Matlab--二次多项式曲面拟合

二次多项式曲面公式

Matlab--二次多项式曲面拟合

总共有6个系数。

绘制曲面图形时，一般给定x和y的取值（一维数组），然后对x和y网格化成二维数组X和Y，将X和Y代入公式，即可得到曲面的数值，最后用surf函数显示。

实例

给定一个二次多项式模型，然后成图




x = 0.1 : 0.1 : 5;



y = 0.1 : 0.1 : 5;



[X, Y] = meshgrid(x, y);



Z = X.^2 + Y.^2 + X.*Y + X + Y + 1;



surf(x, y, Z);



colormap hot



shading interp

Matlab--二次多项式曲面拟合

添加噪声，随机选择 100 个点用于拟合




rng('default');



Zn = awgn(Z, 30, 'measured');



xfit = randi(50, 100, 1) / 10;



yfit = randi(50, 100, 1) / 10;



zfit = zeros(100, 1);




for i = 1 : 100




    zfit(i) = Zn(xfit(i) * 10, yfit(i) * 10);



end




surf(x, y, Zfit);



colormap hot



shading interp








scatter3(xfit, yfit, zfit, 50, 'MarkerFaceColor', [0 0 0]);

Matlab--二次多项式曲面拟合

用lsqcurvefit函数拟合：拟合结果精度高





func = @(var,x) var(1)*x(:,1).^2 + var(2)*x(:,2).^2 + var(3)*x(:,1).*x(:,2) + var(4)*x(:,1) + var(5)*x(:,2) + var(6);



a = lsqcurvefit(func,ones(1, 6),[xfit, yfit],zfit);




Zfit = a(1) * X.^2 + a(2) * Y.^2 + a(3) * X.*Y + a(4) * X + a(5) * Y + a(6);



ya = func(a,[xfit, yfit]);



surf(x, y, Zfit);



colormap hot



shading interp



hold on



surf(x, y, Zfit);



shading interp



scatter3(xfit, yfit, ya, 50, 'MarkerFaceColor', [0 0 1]);



hold off

Matlab--二次多项式曲面拟合

完整程序




clear;clc;



x = 0.1 : 0.1 : 5;



y = 0.1 : 0.1 : 5;



[X, Y] = meshgrid(x, y);



Z = X.^2 + Y.^2 + X.*Y + X + Y + 1;



rng('default');



Zn = awgn(Z, 30, 'measured');



xfit = randi(50, 100, 1) / 10;



yfit = randi(50, 100, 1) / 10;



zfit = zeros(100, 1);




for i = 1 : 100




    zfit(i) = Zn(xfit(i) * 10, yfit(i) * 10);



end



func = @(var,x) var(1)*x(:,1).^2 + var(2)*x(:,2).^2 + var(3)*x(:,1).*x(:,2) + var(4)*x(:,1) + var(5)*x(:,2) + var(6);



a = lsqcurvefit(func,ones(1, 6),[xfit, yfit],zfit);



Zfit = a(1) * X.^2 + a(2) * Y.^2 + a(3) * X.*Y + a(4) * X + a(5) * Y + a(6);



ya = func(a,[xfit, yfit]);



surf(x, y, Zn);



shading interp



freezeColors



hold on




surf(x, y, Zfit);



colormap hot



shading interp




scatter3(xfit, yfit, zfit, 50, 'MarkerFaceColor', [0 0 0]);



scatter3(xfit, yfit, ya, 50, 'MarkerFaceColor', [0 0 1]);



surf(x, y, Z - Zfit);



shading interp



hold off




legend('Model','Fits','Fit','fit','Error');

Matlab--二次多项式曲面拟合

使用cftool工具拟合

Matlab--二次多项式曲面拟合

附录：freezeColors函数可从https://ww2.mathworks.cn/matlabcentral/fileexchange/7943-freezecolors-unfreezecolors下载




function freezeColors(varargin)



% freezeColors  Lock colors of plot, enabling multiple colormaps per figure. (v2.3)



%




%   Problem: There is only one colormap per figure. This function provides




%       an easy solution when plots using different colomaps are desired 



%       in the same figure.



%



%   freezeColors freezes the colors of graphics objects in the current axis so 



%       that subsequent changes to the colormap (or caxis) will not change the



%       colors of these objects. freezeColors works on any graphics object 




%       with CData in indexed-color mode: surfaces, images, scattergroups, 



%       bargroups, patches, etc. It works by converting CData to true-color rgb



%       based on the colormap active at the time freezeColors is called.



