Underlying idea

All the preceding methods are a perfect plea for the use of a fundamental tool of multiscale filtering: wavelet analysis. We will show how the use of wavelet can speed up the computation of an optic flow field.

Given a family of wavelets built as dilates and translates of a set of mother wavelets :

where k=(k₁, k₂) is a translation index, j is a scale index and x=(x₁, x₂) the position vector.

The principle of the method is to do an inner product of the optic flow equation with S different vectors for a given scale j and position k, to get S equations:

Assuming that the optic flow field is constant over the support of our wavelets, we have:

and after an integration by parts:

All coefficients that appear in these S equations can be computed with a fast wavelet transform, under some provisions on the wavelet design. This is where we have a computational speedup.

Technical difficulties

As can be read in the standard literature on wavelets, the above coefficients can be computed with a fast wavelet transform only in the case where our wavelet are scale separable, that is if each mother wavelet has a Fourier transform that fulfills:

where the m_j are trigonometric polynomials, and are up to a small subset of them all the same functions.

For stability purposes, the wavelets we use are analytic, ie their spectrum has a single maximum. The computation of such wavelet coefficients relies on an adapted approximation of the Hilbert transform with scale separable filters. For more technical details, please refer to the lab report or go there.

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