  Sub-pixel registration is a crucial step for applications such as
super-resolution in remote sensing, motion compensation in magnetic resonance
imaging, and non-destructive testing in manufacturing, to name a few. Recently,
these technologies have been trending towards wavelet encoded imaging and
sparse/compressive sensing. The former plays a crucial role in reducing imaging
artifacts, while the latter significantly increases the acquisition speed. In
view of these new emerging needs for applications of wavelet encoded imaging,
we propose a sub-pixel registration method that can achieve direct wavelet
domain registration from a sparse set of coefficients. We make the following
contributions: (i) We devise a method of decoupling scale, rotation, and
translation parameters in the Haar wavelet domain, (ii) We derive explicit
mathematical expressions that define in-band sub-pixel registration in terms of
wavelet coefficients, (iii) Using the derived expressions, we propose an
approach to achieve in-band subpixel registration, avoiding back and forth
transformations. (iv) Our solution remains highly accurate even when a sparse
set of coefficients are used, which is due to localization of signals in a
sparse set of wavelet coefficients. We demonstrate the accuracy of our method,
and show that it outperforms the state-of-the-art on simulated and real data,
even when the data is sparse.