%



%   The original indexed color data is saved, and can be restored using



%       unfreezeColors, making the plot once again subject to the colormap and




%       caxis.



%



%



%   Usage:



%       freezeColors        applies to all objects in current axis (gca),



%       freezeColors(axh)   same, but works on axis axh.



%



%   Example:



%       subplot(2,1,1); imagesc(X); colormap hot; freezeColors



%       subplot(2,1,2); imagesc(Y); colormap hsv; freezeColors etc...



%



%       Note: colorbars must also be frozen. Due to Matlab 'improvements' this can



%				no longer be done with freezeColors. Instead, please



%				use the function CBFREEZE by Carlos Adrian Vargas Aguilera




%				that can be downloaded from the MATLAB File Exchange



%				(http://www.mathworks.com/matlabcentral/fileexchange/24371)



%




%       h=colorbar; cbfreeze(h), or simply cbfreeze(colorbar)




%



%       For additional examples, see test/test_main.m



%




%   Side effect on render mode: freezeColors does not work with the painters



%       renderer, because Matlab doesn't support rgb color data in




%       painters mode. If the current renderer is painters, freezeColors



%       changes it to zbuffer. This may have unexpected effects on other aspects



%	      of your plots.



%



%       See also unfreezeColors, freezeColors_pub.html, cbfreeze.



%



%



%   John Iversen ([email protected]) 3/23/05



%







%   Changes:



%   JRI ([email protected]) 4/19/06   Correctly handles scaled integer cdata



%   JRI 9/1/06   should now handle all objects with cdata: images, surfaces, 



%                scatterplots. (v 2.1)



%   JRI 11/11/06 Preserves NaN colors. Hidden option (v 2.2, not uploaded)



%   JRI 3/17/07  Preserve caxis after freezing--maintains colorbar scale (v 2.3)




%   JRI 4/12/07  Check for painters mode as Matlab doesn't support rgb in it.



%   JRI 4/9/08   Fix preserving caxis for objects within hggroups (e.g. contourf)



%   JRI 4/7/10   Change documentation for colorbars



 



% Hidden option for NaN colors:



%   Missing data are often represented by NaN in the indexed color



%   data, which renders transparently. This transparency will be preserved



%   when freezing colors. If instead you wish such gaps to be filled with 



%   a real color, add 'nancolor',[r g b] to the end of the arguments. E.g. 



%   freezeColors('nancolor',[r g b]) or freezeColors(axh,'nancolor',[r g b]),



%   where [r g b] is a color vector. This works on images & pcolor, but not on




%   surfaces.



%   Thanks to Fabiano Busdraghi and Jody Klymak for the suggestions. Bugfixes 



%   attributed in the code.



 



% Free for all uses, but please retain the following:



%   Original Author:



%   John Iversen, 2005-10




%   [email protected]




 



appdatacode = 'JRI__freezeColorsData';



 



[h, nancolor] = checkArgs(varargin);



 



%gather all children with scaled or indexed CData



cdatah = getCDataHandles(h);



 



%current colormap



cmap = colormap;



nColors = size(cmap,1);



cax = caxis;



 



% convert object color indexes into colormap to true-color data using 



%  current colormap




for hh = cdatah',




    g = get(hh);



    



    %preserve parent axis clim



    parentAx = getParentAxes(hh);




    originalClim = get(parentAx, 'clim');    




   



    %   Note: Special handling of patches: For some reason, setting



    %   cdata on patches created by bar() yields an error,




    %   so instead we'll set facevertexcdata instead for patches.



    if ~strcmp(g.Type,'patch'),



        cdata = g.CData;



    else




        cdata = g.FaceVertexCData; 



    end




    



    %get cdata mapping (most objects (except scattergroup) have it)



    if isfield(g,'CDataMapping'),



        scalemode = g.CDataMapping;



    else




        scalemode = 'scaled';



    end




    



    %save original indexed data for use with unfreezeColors



    siz = size(cdata);



    setappdata(hh, appdatacode, {cdata scalemode});



 



    %convert cdata to indexes into colormap



    if strcmp(scalemode,'scaled'),



        %4/19/06 JRI, Accommodate scaled display of integer cdata:



        %       in MATLAB, uint * double = uint, so must coerce cdata to double



        %       Thanks to O Yamashita for pointing this need out




        idx = ceil( (double(cdata) - cax(1)) / (cax(2)-cax(1)) * nColors);



    else %direct mapping



        idx = cdata;



        %10/8/09 in case direct data is non-int (e.g. image;freezeColors)



        % (Floor mimics how matlab converts data into colormap index.)



