Fitting a spline: find coefficients using Fourier Transform?
I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way.
Given that
$$s(x)=sum_kc(k)beta^n(x-k)$$ where $k$ is an integer and $beta^n(x)$ is defined as a ($n+1$) times convolution :$$beta_0*beta_0*...*beta_0(x)$$
where $beta_0$ is a rectangular pulse of length 1 centred at zero.
One could hence write $s(x)=(c*beta^n)(x)$.
Applying the Fourier Transform we obtain.
$s(omega)=c(omega)cdot beta^n(omega)=c(omega)cdot beta_0(omega)^{n+1}$
From here I would like to obtain $c(omega)$ to then apply the inverse FFT.
I've seen some papers dealing with a similar methodology but without explaining in detail what are the condition in order for this to work.
Any suggestion? Thanks in advance
fourier-transform regression convolution spline
asked Nov 26 at 12:16
Ramiro Scorolli
655113
add a comment |
I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way.
Given that
$$s(x)=sum_kc(k)beta^n(x-k)$$ where $k$ is an integer and $beta^n(x)$ is defined as a ($n+1$) times convolution :$$beta_0*beta_0*...*beta_0(x)$$
where $beta_0$ is a rectangular pulse of length 1 centred at zero.
One could hence write $s(x)=(c*beta^n)(x)$.
Applying the Fourier Transform we obtain.
$s(omega)=c(omega)cdot beta^n(omega)=c(omega)cdot beta_0(omega)^{n+1}$
From here I would like to obtain $c(omega)$ to then apply the inverse FFT.
I've seen some papers dealing with a similar methodology but without explaining in detail what are the condition in order for this to work.
Any suggestion? Thanks in advance
fourier-transform regression convolution spline
asked Nov 26 at 12:16
Ramiro Scorolli
655113
add a comment |
I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way.
Given that
$$s(x)=sum_kc(k)beta^n(x-k)$$ where $k$ is an integer and $beta^n(x)$ is defined as a ($n+1$) times convolution :$$beta_0*beta_0*...*beta_0(x)$$
where $beta_0$ is a rectangular pulse of length 1 centred at zero.
One could hence write $s(x)=(c*beta^n)(x)$.
Applying the Fourier Transform we obtain.
$s(omega)=c(omega)cdot beta^n(omega)=c(omega)cdot beta_0(omega)^{n+1}$
From here I would like to obtain $c(omega)$ to then apply the inverse FFT.
I've seen some papers dealing with a similar methodology but without explaining in detail what are the condition in order for this to work.
Any suggestion? Thanks in advance
fourier-transform regression convolution spline
asked Nov 26 at 12:16
Ramiro Scorolli
655113
I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way.
Given that
$$s(x)=sum_kc(k)beta^n(x-k)$$ where $k$ is an integer and $beta^n(x)$ is defined as a ($n+1$) times convolution :$$beta_0*beta_0*...*beta_0(x)$$
where $beta_0$ is a rectangular pulse of length 1 centred at zero.
One could hence write $s(x)=(c*beta^n)(x)$.
Applying the Fourier Transform we obtain.
$s(omega)=c(omega)cdot beta^n(omega)=c(omega)cdot beta_0(omega)^{n+1}$
From here I would like to obtain $c(omega)$ to then apply the inverse FFT.
I've seen some papers dealing with a similar methodology but without explaining in detail what are the condition in order for this to work.
Any suggestion? Thanks in advance
fourier-transform regression convolution spline
fourier-transform regression convolution spline
asked Nov 26 at 12:16
Ramiro Scorolli
655113
asked Nov 26 at 12:16
Ramiro Scorolli
655113
asked Nov 26 at 12:16
Ramiro Scorolli
655113
asked Nov 26 at 12:16
Ramiro Scorolli
655113
asked Nov 26 at 12:16
Ramiro Scorolli
655113
655113
add a comment |
add a comment |
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