Abstract
Nonlinear functions, including nonlinear iterated function systems, have interesting fixed points. We present a non-Lipschitz theoretical approach to nonlinear function system fixed points which generalizes to noncontractive functions, compare several methods for evaluating such fixed points on modern graphics hardware, and present a nonlinear generalization of Barnsley’s Deterministic Iteration Algorithm. Unlike the many existing randomized rendering algorithms, this deterministic method avoids noncoherent branching and memory access and takes advantage of programmable texture mapping hardware. Together with the performance potential of modern graphics hardware, this allows us to animate high-quality and high-definition fixed points in real time.
Highlights
Iterated function systems are a method to generate controlled, infinitely detailed fractal images such as Figure 1 from the repeated application of simple mathematical functions.1.1
An image-to-image transform F : I → I has a fixed point image a ∈ I when F(a) = a that is, the fixed point image remains unchanged under the image transform
With some graphics interfaces, such as GLSL or Open CL, we do generate the Graphics Processing Unit (GPU) functions at runtime; this means each time we find an inverse, at function generation time, we can paste in a call to the function needing the inverse value
Summary
Iterated function systems are a method to generate controlled, infinitely detailed fractal images such as Figure 1 from the repeated application of simple mathematical functions. The majority of iterated function system works uses the well-known Banach fixed point theorem, which gives both the existence and uniqueness of the fixed point and merely requires X and C to be complete, but requires the image transform F to be Lipschitz contractive. This theorem has been used for much iterated function system work [1], but it does require contractivity. The Jacobian determinant |Jwi−1 (p)|, as discussed, is present in order to preserve the integral of the transformed image, at least for well-behaved map functions This can be seen via the substitution method for multiple integrals, where q = wi−1(p). If the distortion function wi−1 is continuous, the geometric distortions in F will be continuous and if the Jacobians |Jwi−1 | change continuously, the color intensity changes in F will be continuous; by Schauder-Tychonoff, F has a fixed point
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