Content-Based Image Querying with Complex Wavelets: The Complex Discrete Wavelet Transform 1.3 2003/12/11 01:23:06 US/Central 2003/12/16 00:22:15.239 US/Central Tom Mowad tm@rice.edu Venkat Chandrasekaran venkatc@rice.edu Tom Mowad tm@rice.edu Venkat Chandrasekaran venkatc@rice.edu cdwt complex discrete wavelet transform Introduction to the complex discrete wavelet transform and its properties that make it an appropriate transform for the image querying. 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.

A 1-dimensional complex wavelet Figure generated using Ivan Selesnick's code.