Amortized Set Prediction for Inverse IFS Reconstruction from Density Maps

Abstract

Iterated Function Systems (IFS) generate self-similar fractals from a few contractive affine maps. The forward map from parameters to images is computationally inexpensive and well understood, whereas the inverse problem of estimating maps from an image is difficult and is typically handled by per-image optimization. We replace this loop with a single forward pass of a learned estimator that predicts the affine-map set directly from a visit-frequency density map, thereby amortizing the inverse problem. The design follows two constraints. First, density maps do not uniquely identify IFS parameters, so evaluation is based on reconstruction rather than parameter recovery; unordered map sets are handled by Hungarian matching, and ground-truth parameters provide a stable training surrogate. Second, the fully known forward model lets us generate exact synthetic training pairs and also supports image-only test-time refinement. On in-distribution tests, amortized initialization plus a few refinement steps lies on a better quality--speed frontier than equal-budget random-initialized per-image optimization, and a 30-step refinement (about $0.56$ s per sample) remains better than a doubled-budget baseline. Extending optimization to 1000 steps shows that the benefit is not only speed: amortized initialization reaches high-quality reconstructions more frequently than random starts. On real images (MNIST and Fashion-MNIST), it improves density metrics on average over a published per-image optimizer while being roughly 12 to 2600 times faster.