Multiscale Analysis vs Multiresolution Analysis: MSA or MRA?

The two names ``multiscale analysis'' (=MSA) and ``multiresolution analysis''(=MRA) refer to the same concept. Both are characterized by the discrete set of powers of the number 2,

$$2^k,\quad k=0,\pm1,\pm2,\cdots~,$$

and the corresponding set of orthonormal basis functions

$$\{ \sqrt{2^{-k}}\,\phi(2^{-k}t-\ell):~\ell=0,\pm 1,\cdots \},~k=0,\pm1,\pm2,\cdots~.$$

The difference is that MRA and MSA highlight different aspects of the same thing. As $k$ increases, the scale increases but the resolution decreases. This is like stepping away from a picture.

Consider the array of functions

$$\phi(2^{-k}t-\ell)=\phi\left( \frac{t-2^k\ell}{2^k}\right),\quad\ell=\cdots, -2,-1,0,1,2,\cdots~.$$

This array is a set of identical localized graphs, each one displaced by the amount $2^k$ from its nearest neighbor. Thus each of these graphs serves as marker on the real line and $2^k$ is the distance between successive makers. In brief, the real line equipped with this set of markers constitutes a new kind of measuring rod for measuring signals. The integer $k$ specifies the nature - the resolution - of this measuring rod. Every integral increase in $k$ increases (decreases) the scale for performing these mesurements, and hence decreases (increases) the resolution of the measuring rod. Figure 2.24 depicts several such measuring rods.

Figure 2.24: Nine different measuring rods. Each is graduated with its own set of markers, i.e. shifted scaling functions $\phi (2^{-k}(t-m))$ , ranging from a set of very high resolution ( $2^{9}:k=-9$ ) markers, through a set of medium resolution ( $2^{5}:k=-5$ ), to the set of lowest resolution ( $2^{1}:k=-1$ ) markers. A high resolution measuring rod accomodates additional high resolution markers, which are, however, not depicted in this figure. The markers of each rod are uniformly spaced, as they must. The novelty of these rods is that each marker has the mathematically precise internal structure of a wavepacket. This novelty permits one to measure not only the locations of specific features in a given signal but also their amplitudes.