# Energy-preserving Laplacian Pyramid

Both Discrete Wavelet Transform (DWT) and Undecimated DWT possess an important property of energy preservation: on each level

$$\sum_i W_i^2 + \sum_j V_j^2 = \sum_k X_k^2$$

where $W$ and $V$ are detail and approximation coefficients, respectively.

It turns out that Laplacian Pyramid (LP) generally lacks this (important) property. Is it possible to construct a filter for LP decomposition that would preserve energy? I'm ready to sacrifice the simplicity of backward transform.

• I believe my question can be formulated also as: Does energy preservation require the orthogonality of filter basis (as in DWT)? – Andrey Paramonov Dec 19 '16 at 20:45