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

[Ansley Manke <ansley.b.manke@xxxxxxxx>]

Hi Ryo,
Yes, good suggestion. We should be able to write a set of new functions 
that would parallel the smoothing transforms

@SBX        box smoothed      
@SBN        binomial smoothed
@SWL        Welch smoothed
@SHN        Hanning smoothed
@SPZ        Parzen smoothed


but with the edges of the "windows" filled in so that these smoothers 
use filled in so that data isn't lost at the edges.

Ansley

Ryo Furue wrote:

> 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.
>