# Basic difference between Multi-scale and Multi-resolution images

While working on Convolutional Neural Network (CNN) I saw many research articles working on the Multi-scale images. In the literature of image processing I saw there is also Multi-resolution analysis of image which is at different level of images. Whereas in Multi-scale it seems that size of images are not changed instead it gets more blur and edges are getting more clear of the objects.

I am not using any wavelet transform. So in terms of general images how can we explain to beginners about the basic difference between Multi-scale and Multi-resolution images?

In scale space the square of width of blurring kernel considered as the scale (variance of Gaussian and not its Standard deviation because if you sequentially blur an image with 2 Gaussian kernel variance of effective blurring kernel will be sum of variance of each kernel) but in multi-resolution analysis the resolution defined by the scaling parameter.

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.