This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by incorporating parameters of refinement multiplicity, dispersion, and rotation. We rigorously establish key properties of the transform including completeness, uniqueness, invertibility, closure, and stability using frame bounds over functions of bounded variation and define a natural inner product structure emerging in L2. We identify regions of parameter space that recover known transforms, including multiscale wavelet decompositions and the generalized Radon transform. Applications are discussed across a range of disciplines, with particular emphasis on entropy formulations. Notably, the transform remains well behaved on geometrically complex and even non convex domains, where traditional methods may struggle. Despite the complexity of the underlying geometry, the coefficient spectrum reveals structure, offering insight even in highly irregular settings.

翻译：本文引入了一类新颖且通用的连续变换，其基于分层Voronoi细化方案。所得变换空间通过引入细化多重性、分散性和旋转等参数，推广了诸如小波和Radon变换等经典方法。我们严格建立了该变换的关键性质，包括完备性、唯一性、可逆性、封闭性以及基于有界变差函数框架界的稳定性，并定义了L2空间中自然涌现的内积结构。我们识别了参数空间中可恢复已知变换的区域，包括多尺度小波分解和广义Radon变换。本文讨论了该变换在多个学科中的应用，并特别关注熵表述。值得注意的是，该变换在几何复杂甚至非凸域上仍保持良好性质，而传统方法在此类域上可能面临困难。尽管底层几何结构复杂，系数谱仍能揭示结构特征，为高度不规则场景提供新的见解。