Matlab--二次多项式曲面拟合

二次多项式曲面公式

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

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]);

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

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');

使用cftool工具拟合

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 (iversen@nsi.edu) 3/23/05
%
% Changes:
% JRI (iversen@nsi.edu) 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
% john_iversen@post.harvard.edu
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

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Matlab--二次多项式曲面拟合

二次多项式曲面公式

实例

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