Hidden Variable Fractal Interpolation

• The paper introduces CHFIF, a method that integrates hidden parameters into IFS to transition from self-affine to non-self-affine fractal behaviors.
• It employs free and constrained parameters within affine maps to finely control regularity and fractal dimension, ensuring unique attractors under strict contractivity conditions.
• Practical applications include signal modeling, surface fitting, and multiresolution analysis, with rigorous analytical, geometric, and stability results.

A hidden variable fractal interpolation function (HVFIF), and in particular its coalescence variant (CHFIF), is a continuous function whose graph arises as the projection of the attractor of an iterated function system (@@@@2@@@@) or recurrent IFS (RIFS) defined on an extended state space $I \times \mathbb{R}^2$, where $I$ is an interval and the hidden variable provides additional functional freedom. This framework generalizes classical Barnsley-type fractal interpolation functions, enabling both self-affine and non-self-affine behavior. CHFIFs exploit the structure of free and constrained parameters in the defining transformations, allowing fine control over both regularity and fractal dimension; extensions to graph-directed systems accommodate simultaneous interpolation of multiple data sets. Analytical, geometric, and wavelet-theoretic properties of CHFIFs have been rigorously characterized, with practical applications in signal modeling, surface fitting, and multiresolution analysis.

1. Generalized Data, Hidden Variable, and Interpolant Construction

The construction begins from a “generalized” data set in $\mathbb{R}^3$: $$\mathcal{D} = \{ (x_i, y_i, z_i) : i = 0, 1, \dots, N \}, \qquad x_0 < x_1 < \cdots < x_N,\; y_i, z_i \in \mathbb{R}.$$ Here $z_i$ is the “hidden variable” auxiliary to the observed interpolation $(x_i, y_i)$. The goal is a continuous function $f: [x_0, x_N] \to \mathbb{R}^2$, $f(x_i) = (y_i, z_i)$ for all knots.

The associated IFS acts on $K = I \times \mathbb{R}^2$, with maps

$$w_i: K \to I_i \times \mathbb{R}^2, \qquad w_i(x, y, z) = \big(L_i(x), F_i(x, y, z)\big),$$

where $I_i = [x_{i-1}, x_i]$ and $L_i$ is affine: $$L_i(x) = a_i x + b_i, \quad a_i = \frac{x_i - x_{i-1}}{x_N - x_0}, \quad b_i = \frac{x_N x_{i-1} - x_0 x_i}{x_N - x_0}.$$ The vertical (vector-valued) component is defined by \begin{align*} F_i^1(x, y, z) &= \alpha_i y + \beta_i z + c_i x + d_i, \ F_i^2(x, y, z) &= \gamma_i z + e_i x + f_i, \end{align*} with the parameters $\alpha_i,\gamma_i \in (-1, 1)$ (free), and $\beta_i$ subject to $|\beta_i| + |\gamma_i| < 1$ (constrained free). The coefficients $c_i, d_i, e_i, f_i$ resolve the join-up conditions

$$F_i(x_0, y_0, z_0) = (y_{i-1}, z_{i-1}), \quad F_i(x_N, y_N, z_N) = (y_i, z_i),$$

via explicit linear equations (Akhtar et al., 2015, Kapoor et al., 2012).

2. Self-Affine versus Non-Self-Affine CHFIF Structure

Setting all $\beta_i = 0$ recovers the classical self-affine fractal interpolation function, for which $F_i^1$ depends solely on $(x, y)$. In the CHFIF, any nonzero $\beta_i$ enables $F_i^1$ to “coalesce” the hidden variable $z$, breaking strict self-affinity and leading to non-self-affine or more general fractal behavior. The first graph-component $f_1$ of the solution $f$ thus delivers a richer set of interpolants, whose shape and regularity are controlled by both $\alpha_i$ (vertical contraction), $\gamma_i$ (hidden-variable contraction), and $\beta_i$ (coupling strength) (Akhtar et al., 2015).

3. Existence, Uniqueness, Regularity, and Functional Equations

Under the contractivity conditions

$$\max_i |\alpha_i| < 1, \quad \max_i |\gamma_i| < 1, \quad \max_i (|\beta_i| + |\gamma_i|) < 1,$$

there exists an equivalent norm on $\mathbb{R}^3$ so that each map $w_i$ is strictly contractive; the IFS (or RIFS for generalizations) admits a unique compact attractor graph $G \subset K$ by Hutchinson–Barnsley theory. $G$ is the graph of a continuous vector-valued interpolant $f$, with $f(x_i) = (y_i, z_i)$, and $f_1$ is the CHFIF through the original data. The CHFIF satisfies a fixed-point equation of the Read–Bajraktarević type: $$f(x) = F_i(L_i^{-1}(x), f(L_i^{-1}(x))), \quad x \in [x_{i-1}, x_i].$$ Perturbation analysis demonstrates Lipschitz and Hölder regularity of $f_1$ depending on the parameter regime, with explicit bounds for modulus of continuity and stability under data variation (Prasad, 2012, Yun et al., 2019, Ri et al., 2019).

