Multiresolution Techniques

Multiresolution Techniques

Multiresolution Techniques span an exceptionally broad range of algorithms, models, methods, and concepts.  Central to the multiresolution approach is to somehow express short-range, mid-range, and long-range relationships explicitly.  The main reasons for a multiresolution approach is one of

  • Improving performance, by capturing long-range phenomena that would otherwise not be utilized
  • Reducing computational complexity, by allowing algorithms to work on both fine and coarse scales, rather than waiting for local pixel-level operations to converge at large scales
  • Improving numerical robustness (reducing problem conditioning), whereby a multiresolution transformation is essentially an algebraic pre-conditioner
  • Simplifying the algorithm, by making accessible long-range features that might, in some problems, be much easier to work with than pixel-level features
  • Improving intuition, by modeling or analyzing the problem over multiple scales, getting deeper insights into the phenomenon at hand.

Although there are, for sure, many mutiresolution approaches and algorithms which have been proposed, broadly these fall into a few groups:

Wavelet Methods

Problems in which a wavelet transform is used to decompose an image or video into multiple scales, very commonly for image/video denoising, or for feeding the coefficients at multiple scales into a classifier for image classification and segmentation. 

Hierarchical Models

A model in which a pixellated, finest-scale random field is explicitly represented using a set of random fields over scales.  In many cases the multi-scale model may be simpler, using principles of Markov decomposition to decouple the problem into pieces.  A multi-scale model allows different models to be asserted at different scales, usually simpler or more meaningful than having a single-scale model which needs to assert all of the various sccale-dependent behaviours simultaneously.

Hierarchical Algorithms

Even if there is no explicitly hierarchical model, it is possible for the processing algorithm to be hierarchical.  Best known examples include multigrid methods, whereby a single-scale linear system is solved by casting the problem onto a hierarchy, and wavelet methods in image processing, whereby the image is transferred into a set of multiscale coefficients in the wavelet domain, in which certain operations (like image compression or image denoising) are relatively simple.