基于SVD方法设计几何变形可控的金字塔变换

Design Deformable Pyramid Transform via SVD Approach

摘要: 本文设计了一种具有平移不变性、方向和尺度联合可控特性的金字塔变换，称为几何变形可控金字塔变换(DPT)。此DPT从一种数值形式表示的方向可控金字塔变换(SPT)发展而来。我们以SPT的每一个方向可控基滤波器作为核函数，并通过奇异值分解(SVD)设计针对该方向的尺度可控基滤波器和插值函数。我们以此尺度可控基滤波器作为DPT的分析滤波器，进而在理想重构约束下通过理论推导得到DPT的综合滤波器。另外，我们还通过理论推导获得了实现方向可控特性的插值函数，并通过定量表达式分析了采用不同数量的尺度可控基滤波器时对DPT重构性能的影响。数值仿真表明，本文设计的尺度可控基滤波器能满足理想重构约束，且能在误差不超过1 dB的情况下以一半数量的尺度可控基滤波器逼近最优重构性能。

Abstract: This paper presents a deformable pyramid transform (DPT) with shift-invariance, steerability, and scalability. This DPT is extended from a numerical steerable pyramid transform (SPT). We take each steerable basis filter of the SPT as the kernel. For each kernel, we employ the singular value decomposition (SVD) approach to construct scalable basis filters and their corresponding interpolation functions. These scalable basis filters are used as analysis filters of the DPT. Its synthesis filters are then theoretically derived under the constraint of perfect reconstruction for analysis and synthesis filters. In addition, we theoretically derive the interpolation function for steerability, and quantitatively analyze the relationship between the number of scalable basis filters and the reconstruction performance. Numerical simulations demonstrate that the proposed scalable basis filters satisfy the constraint of perfect reconstruction. Also, it is observed that merely using half of the number of scalable basis filters can approximate the optimum reconstruction performance at a cost of reconstruction error within 1 dB.