The definitions behind the concepts of multiresolution or multiscale may overlap somehow, and are sometimes used interchangeably.
Let me provide the following distinction: resolution encompasses spatial discretization, while scale relates to a more continuous framework.
The real world can be considered as continuous, and a true image $I(x,y)$ would have continuous spatial locations $(x,y)$. The concept of scale can then be related to different ideas. Two of them are:
- sensors (your eyes included) have limitations, allowing them to only seize real images at a certain scale, i.e. through some blur operator.
- objects are generally distinct over a limited range of scales: take a standard paper sheet on a wooden desk: from far away, it looks like some indistinct white shape, with a close-up you see fibers. In-between, it can be interpreted as a sheet. The same happens with the desk, but since bigger, its distinctive shape remains interpretable at a distance where the sheet is not anymore.
Having a continuous image seen at different scales may help interpret its content. The notion of scale-space representations is very related to this notion, and it can be axiomatized, see the works of Witkin, Koenderink or Lindeberg, and the Gaussian kernels play an important role. An image at different (Gaussian blurred) scales (hence multiscale) is depicted below:
The wikipedia entry on Multiscale modeling quotes:
The recent surge of multiscale modeling from the smallest scale
(atoms) to full system level (e.g., autos) related to solid mechanics
that has now grown into an international multidisciplinary activity
was birthed from an unlikely source. Since the US Department of Energy
(DOE) national labs started to reduce nuclear underground tests in the
mid 1980s, with the last one in 1992, the idea of simulation-based
design and analysis concepts were birthed. Multiscale modeling was a
key in garnering more precise and accurate predictive tools. In
essence, the number of large scale systems level tests that were
previously used to validate a design was reduced to nothing, thus
warranting the increase in simulation results of the complex systems
for design verification and validation purposes
The concept of resolution discretizes the space: an object with a certain width can be represented by a certain sampling of the physical space, in how many pixels per width. Sampling may induces loss from the continuous space, but if the continuous image is smoothed by well-chosen kernels, then it can be (re)sampled at lower rates. From a continuous or a originally discrete image, one can build a hierarchy of discretized versions at different resolutions (and thus size), and generally those can be used to recreate images at scales (and orientation) different from that of the original one. Multiresolution analysis can be axiomatized as well (as nested orthogonal projection subspace), and discrete wavelets are an avatar of those. A Gaussian pyramid is shown below, corresponding to different samplings of the above blurred images (hence multiresolution):
So multiresolution can be seen as a discretization (invertible or not) of some multiscale representation. In practice however, since the data we are dealing with is always discrete, and since multiresolution can recreate a image at different scales, it happens that people use the term interchangeably.
For images, a fully referenced version of that story (for images, in 2D) is to be found at the beginning of A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity (Signal Processing, 2011).
Recently, the relationship between some wavelets schemes and deep learning has been made more explicit by S. Mallat, with scattering networks.
Additional literature:
- Multiresolution analysis on the symmetric group, 2012, R. Kondor
- Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space, 2010, T. Lindeberg
- On the Axioms of Scale Space Theory, 2004, R. Duits
- Gaussian Scale Space, 2002, A. Kuijper
- Discrete Scale-Space Theory and the Scale-Space, 1991, T. Lindeberg
