# Wavelet transform of a spatial convolution

Does anyone know if there exist a kind of convolution theorem for the discrete wavelet transform (decimated or undecimated)?

In other words can I find a simple form of $$W\left[ \int f(t) g(x-t) \, dt\right]$$ where $$W$$ is the discrete wavelet transform operator?

## 1 Answer

I cannot say I have a clear understanding of this at this time. However, a few pointers. I'd love to see somebody provide a detailed account. Others bits at: Multiplication in the wavelet domain, what does it look like in real space?

• The nonexistence of a wavelet function admitting a wavelet convoluton theorem of the Fourier type, 1994, A. R. Lindsey, unpublished report

Convolution is one of the most widely used digital signal processing operations. It can be implemented using the fast Fourier transform (FFT), with a computationalcomplexity of $$O(N \log N)$$. The undecimated discrete wavelet transform (UDWT) is linear and shift invariant, so it can also be used to implement convolution. In this paper, we propose a scheme to implement the convolution using the UDWT, and study its advantages and limitations.

This correspondence explores the existence of convolution theorem for linear transformations under a variety of different assumptions. There are eight convolution theorems, all Fourier-related with only N operations in the transform domain and no ordering constraints on the convolution components in the result. They include circular convolutions and correlations.

We study the application of the continuous wavelet transform to perform signal filtering processes. We first show that the convolution and correlation of two wavelet functions satisfy the required admissibility and regularity conditions. By using these new wavelet functions to analyze both convolutions and correlations, respectively, we derive convolution and correlation theorems for the continuous wavelet transform and show them to be similar to that of other joint spatial/spatial–frequency or time/frequency representations. We then investigate the effect of multiplying the continuous wavelet transform of a given signal by a related transfer function and show how to perform spatially variant filtering operations in the wavelet domain. Finally, we present numerical examples showing the usefulness of applying the convolution theorem for the continuous wavelet transform to perform signal restoration in the presence of additive noise.

This paper shows that members of the fourier transform family are the only linear transforms that have a convolution theorem, that is, that can replace $$O(N^2)$$ operations of a convolution in a time domain by $$O(N)$$ operations in a transform domain. Generally, there is an additional cost to compute the transform itself. Our observation is motivated by recent activity in wavelet and subband decompositions and related spectral analyses, which are attractive alternatives for signal compression applications. A natural question when using such techniques is to determine if convolutions of $$N$$-point signals can be calculated with fewer operations in a compressed transform domain than in an uncompressed time domain. The answer is negative for a broad set of assumptions. This paper indicates what assumptions must be relaxed in seeking a linear transform that has a convolution theorem comparable to the convolution theorem for Fourier transforms.