Learnable Curvelet Transform

This repository provides a MATLAB implementation of a **le

Curvelets are a canonical multiscale directional representation with strong theoretical guarantees, yet their fixed frequency tiling limits adaptability across diverse data and tasks. This paper develops a principled framework for learnable curvelet transforms that preserves the analytic structure of classical curvelets while enabling task-specific adaptation. Learning is performed through a structured parameterisation of the frequency-domain windows within the FDCT-wrapping construction, ensuring compact support, bounded overlap, and parabolic scaling. We establish rigorous analysis--synthesis guarantees, including exact and approximate perfect reconstruction, uniform stability, and finite frame bounds. Furthermore, we prove that key classical properties of curvelets---directional localisation and optimal sparse approximation rates for cartoon-like images---are preserved under admissible learned deformations. The learned curvelets are then specialised to three tasks: sparse recovery, classification, and clustering, yielding task-conditioned analysis operators with provable recovery guarantees in the analysis $\ell_1$ setting. Extensive experiments on images demonstrate consistent improvements over fixed and learned wavelet and contourlet representations, while a cross-task transfer analysis reveals the non-universality of learned transforms. Overall, the results show that learning, when guided by analytic constraints, can enhance performance without sacrificing the theoretical foundations of multiscale harmonic analysis.