Spatial aliasing and scale invariance

MULTISCALE, SELF-SIMILAR FITTING

Large objects often resemble small objects. To express this idea we use axis scaling and we apply it to the basic theory of prediction-error filter (PEF) fitting and missing-data estimation.

Equations (3) and (4) compute the same thing by two different methods, $\bold r = \bold Y \bold a$ and $\bold r = \bold A \bold y$ . When it is viewed as fitting goals minimizing $\vert\vert\bold r\vert\vert$ and used along with suitable constraints, (3) leads to finding filters and spectra, while (4) leads to finding missing data.

$$\left[ \begin{array}{c} r_1 \\ r_2 \\ r_3 \\ r_4 \\ r_5 \\ \hline... ...\ \left[ \begin{array}{c} \bold Y_1 \\ \bold Y_2 \end{array} \right] \; \bold a$$ (3)

$$\left[ \begin{array}{c} r_1 \\ r_2 \\ r_3 \\ r_4 \\ r_5 \\ \hline... ...\ \left[ \begin{array}{c} \bold A_1 \\ \bold A_2 \end{array} \right] \; \bold y$$ (4)

A new concept embedded in (3) and (4) is that one filter can be applicable for different stretchings of the filter's time axis. One wonders, ``Of all classes of filters, what subset remains appropriate for stretchings of the axes?''

Spatial aliasing and scale invariance