Reconstruction and Approximation - Vision with Direction - page 105

is called synthesis or reconstruction , and is illustrated by Fig. 8.6 for tent functions.
Since the sum of two linear functions is another linear function, and because the in-
terpolation functions are exact on the grid (integers of the x -axis, in the figure), the
approximation using Eq. (8.12) results in a piecewise linear function. Continuous
by construction, the approximating function is delivered by the weighted tent sum-
mation, which “automatically” joins the values of the original function on a set of
discrete points. Thus, the approximation is error free if the function to be approxi-
mated is piecewise linear. If the constant “1” is to be approximated, then the entire
set of interpolation functions is summed and the result is error free as shown in the
figure. However, the linear interpolator suffers from the following drawbacks:
i) Extending them to 2D via μ ( x ) μ ( y ) will make them anisotropic because this 2D
function has isocurves that are not circles (but biased by squares). This means
that certain directions of edges will be artificially favored over the others.
ii) It is not differentiable everywhere. As a result we cannot construct all partial
derivatives easily everywhere. We discuss the partial derivative operator in the
next section.
Another function family, called B-splines , can be obtained by successive convolution
of the linear interpolator by itself. The result is a piecewise polynomial interpolator
that is smooth. Because they do not suffer from the second point above, B-splines are
often preferred over the linear interpolator [123]. They have found many applications
because of their speed thanks to their separability in x and y . However, they too suffer
from being anisotropic.
A nonpolynomial interpolator, that we will study further in Sect. 9.2 is the Gaus-
sian interpolator, which can be shown to be the asymptotic limit of B-splines [221].
In Fig. 8.7 we show a set of shifted Gaussians that are used to illustrate a synthesis
process. A Gaussian is isotropic when extended to 2D via μ ( x ) μ ( y ), and it is in-
finitely differentiable, everywhere. In the next section, it suffices to know that there
exists an interpolation function that has a localized support (it decreases rapidly to
zero), and that the choice is not restricted to the sinc functions.

8.2 Sampling Band-Preserving Linear Operators

We will be concerned here with continuous operators acting on vector spaces, in
particular, function spaces. Such operators act on functions and deliver elements that
stay in the same function space, as a result. The continuity of an operator means that
a small change of its argument (a function) results in a small change of its result (also
a function). Derivation, convolving with a particular filter, and taking the square root
of a function are examples of operators. We start by defining linear operators.
Definition 8.2. An operator T is a linear operator if it satisfies

T ( f + g )= T ( f )+ T ( g ) (8.13)

T ( λf )= λ T ( f ) (8.14)

where λ is a scalar.