# 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

The bad news is that I disagree with the first assertion: there exist non-orthogonal DWT (biorthogonal for instance).

But the good news is that this is not important here, and the answer is yes. A Laplacian pyramid is an overcomplete expansion. It can not be treated as a basis, but as a frame. And the concept akin to orthogonal bases in frames is the tight frame.

You can get details on such constructions in Framing Pyramids (2003), whose abstract is:

In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous-domain frames associated with the LP.

And yes, the pseudo-inverse can be less simple, but you will gain more optimality in the presence of noise.

Notably, such design allowed the construction of contourlets, a multiresolution directional tight frame designed to efficiently approximate images made of smooth regions separated by smooth boundaries.