generating and plotting a matrix of bsplines with different positions and width

I am able to create a set of bsplines in a single plot. However, I would like to create a matrix of individual bsplines, whereby each matrix row contains a single bspline from range 1950 to 20150, in the way depicted in the drawing. The bsplines should have the same height with coefficient of 1. The next bspline in the library will be spaced one knot apart from the first bspline and so on until the entire width is taken, depending on the width of the bspline. Then, starting from the left-most again, a bspline of twice the width of the previous bspline is added to the reference set. Then a bspline for the same width is added 1 knot apart from the first and so on… This process will continue until the bspline become so wide that one bspline is able to fit the entire width.

Also, a bonus question: is there any nice functions for plotting all the functions described above in a way similar to the drawing?

B = spcol([0, 0, 0, linspace(0,1,20), 1, 1, 1], 4, linspace(0,1,100)); plot(B);

Answer 1

[Matt J]

See Example1D.m in

https://www.mathworks.com/matlabcentral/fileexchange/26292-regular-control-point-interpolation-matrix-with-boundary-conditions?s_tid=srchtitle

[Tim Dong]

Hello Matt, this sounds promising. This is what I am doing:

cubicBspline = @(t) (t>-1 & t<1).*(2/3 - t.^2 +abs(t).^3/2) +...
                        (abs(t)>=1 & abs(t)<2).*((2-abs(t)).^3/6);
kernel=cubicBspline(-2:1/SampRatio:2 );
  nCtrlPts=length(tCoarse);
  
  %create interpolation system matrices
  BasisFine=interpMatrix(kernel, 'max', nCtrlPts, SampRatio, 'mirror');
  BasisCoarse=BasisFine(1:SampRatio:end,:);

 subplot(1,1,1) 
  hold on;  plot(tFine,sFit,'-b'); hold off
  hold on;  plot(2* tFine + 1,sFit,'-r'); hold off
  hold on;  plot(3* tFine + 2,sFit,'-g'); hold off
  hold on;  plot(4* tFine + 3,sFit,'-p'); hold off

However, it appears that there are wiggly basis that should not be there apart from at the two extremeties. Please advise if and how these should be removed?

Do you have any references for how you derive the cubicBspline equation?

[Matt J]

Your code doesn't show how you create tFine or sFit.

[Tim Dong]

Hello Matt,

This is the entirety of the code, although not all parts are used.

function Example1D

%One dimensional cubic B-Spline fitting example using interpMatrix()

%and comparing different extrapolation schemes.

%%Data

s = @(t) cos(2*pi*t).*exp(-abs(2*t))+ 2; %signal to fit

cubicBspline = @(t) (t>-1 & t<1).*(2/3 - t.^2 +abs(t).^3/2) +...

(abs(t)>=1 & abs(t)<2).*((2-abs(t)).^3/6);

tCoarse=linspace(-1.2, 1.2,9); %Coarse sample locations on t-axis

dtCoarse=tCoarse(2)-tCoarse(1);

tFine=linspace(-1.2, 1.2,81); %Fine sample locations on t-axis

dtFine=tFine(2)-tFine(1);

SampRatio=round(dtCoarse/dtFine); %Sampling ratio

%sample the signal

sCoarse=s(tCoarse(:));

sFine=s(tFine(:));

figure; subplot(1,2,1)

plot(tFine,sFine,'-b', tCoarse,sCoarse,'r*');

legend('True Signal', 'Control Points');

title 'Signal Samples'

%%Engine

kernel=cubicBspline(-2:1/SampRatio:2 );

nCtrlPts=length(tCoarse);

%create interpolation system matrices

BasisFine=interpMatrix(kernel, 'max', nCtrlPts, SampRatio, 'mirror');

BasisCoarse=BasisFine(1:SampRatio:end,:);

%up to here

%%Do the fit!!!

sFit = BasisFine*(BasisCoarse\sCoarse);

subplot(1,2,2)

hold on; plot(tFine,sFine,'-b',tFine,sFit,'--m+'); hold off

%Repeat the above with zero padded extrapolation

BasisFine=interpMatrix(kernel, 'max', length(tCoarse), SampRatio, 'zero');

BasisCoarse=BasisFine(1:SampRatio:end,:);

sFit = BasisFine*(BasisCoarse\sCoarse);

hold on; plot(tFine,sFit,'-.ko'); hold off

legend('True Signal', 'Mirror Extrap.','Zero Extrap.')

title 'Cubic B-Spline Reconstructions'

