Design of Deformable Pyramid Transform with Shift-invariance, Steerability, and Scalability

摘要

This paper presents a closed-form Deformable Pyramid Transform (DPT) with shift-invariance, steerability, and scalability. This DPT is extended from a Steerable Pyramid Transform (SPT) that is represented in a polar-separable form. Specifically, the radial component of the SPT's basis filters is taken as the kernel for designing scalable basis filters. These are further combined with the SPT's steerable basis filters corresponding to the angular components, achieving the joint steerability and scalability. The shift-invariance is inherited from the SPT by retaining undecimated high-pass and band-pass basis subbands. Based on the designed DPT, we theoretically derive both interpolation functions for steerability and scalability and synchronization mechanisms for translation, rotation, and scaling. Numerical results demonstrate perfect reconstruction of the DPT and the synchronization for translation, rotation, and scaling, which justify the DPT design and synchronization mechanisms.