Inverse Digital Marbling: Recovering Gesture Programs with a Replay Adjoint
Tianqi Liu, Yushan Han, Hang Liu
Abstract
Pigment deposition in paper marbling displaces the pattern already present, coupling the appearance of each gesture to later actions. We recover executable programs for a deposition-based digital marbling model: given a target image, we optimise an ordered program of capsule insertions whose replay approximates it. The capsule primitive continuously joins circular drops to elongated deposits. Its transport is exactly area-preserving and has a closed-form inverse on the exterior of the deposited region. A replay adjoint reconstructs intermediate states, retaining coordinates lost inside deposits and periodic position checkpoints. At 2000 gestures and 1024^2 pixels, the PyTorch replay implementation uses 8.7x less memory than the tested checkpointed-autograd configuration at comparable step time; the fused implementation fits a program in about four minutes on one workstation GPU. We evaluate image reconstruction on five marbled sheets, compare against transport-disabled fitting, one-pass geometric compensation and a published stroke-based fitter at matched stroke count, and measure sensitivity to an alternative ordered-drop transport. Recovered programs replay across a 4x range of linear resolution. Edits specified in program order or in palette space -- inserting a gesture, recolouring a stage, translating a stage -- replay correctly under the same model; edits specified by image content, such as moving a motif, do not. On synthetic targets with known generating programs, the recovered programs match the images but not the generating gestures under a positional matching statistic. The output is an editable program in the specified digital medium.