%TD: multi-resolution bspline plot

subplot(1,1,1)

hold on; plot(tFine,sFit,'-b'); hold off

hold on; plot(2* tFine + 1,sFit,'-r'); hold off

hold on; plot(3* tFine + 2,sFit,'-g'); hold off

hold on; plot(4* tFine + 3,sFit,'-p'); hold off

[Matt J]

The matrix BasisFine should be very close to what you want:

   
    cubicBspline = @(t) (t>-1 & t<1).*(2/3 - t.^2 +abs(t).^3/2) +...
                        (abs(t)>=1 & abs(t)<2).*((2-abs(t)).^3/6);
       tCoarse=linspace(-1.2, 1.2,9);     %Coarse sample locations on t-axis
       dtCoarse=tCoarse(2)-tCoarse(1);
    tFine=linspace(-1.2, 1.2,81);      %Fine sample locations on t-axis
       dtFine=tFine(2)-tFine(1);
    SampRatio=round(dtCoarse/dtFine); %Sampling ratio


%%Engine
  kernel=cubicBspline(-2:1/SampRatio:2 );
  nCtrlPts=length(tCoarse);
  
  %create interpolation system matrices
  BasisFine=interpMatrix(kernel, 'max', nCtrlPts, SampRatio);
  
  plot(BasisFine)

[Tim Dong]

@Matt J, I think I am getting closer, but there is still the issue of the too much hiding of the curve on the right

part of the splines, when I try to double each consecutive spline width?

%TD: multi-resolution bspline plot

subplot(1,1,1)

hold on; plot(1:10, BasisFine(1:10,1),'-b'); hold off

hold on; plot(1:10 , BasisFine(1:10,2),'-r'); hold off

hold on; plot(2 * (1:10), BasisFine(1:10,3),'-g'); hold off

hold on; plot(3 * (1:10), BasisFine(1:10,4),'Color', plotColors(1,:)); hold off

hold on; plot(4 * (1:10) , BasisFine(1:10,5),'Color', plotColors(2,:)); hold off

[Matt J]

You have not shown what you are doing, but you might wish to review the documentation for interpMatrix. If you do not want the basis functions truncated to a particular support, the 'allcontrib' extrapolation option may be what you need.

[Tim Dong]

@Matt J, this is great work. I think I am getting the bsplines I need. However, I would like the height of the bsplines to be set to 1 instead of it currently being ~0.67. My code are shown below as well.

cubicBspline = @(t) (t>-1 & t<1).*(2/3 - t.^2 +abs(t).^3/2) +...

(abs(t)>=1 & abs(t)<2).*((2-abs(t)).^3/6);

tCoarse=linspace(-1.2, 1.2,9); %Coarse sample locations on t-axis

dtCoarse=tCoarse(2)-tCoarse(1);

tFine=linspace(-1.2, 1.2,81); %Fine sample locations on t-axis

dtFine=tFine(2)-tFine(1);

SampRatio=round(dtCoarse/dtFine); %Sampling ratio

%sample the signal

% sCoarse=s(tCoarse(:));

% sFine=s(tFine(:));

%

% figure; subplot(1,2,1)

% plot(tFine,sFine,'-b', tCoarse,sCoarse,'r*');

% legend('True Signal', 'Control Points');

% title 'Signal Samples'

%%Engine

kernel=cubicBspline(-2:1/SampRatio:2 );

nCtrlPts=length(tCoarse);

%create interpolation system matrices

%BasisFine=interpMatrix(kernel, 'max', nCtrlPts, SampRatio, 'mirror');

BasisFine=interpMatrix(kernel, 'max', nCtrlPts, SampRatio);

%BasisCoarse=BasisFine(1:SampRatio:end,:);

plot(BasisFine)

kernel=cubicBspline(-2:1/5:2 );

nCtrlPts=length(tCoarse);

%BasisFine=interpMatrix(kernel, 'max', nCtrlPts, 5);

BasisFine=interpMatrix(kernel, 'max', nCtrlPts, 5, 'allcontrib');

%BasisCoarse=BasisFine(1:SampRatio:end,:);

plot(1:10, BasisFine(1:10,1))

plotColors = jet(20);

%TD: multi-resolution bspline plot

subplot(1,1,1)

hold on; plot(((1:19)./4 ), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +1), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +2), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +3), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +4), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +5), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +6), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +7), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +8), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +9), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +10), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +11), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +12), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +13), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +14), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

hold on; plot(((1:19)./4 +15), BasisFine(2:20,5),'Color', plotColors(2,:)); hold off

[Matt J]

However, I would like the height of the bsplines to be set to 1 instead of it currently being ~0.67

Could you not just multiply the matrix by 3/2