        % Thanks to D Armyr for the catch



        idx = floor(idx);



    end




    



    %clamp to [1, nColors]



    idx(idx<1) = 1;



    idx(idx>nColors) = nColors;



 



    %handle nans in idx



    nanmask = isnan(idx);



    idx(nanmask)=1; %temporarily replace w/ a valid colormap index




 



    %make true-color data--using current colormap



    realcolor = zeros(siz);



    for i = 1:3,



        c = cmap(idx,i);



        c = reshape(c,siz);



        c(nanmask) = nancolor(i); %restore Nan (or nancolor if specified)



        realcolor(:,:,i) = c;



    end




    



    %apply new true-color color data



    



    %true-color is not supported in painters renderer, so switch out of that



    if strcmp(get(gcf,'renderer'), 'painters'),



        set(gcf,'renderer','zbuffer');



    end




    



    %replace original CData with true-color data



    if ~strcmp(g.Type,'patch'),



        set(hh,'CData',realcolor);



    else




        set(hh,'faceVertexCData',permute(realcolor,[1 3 2]))



    end




    



    %restore clim (so colorbar will show correct limits)



    if ~isempty(parentAx),



        set(parentAx,'clim',originalClim)



    end




    




end %loop on indexed-color objects



 



 



% ============================================================================ %



% Local functions



 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%% getCDataHandles -- get handles of all descendents with indexed CData



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



 



function hout = getCDataHandles(h)



% getCDataHandles  Find all objects with indexed CData




 



%recursively descend object tree, finding objects with indexed CData




% An exception: don't include children of objects that themselves have CData:




%   for example, scattergroups are non-standard hggroups, with CData. Changing




%   such a group's CData automatically changes the CData of its children, 



%   (as well as the children's handles), so there's no need to act on them.



 



error(nargchk(1,1,nargin,'struct'))



 



hout = [];




if isempty(h),return;end




 



ch = get(h,'children');




for hh = ch'




    g = get(hh);




    if isfield(g,'CData'),     %does object have CData?




        %is it indexed/scaled?



        if ~isempty(g.CData) && isnumeric(g.CData) && size(g.CData,3)==1, 



            hout = [hout; hh]; %#ok<AGROW> %yes, add to list



        end



    else %no CData, see if object has any interesting children



            hout = [hout; getCDataHandles(hh)]; %#ok<AGROW>



    end



end







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%% getParentAxes -- return handle of axes object to which a given object belongs



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



function hAx = getParentAxes(h)



% getParentAxes  Return enclosing axes of a given object (could be self)








error(nargchk(1,1,nargin,'struct'))




%object itself may be an axis




if strcmp(get(h,'type'),'axes'),




    hAx = h;



    return



end








parent = get(h,'parent');





if (strcmp(get(parent,'type'), 'axes')),




    hAx = parent;



else



    hAx = getParentAxes(parent);



end











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%% checkArgs -- Validate input arguments



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%







function [h, nancolor] = checkArgs(args)



% checkArgs  Validate input arguments to freezeColors







nargs = length(args);




error(nargchk(0,3,nargs,'struct'))








%grab handle from first argument if we have an odd number of arguments



if mod(nargs,2),



    h = args{1};



    if ~ishandle(h),




        error('JRI:freezeColors:checkArgs:invalidHandle',...





            'The first argument must be a valid graphics handle (to an axis)')




    end



    % 4/2010 check if object to be frozen is a colorbar




    if strcmp(get(h,'Tag'),'Colorbar'),





      if ~exist('cbfreeze.m'),





        warning('JRI:freezeColors:checkArgs:cannotFreezeColorbar',...





            ['You seem to be attempting to freeze a colorbar. This no longer'...





            'works. Please read the help for freezeColors for the solution.'])




      else



        cbfreeze(h);



        return



      end



    end



    args{1} = [];



    nargs = nargs-1;



else



    h = gca;



end







%set nancolor if that option was specified



nancolor = [nan nan nan];



if nargs == 2,




    if strcmpi(args{end-1},'nancolor'),




        nancolor = args{end};



        if ~all(size(nancolor)==[1 3]),




            error('JRI:freezeColors:checkArgs:badColorArgument',...





                'nancolor must be [r g b] vector');




        end



        nancolor(nancolor>1) = 1; nancolor(nancolor<0) = 0;



    else




        error('JRI:freezeColors:checkArgs:unrecognizedOption',...





            'Unrecognized option (%s). Only ''nancolor'' is valid.',args{end-1})




    end



end

转载自：https://blog.csdn.net/u012366767/article/details/82703670