[Paper Review] Construction of Recurrent Fractal Interpolation Surfaces with Function Scaling Factors and Estimation of Box-counting Dimension on Rectangular Grids

Chol-Hui Yun, HuiChol Choi+1 more|arXiv (Cornell University)|Jul 10, 2013

Mathematical Dynamics and Fractals17 references3 citations

TL;DR

This paper constructs recurrent fractal interpolation surfaces (RFIS) on rectangular grids using function scaling factors within a recurrent iterated function system (RIFS), proving the existence of such surfaces as attractors. It establishes tight lower and upper bounds for the box-counting dimension of the constructed RFISs, showing that when the spectral radius exceeds the scaling factor, the dimension is bounded by $1 + \log_a \underline{\lambda}$ and $1 + \log_a \bar{\lambda}$, generalizing prior results with constant scaling factors.

ABSTRACT

We consider a construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of their box-counting dimension. A recurrent fractal interpolation surface (RFIS) is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given data set on rectangular grids, we construct general recurrent iterated function systems with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.

Motivation & Objective

• To extend existing fractal interpolation surface constructions by incorporating function-dependent vertical scaling factors instead of constant ones.
• To develop a general framework for constructing recurrent fractal interpolation surfaces (RFIS) using recurrent iterated function systems (RIFS) on rectangular grids.
• To prove the existence of bivariate fractal interpolation functions whose graphs are attractors of the constructed RIFSs for any given data set on rectangular grids.
• To estimate sharp lower and upper bounds for the box-counting dimension of the resulting RFISs, improving upon previous results with constant scaling factors.

Proposed method

• Define a recurrent iterated function system (RIFS) on rectangular domains $E_{ij} = I_i \times J_j$ using domain contraction maps $L_{x,ij}, L_{y,ij}$ and function scaling factors $s_{ij}(x,y)$.
• Construct the RIFS maps $\Psi_{ij}$ as affine transformations with variable vertical scaling factors $s_{ij}(x,y)$, ensuring the system is contractive on the product space.
• Prove the existence of a unique attractor $\mathcal{A}$ as the graph of a continuous bivariate function via the contraction mapping principle in the complete metric space of continuous functions.
• Estimate the box-counting dimension using covering numbers $N(\varepsilon_r)$, where $\varepsilon_r = a^{-r}$, and analyze the asymptotic growth rate via spectral radii $\underline{\lambda}, \bar{\lambda}$ of coefficient matrices.
• Use Frobenius theorem and matrix norm analysis to bound the covering number growth, leading to dimension bounds in terms of $\log_a \underline{\lambda}$ and $\log_a \bar{\lambda}$.
• Establish dimension bounds: $1 + \log_a \underline{\lambda} \leq \dim_B \mathcal{A} \leq 1 + \log_a \bar{\lambda}$ when $\underline{\lambda} > a$, and $\dim_B \mathcal{A} = 2$ if $\bar{\lambda} \leq a$.

Experimental results

Research questions

• RQ1 How can recurrent fractal interpolation surfaces be constructed with function-dependent vertical scaling factors on rectangular grids?
• RQ2 What conditions ensure the existence of a unique attractor surface as the graph of a bivariate continuous function in such a system?
• RQ3 What are the tightest possible lower and upper bounds for the box-counting dimension of the constructed RFISs?
• RQ4 How do variable scaling factors affect the dimension bounds compared to systems with constant scaling factors?
• RQ5 Under what conditions does the box-counting dimension of the RFIS equal 2?

Key findings

• The constructed RFISs are well-defined attractors of a recurrent iterated function system (RIFS) with function scaling factors, ensuring existence and uniqueness of the interpolation surface.
• The box-counting dimension of the RFIS is bounded below by $1 + \log_a \underline{\lambda}$ and above by $1 + \log_a \bar{\lambda}$, where $\underline{\lambda}$ and $\bar{\lambda}$ are the spectral radii of coefficient matrices derived from the system.
• When the lower spectral radius $\underline{\lambda} > a$, the dimension is strictly greater than 1 and lies within the derived logarithmic bounds.
• If the upper spectral radius $\bar{\lambda} \leq a$, the box-counting dimension of the RFIS is exactly 2, indicating a surface of full topological dimension in $\mathbb{R}^3$.
• The results generalize prior work on constant scaling factors, providing tighter and more flexible dimension estimates for fractal surfaces with variable scaling behavior.
• The method allows for arbitrary data sets on rectangular grids and provides a systematic way to control the fractal dimension through the choice of scaling functions and contraction maps.