# Free-knot spline approximation (BSFK) problem

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@BrunoLuong

My data acquisition system produce periodically 1-D measured noised data with the fixed time window length W. I want to produce smoothed data for each window W separately, with specific constraints on continuous (k=2) or smooth (k = 3 or 4) processed signal connections between consecutive time measurement windows. So, for each window W I get finally separate "pp" structure. How to set proper BSFK options setting to fulfil these constraints?

The second question is: Is there any method how to merge separate "pp" structures to one "pp" structure for several processing windows at one?

Add note: May by some processing windows overlap could be required. Do you have any experiance with using BSFK in streaming regime?

##### 13 Comments

Bruno Luong
on 23 Sep 2022

Edited: Bruno Luong
on 23 Sep 2022

### Accepted Answer

Bruno Luong
on 23 Sep 2022

Edited: Bruno Luong
on 12 Jan 2023

Reference to BSFK function https://fr.mathworks.com/matlabcentral/fileexchange/25872-free-knot-spline-approximation

Here is the recursive pointwise constraint. You'll see it does the job (zoom in) the transition is not nice

data=load('result_4_8.mat')

data=data.result;

[m,n] = size(data);

x = cellfun(@(data) data.x, data, 'unif', 0);

y = cellfun(@(data) data.y, data, 'unif', 0);

j = 1; % n

close all

figure

hold on

for i = 1:m

% Normalize data so that dy/dx is comparable to y

xij = x{i,j}/10000;

yij = y{i,j}/10000;

options = struct('lambda', 1e-8);

if i >= 2

xleft = pp.breaks(end);

yleft = ppval(pp,xleft);

ydleft = ppval(ppder(pp),xleft);

xij = [xleft; xij];

yij = [yleft; yij];

pntcon = struct('p', {0 1}, 'x', {xleft,xleft}, 'v', {yleft ydleft});

options.pntcon = pntcon;

end

pp =BSFK(xij, yij, 4, [], [], options);

xi = linspace(min(xij),max(xij),1000);

yi = ppval(pp, xi);

plot(xij, yij,'c.');

plot(xi, yi, 'r', 'Linewidth', 2);

drawnow

end

function ppd = ppder(pp)

ppd = pp;

coefs = ppd.coefs;

n = size(coefs,2);

ppd.coefs = coefs(:,1:n-1).*(n-1:-1:1);

ppd.order = ppd.order-1;

end

##### 5 Comments

Bruno Luong
on 29 Sep 2022

I haven't not studied the complexity of BSFK.

But it seems to me the linear dependency to number of knots is not quite true, I would say it is more like quadratic.

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