4. Graph-Directed IFS Generalization and Multivariate CHFIFs

CHFIFs generalize naturally to settings where multiple data sets are to be interpolated simultaneously. Given, for example, two data sets $$\mathcal{D}^r = \{ (x_i^r, y_i^r, z_i^r) : i = 0, \dots, N_r \}$$, one constructs a graph-directed IFS (GDIFS) by assigning a directed graph $G=(\{1,2\}, E)$, and constructing edge maps $w_n^{rs}: X^s \to X^r$ controlled by the graph structure (Akhtar et al., 2015). The Mauldin–Williams theorem secures the existence and uniqueness of attractors, with the projection onto $\mathbb{R}^2$ of each $G^r$ yielding the CHFIF for the respective dataset.

Analogous constructions exist for bivariate or grid-based datasets, leading to hidden variable bivariate FIFs (HVBFIFs) with multidimensional nonlinear contractivity functions as in (Yun, 2019, Yun et al., 2019).

5. Analytical Properties: Box-Counting Dimension, Smoothness, and Stability

The box-counting dimension and geometric properties of CHFIFs and HVFIFs are governed by the magnitude and variability of the contractivity factors. For CHFIFs constructed by affine IFS, the dimension of the graph $G(f_1)$ satisfies explicit bounds depending on contraction rates and coupling strength; variable contractivity factors allow for local or global adjustment of fractal dimension (Yun et al., 2019, T et al., 22 Jan 2026). If the Hölder exponent is $\alpha$, the box-counting dimension obeys

$\overline{\dim}_B G(f_1) \leq 2 - \alpha.$

More generally, for RIFS-based HVRFIFs, the dimension is bounded in terms of the Perron–Frobenius eigenvalues associated to matrices of maximal and minimal vertical scaling, and can range strictly between $1$ and $2$ for functions, or $2$ and $3$ for bivariate surfaces (Ri et al., 2019). Stability results establish that the supremum norm error under perturbation of contractivity factors or data is finite and proportional to the perturbation magnitude (Yun et al., 2019, Ri et al., 2019, Ri et al., 2019).

6. Multiresolution Analysis and Wavelet Construction with CHFIFs

CHFIFs form a vector space whose dimension is determined by the number of free parameters and data points. By defining subspaces $\mathbb{V}_0$ of $L^2(\mathbb{R})$ generated by CHFIFs supported on intervals, one constructs multiresolution analyses (MRA) analogous to classical splines and wavelets, but with greater functional diversity due to the hidden variable mechanism. For $N$ interpolation points, the subspace of CHFIFs through the data has dimension $2N$; orthogonal and orthonormal bases can be explicitly constructed, often involving Gram–Schmidt procedure and the resolution of systems of nonlinear equations among the interpolation heights and contraction parameters (Kapoor et al., 2012). Compactly supported continuous orthonormal wavelets are generated within this CHFIF-MRA framework, with enhanced flexibility in designing basis functions suitable for applications requiring fractal regularity or controlled roughness.

7. Practical Applications and Example Scenarios

CHFIFs and their hidden variable generalizations have strong utility in fields requiring flexible interpolation of irregular or scattered data, including:

• Image and signal modeling, exploiting non-self-affine structures to match natural irregularities.
• Geosciences, especially for terrain profile fitting where multifractal textures are prevalent.
• Computer graphics, providing alternatives to traditional spline curves or surfaces, with controllable roughness and local geometry.
• Wavelet analysis and functional approximation, due to the existence of orthonormal and compactly supported basis functions (Akhtar et al., 2015, Kapoor et al., 2012, Yun, 2019).

Numerical examples confirm that insertion of new interpolating nodes and adjustments of the hidden variable permit fine modulation of regularity, box-dimension, and local geometry (Prasad, 2012). The contractivity factors and hidden variable coupling ($\beta$) afford localized control over both the analytic and geometric properties of the resulting fractal interpolant.

8. Summary Table: Parameter Roles in CHFIF Construction

Parameter	Type	Role in CHFIF Definition
$\alpha_i$	Free	Vertical contraction, self-affinity
$\beta_i$	Constrained	'Coalescence' (hidden-variable)
$\gamma_i$	Free	Hidden-variable contraction
$c_i, d_i, e_i, f_i$	Derived	Join-up/interpolation conditions

The interplay among these parameters controls the overall regularity, self-affine or non-self-affine nature, and shape of the CHFIF and its generalizations.

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The CHFIF framework and its hidden variable extensions represent a rigorously analyzed, analytically flexible, and broadly applicable generalization of fractal interpolation theory. The integration of multi-parameter contractivity, recursive operator structure, and functional analytic techniques has resulted in robust tools suitable for modeling, approximation, and analysis across disciplines where complex, fractal, or irregular data arise.