$ is recursively partitioned (dyadically), and for each cube $Q$, $g$ is evaluated at its center $c_Q$. If $g(c_Q)$ lies above/below the threshold $T$ by more than $L(h/2)$ ($h$ is side-length), one can classify $Q$ as fully contained in/freed from the failure region. Only "uncertain" cubes near the threshold are split further.

The tree construction proceeds until a computational or accuracy budget is met, and produces lower/upper bounds $p_n^-, p_n^+$ for $p = \mathbb{P}(g(X) > T)$. For $d=1$, error decays exponentially in $n$; for $d>1$, decay is polynomial $n^{-1/(d-1)}$, which is provably optimal (Bernard et al., 2021).

Probabilities of cubes under $X$ can be recursively estimated via splitting or MCMC within each partition for generalized densities.

4. Recursive Threshold Optimal Stopping Rules

In stochastic control, particularly for Lévy processes, recursive threshold algorithms emerge as optimal policies for (possibly multi-stage) stopping problems under general discounting. The problem

$$V(x) = \sup_{\tau} \mathbb{E}_x[e^{-A_\tau}f(X_\tau)\mathbf{1}_{\tau<\infty}]$$

is reformulated under the average-problem approach: if $f(x) = \mathbb{E}_x h(\overline{X}_\zeta)$ (Condition M, for nondecreasing $h$ changing sign at a unique $b$), then stop at first up-crossing of threshold $b$. Multi-stage (recursive) problems involve value iteration: $f^{(\ell+1)}(x) = f(x) + e^{-r\delta}v^{(\ell)}(x)$ with each value function admitting a threshold-based stopping rule. The optimal thresholds $\{b_\ell\}$ satisfy monotonicity ($b_N \leq \cdots \leq b_1$) and admit efficient computation in spectrally negative models via scale functions (Long et al., 2017).

5. Piecewise Recursive Dynamics with Adaptive Thresholds

Piecewise-recursive dynamical systems feature recursive application of update maps $f, g$ with regime selection via a dynamic, recursively updated threshold $c_n$: $$a_{n+1} = \begin{cases} f(a_n) & \text{if } a_n \leq c_n \ g(a_n) & \text{if } a_n > c_n \end{cases}, \quad c_{n+1} = h(a_n, c_n)$$ This structure generates a rich spectrum of behaviors: global convergence, bistability, periodic orbits, spirals, and chaos, as characterized by affine parameter regimes. The Common Limit Theorem proves that, under infinite switching and convergence of all sequences, both $a_n$ and $c_n$ must converge to the common limit (a boundary fixed point).

Convergence criteria include:

• Global contractiveness of all update maps.
• Monotone evolution of $c_n$ and contraction of $f$.

Empirical application to central bank policy models, where inflation and threshold dynamically interact, reveals convergence to common policy-inflation limiting values that align with observed macroeconomic data (Valenti, 25 Jul 2025).

6. Recursive Threshold Secret Sharing

In visual cryptography and secret sharing, recursive threshold algorithms enable the embedding of multiple "secrets" into the shares of a larger secret using 2-out-of-3 (or more generally, 2-out-of-n) threshold schemes. The key technical detail is the use of randomness in the encoding—the "Split" operation gives multiple possibilities for share generation per bit, and these degrees of freedom allow embedding previous secrets recursively.

Perfect secrecy is maintained:

• Any single share gives no information.
• Any pair of shares uniquely reconstructs all embedded secrets, by recursive decoding.

Efficiency is increased: total secret bits per symbol approaches 0.44, compared to 0.33 in non-recursive schemes (0902.2487).

7. Computational Complexity and Practical Considerations

Computational performance and trade-offs depend on the specific application domain:

• For median filtering, per-pass complexity is $O(Nm^2 \log m^2)$ (image size $N$, window $m\times m$), scaling linearly with recursive depth. The method is uniquely suited to resource-limited hardware (Boriskov et al., 15 Nov 2025).
• For rare-event probability estimation in $d$ dimensions, the number of required expensive function calls grows as $\epsilon^{-(d-1)}$ for a target bias $\epsilon$, which is optimal and superior to rates achievable by Monte Carlo (Bernard et al., 2021).
• In secret sharing, encoding/decoding is linear in the size of the largest secret, and share sizes are fixed throughout recursive embedding (0902.2487).

8. Summary Table: Domains of Recursive Threshold Algorithms

Application Domain	Core Recursion	Threshold Mechanism Scope
Image Denoising	Iterated median/AE with threshold	Per-pixel, per-iteration
Probability Estimation	Dyadic partition tree	Subregion splitting
Optimal Stopping	Value function iteration	State process crossing
Dynamical Systems	Piecewise update, adaptive thres.	Regime switching
Secret Sharing	Recursive encoding	Bit-selection, embedding

The recursive threshold paradigm thus encapsulates a range of precise algorithms, analytical frameworks, and optimality results across applied probability, statistical inference, signal processing, stochastic control, dynamical systems, and cryptography, with each instantiation reflecting the fundamental algorithmic strategy of recursively partitioning, updating, or fusing based on threshold-driven local rules.