1.3 Content-based image querying with complex wavelets: the complex

Because of its desirable multiresolution properties, the two-dimensional wavelet transform happens to be highly applicable to many areas, very notably to the field of image processing. However, its lack of shift-invariance tends to be a major inconvenience, and a transform that provides multiresolution as well as shift-invariance would be highly useful almost everywhere wavelets are used. Complex wavelets are an answer to this problem, and a solid mathematical foundation that allowed practical use of complex wavelets in image processing was originally set up in 1997 by Nick Kingsbury of Cambridge University.

The complex two-dimensional wavelet transform provides all of the advantages that the separable discrete wavelet transform provides – multiresolution, sparse representation, and useful characterization of the structure of an image. What makes the complex wavelet basis exceptionally useful for our purposes is that it provides a high degree of shift-invariance in its magnitude. A drawback to this transform is that it is four-times redundant. That is, if you have an original N x N image, and take the DWT, you get back N x N numbers, whereas using the CDWT, you get back 4 N x N numbers. So, for the price of four-times redundancy, you get a high degree of shift-invariance in magnitude – which seems like a reasonable tradeoff for applications that need a shift-invariant, multiresolution transform.