1D Function into 2D Function Interpolation
in [Matlab]
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From: Bjorn Gustavsson on 29 Mar 2010 10:06 "Royi Avital" <RoyiREMOVEAvital(a)yahoo.com> wrote in message <hoqb55$fd8$1(a)fred.mathworks.com>... > "Matt J " <mattjacREMOVE(a)THISieee.spam> wrote in message <hop5s0$a8j$1(a)fred.mathworks.com>... > > "Royi Avital" <RoyiREMOVEAvital(a)yahoo.com> wrote in message <hoot0q$4bp$1(a)fred.mathworks.com>... > > > Hello. > > > I have a 1D function which I want to interpolate into 2D function. > > > I know the function should have "Polar Symmetry". > > > Hence I use the following code: > > > Assuming the 1D function is LSF of the length 15. > > > > > > [x, y] = meshgrid([-7:7]); > > > r = sqrt(x.^2 + y.^2); > > > PSF = interp1([-7:7], LSF, r(:)); % Sometimes using 'spline' option, same results. > > > > > > PSF = reshape(PSF, [7, 7]); > > > > > > I have few problems: > > > 1. Got some overshoot at the edges. > > > 2. Can't enforce some assumptions (Monotonic, Non Negative). > > ================ > > > > Linear interpolation should satisfy all these properties. > > No it doesn't. > I'll show you. > > Let's create a 2d Gaussian Kernel: > h = fspecial('gauss', 15, 3); > > Now let's take a 1D profile of it: > LSF = h(8, :); > > Let's apply the linear algorithms: > > [x, y] = meshgrid([-7:7]); > r = sqrt(x.^2 + y.^2); > PSF = interp1([-7:7], LSF, r(:)); > PSF = reshape(PSF, [7, 7]); > > The results (Added "Spline" and 'Pchip"): > http://i43.tinypic.com/auxyxi.png > > As you can see at the edges there's "Overshoot" (Especially in "Spline"). The linear method gets negative values and changes the form. > > Thanks. Yes, because your PSF is defined for r = [-7:7] and you use interp1 on it for radii up to 7*2^0.5 -> you extrapolate. If you take this into account in some cunning way you should be fine. HTH, Bjoern
From: Matt J on 29 Mar 2010 10:29 "Royi Avital" <RoyiREMOVEAvital(a)yahoo.com> wrote in message <hoqb55$fd8$1(a)fred.mathworks.com>... > The results (Added "Spline" and 'Pchip"): > http://i43.tinypic.com/auxyxi.png > > As you can see at the edges there's "Overshoot" (Especially in "Spline"). The linear method gets negative values and changes the form. ================== You might want to look at my interpMatrix tool, which allows you to specify some different extrapolation rules, as well as to choose whatever interpolation kernels you like http://www.mathworks.com/matlabcentral/fileexchange/26292-regular-control-point-interpolation-matrix-with-boundary-conditions It has some examples showing how to interpolate using cubic B-Splines, but which avoid overshoot by applying mirror boundary conditions. I don't really understand what you mean by "monotonic". Since your Gaussian is a non-monotonic function, you are not looking for a monotonic curve, nor would you want interpolated values between points to be montonic. You mean you want extrapolation at the edges to run montonically down to zero?
From: Royi Avital on 29 Mar 2010 11:59 "Matt J " <mattjacREMOVE(a)THISieee.spam> wrote in message <hoqdfi$sjt$1(a)fred.mathworks.com>... > "Royi Avital" <RoyiREMOVEAvital(a)yahoo.com> wrote in message <hoqb55$fd8$1(a)fred.mathworks.com>... > > > The results (Added "Spline" and 'Pchip"): > > http://i43.tinypic.com/auxyxi.png > > > > As you can see at the edges there's "Overshoot" (Especially in "Spline"). The linear method gets negative values and changes the form. > ================== > > You might want to look at my interpMatrix tool, which allows you to specify some different extrapolation rules, as well as to choose whatever interpolation kernels you like > > http://www.mathworks.com/matlabcentral/fileexchange/26292-regular-control-point-interpolation-matrix-with-boundary-conditions > > It has some examples showing how to interpolate using cubic B-Splines, but which avoid overshoot by applying mirror boundary conditions. > > I don't really understand what you mean by "monotonic". Since your Gaussian is a non-monotonic function, you are not looking for a monotonic curve, nor would you want interpolated values between points to be montonic. > > You mean you want extrapolation at the edges to run montonically down to zero? Yep, This is exactly what I want. To enforce some assumptions on the Extrapolation, it must be Non Negative and Monotonic. Any solution for that? Thanks.
From: Matt J on 30 Mar 2010 16:58 "Royi Avital" <RoyiREMOVEAvital(a)yahoo.com> wrote in message <hoqioe$31d$1(a)fred.mathworks.com>... > Yep, This is exactly what I want. > To enforce some assumptions on the Extrapolation, it must be Non Negative and Monotonic. > Any solution for that? ================= And why, as suggested in an earlier post, don't you just generate LSF data on a wider interval? Then you wouldn't have to extrapolate at all.
From: Royi Avital on 31 Mar 2010 05:05
"Matt J " <mattjacREMOVE(a)THISieee.spam> wrote in message <hotole$cv5$1(a)fred.mathworks.com>... > "Royi Avital" <RoyiREMOVEAvital(a)yahoo.com> wrote in message <hoqioe$31d$1(a)fred.mathworks.com>... > > > Yep, This is exactly what I want. > > To enforce some assumptions on the Extrapolation, it must be Non Negative and Monotonic. > > Any solution for that? > ================= > > And why, as suggested in an earlier post, don't you just generate LSF data on a wider interval? Then you wouldn't have to extrapolate at all. I don't understand what you mean. If the size of the LSF support is 1X15 I need it on 2D domain with support of 15X15. I can assume Monotonic Decreasing and Non Negative and needless to say, Rotational Invariance. How should I do it? |