[ferret_users] boundary condition for smoothing (@SPZ and friends)

[Ryo Furue <furue@xxxxxxxxxx>]

Hello Ferret users,

I noticed that smoothing transformations (@SPZ and friends) leave
endpoints undefined. For example,

  yes? let pi = 4 * atan(1)
  yes? let func sin(pi*x[x=-1:1:0.2])
  yes? plot/line/symbol/hlimits=-1.2:1.2    func
  yes? plot/line/symbol/hlimits=-1.2:1.2/ov func[x=@spz]

Even though func has datapoints at the edges (x = -1 and x = 1),
func[x=@spz] has missing values there.  I'm wondering if
those transformations can be "improved", at least in simple cases.

Let me take @SPZ as an example.  @SPZ is a 1-2-1 weighted moving
average:

    g(i) = [f(i-1) + 2 f(i) + f(i+1)] / 4

where f is defined for i = 1, 2, . . ., N.  What to do with g(1)
and g(N), for which f(0) and f(N+1) would be required?

One solution is to define g(1) == f(1) and g(N) == f(N).
But, I think a better solution is to make the smoothing behave
like diffusion with a no-flux boundary condition(*).  That is,
we define extra gridpoints at i = 0 and i = N + 1 and
define the values of f(i) there as f(0) == f(1) and f(N+1) == f(N).
And then we compute g(1) and g(N) using the formula above:

    g(1) = [f(0) + 2 f(1) + f(2)] / 4
         = [f(1) + 2 f(1) + f(2)] / 4
         = [       3 f(1) + f(2)] / 4

and likewise for g(N).

We could apply the same no-flux condition around "interior"
missing datapoints, too.

Advantages of this approach are that averages are conserved:
g[i=1:N@sum] == f[i=1:N@sum]  (This property corresponds to
the conservation of heat in the diffusion equation with no-flux
boundary condition.), and that it matches our intuitive
understanding of "mixing" neighboring values.

I haven't given a serious thought to other lowpass filters; the above
consideration may or may not apply to them.

A practical reason for proposing this is that I don't like losing
vectors or shading near the boundaries when smoothing is applied.

Regards,
Ryo
======================================
(*)In fact, the 1-2-1 moving average can be cast into the diffusion
equation:

g(i) = [f(i-1) + 2 f(i) + f(i+1)] / 4
<-->
g(i) - f(i) = [f(i-1) - 2 f(i) + f(i+1)] / 4
<-->
[g(i) - f(i)] / delta_t = kappa [f(i-1) - 2 f(i) + f(i+1)] / (delta_x)^2

with a suitable choice of delta_t, kappa, and delta_x.
This is a finite-difference form of the diffusion equation
with g(i) being the value of the next timestep.

I think the 1-4-6-4-1 smoothing can be likewise cast into
a diffusion equation with biharmonic diffusion.