In this section, we give a brief overview of another very important application of wavelets in image processing: image registration. Readers interested in this topic are encouraged to read the references listed in the context.
Image registration is required for many image processing applications. In medical imaging, co-registration problems are important for many clinical tasks:
1. multimodalities study,
2. cross-subject normalization and template/atlas analysis,
3. patient monitoring over time with tracking of the pathological evolution for the same patient and the same modality.
Many registration methods follow a feature matching procedure. Feature points (often referred to as "control points," or CP) are first identified in both the reference image and the input image. An optimal spatial transformation (rigid or nonrigid) is then computed that can connect and correlate the two sets of control points with minimal error. Registration has always been considered as very costly in terms of computational load. Besides, when the input image is highly deviated from the reference image, the optimization process can be easily trapped into local minima before reaching the correct transformation mapping. Both issues can be alleviated by embedding the registration into a "coarse to fine" procedure. In this framework, the initial registration is carried out on a relatively low resolution image data, and sequentially refined to higher resolution. Registration at higher resolution is initialized with the result from the lower resolution and only needs to refine the mapping between the two images with local deformations for updating the transformation parameters.
The powerful representation provided by the multiresolution analysis framework with wavelet functions has lead many researchers to use a wavelet expansion for such "coarse to fine" procedures [104-106]. As already discussed previously, the information representation in the wavelet transform domain offers a better characterization of key spatial features and signal variations. In addition to a natural framework for "coarse to fine" procedure, many research works also reported the advantages of using wavelet subbands for feature characterization. For example, in  Zheng et al. constructed a set of feature points from a Gabor wavelet model that represented local curvature discontinuities. They further required that a feature point should have maximum energy among a neighborhood and above a certain threshold. In , Moigne et al. used wavelet coefficients with magnitude above 13-15% of the maximum value to form their feature space. In , Dinov et al. applied a frequency adaptive thresholding (shrinkage) to the wavelet coefficients to keep only significant coefficients in the wavelet transform domain for registration.
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