## Shiftable Multi-scale Transforms

### Eero P. Simoncelli, William T. Freeman, Edward H. Adelson, and David J. Heeger

Published in
IEEE Trans. Information Theory, Special Issue on Wavelets,
vol. 38, no. 2, pp. 587-607 (1992).

Orthogonal wavelet transforms have recently become a popular
representation for multi-scale signal and image analysis. One of the
major drawbacks of these representations is their lack of translation
invariance: the content of wavelet subbands is unstable under
translations of the input signal. Wavelet transforms are also
unstable with respect to dilations of the input signal, and in two
dimensions, rotations of the input signal. We formalize these
problems by defining a type of translation invariance that we call
"shiftability". In the spatial domain, shiftability corresponds to
a lack of aliasing; thus, the conditions under which the property
holds are specified by the sampling theorem. Shiftability may also be
considered in the context of other domains, particularly orientation
and scale. We explore ``jointly shiftable'' transforms that are
simultaneously shiftable in more than one domain. Two examples of
jointly shiftable transforms are designed and implemented: a
one-dimensional transform that is jointly shiftable in position and
scale, and a two-dimensional transform that is jointly shiftable in
position and orientation. We demonstrate the usefulness of these
image representations for scale-space analysis, stereo disparity
measurement, and image enhancement.