3D gaussian fit

[bill betz]

I have images with somewhat blurry bright circular spots on a dark
background (these are fluorescence microscopic images). For each
spot, I want to fit a gaussian - just a single max value and variance
value, with the x,y coordinate of the center.

[Jack Loveless]

Hi Bill,
You could write a brute force grid search to solve the for the best
fitting parameters A and C in the equation:

S(Xi,Yi) = A exp(-Ri^2/2C^2)

where S(Xi,Yi) is the Gaussian fit value at coordinates (Xi,Yi), A is
the maximum value, centered on (X0,Y0), Ri is the distance between
(Xi,Yi) and (X0,Y0), and C is the variance or curvature of the
solution.

[Nathan Orloff]

function [fitresult, zfit, fiterr, zerr, resnorm] = fmgaussfit(xx,yy,zz)
%fmgaussfit(xx,yy,zz) Custom Guassian Fit algorithm using lsqcurvefit
% A 3D gaussian fitting routine with error propagation and uncertainties.

%% Condition the data
[xData, yData, zData] = prepareSurfaceData( xx, yy, zz );
xyData = {xData,yData};

%% Set up the startpoint
[amp, ind] = max(zData); % amp is the amplitude.
xo = xData(ind); % guess that it is at the maximum
yo = yData(ind); % guess that it is at the maximum
ang = 45; % angle in degrees.
sy = 1;
sx = 1;
zo = median(zData(:))-std(zData(:));
xmax = max(xData)+2;
ymax = max(yData)+2;
xmin = min(xData)-2;
ymin = min(yData)-2;

%% Set up fittype and options.
Lower = [0, 0, 0, 0, xmin, ymin, 0];
Upper = [Inf, 90, Inf, Inf, xmax, ymax, Inf]; % angles greater than 90 are redundant
StartPoint = [amp, ang, sx, sy, xo, yo, zo];%[amp, sx, sxy, sy, xo, yo, zo];

tols = 1e-16;
options = optimset('Algorithm','levenberg-marquardt',...
'Display','off',...
'MaxFunEvals',5e2,...
'MaxIter',5e2,...
'TolX',tols,...
'TolFun',tols,...
'TolCon',tols ,...
'UseParallel','always');

%% perform the fitting
[fitresult,resnorm,residual] = ...
lsqcurvefit(@gaussian2D,StartPoint,xyData,zData,Lower,Upper,options);
[fiterr, zfit, zerr] = gaussian2Duncert(fitresult,residual,jacobian,xyData);

end

function z = gaussian2D(par,xy)
z = par(7) + ...
par(1)*exp(-(((xy{1}-par(5)).*cosd(par(2))+(xy{2}-par(6)).*sind(par(2)))./par(3)).^2-...
((-(xy{1}-par(5)).*sind(par(2))+(xy{2}-par(6)).*cosd(par(2)))./par(4)).^2);
end

function [dpar,zf,dzf] = gaussian2Duncert(par,resid,xy)
J = guassian2DJacobian(par,xy);
parci = nlparci(par,resid,'Jacobian',J);
dpar = (diff(parci,[],2)./2)';
[zf,dzf] = nlpredci(@gaussian2D,xy,par,resid,'Jacobian',J);
end

function J = guassian2DJacobian(par,xy)
x = xy{1}; y = xy{2};
J(:,1) = exp(- (cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2./par(3).^2 - (cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2./par(4).^2);
J(:,2) = -par(1).*exp(- (cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2./par(3).^2 - (cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2./par(4).^2).*((2.*(cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).*(cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))))./par(3).^2 - (2.*(cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).*(cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))))./par(4).^2);
J(:,3) = (2.*par(1).*exp(- (cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2./par(3).^2 - (cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2./par(4).^2).*(cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2)./par(3)^3;
J(:,4) = (2.*par(1).*exp(- (cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2./par(3).^2 - (cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2./par(4).^2).*(cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2)./par(4)^3;
J(:,5) = par(1).*exp(- (cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2./par(3).^2 - (cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2./par(4).^2).*((2.*cosd(par(2)).*(cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))))./par(3).^2 - (2.*sind(par(2)).*(cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))))./par(4).^2);
J(:,6) = par(1).*exp(- (cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))).^2./par(3).^2 - (cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))).^2./par(4).^2).*((2.*cosd(par(2)).*(cosd(par(2)).*(y - par(6)) - sind(par(2)).*(x - par(5))))./par(4).^2 + (2.*sind(par(2)).*(cosd(par(2)).*(x - par(5)) + sind(par(2)).*(y - par(6))))./par(3).^2);
J(:,7) = ones(size(x));
end

function [sse, zf] = gaussian2Derr(par,xy,z)
zf = gaussian2D(par,xy);
zerr = zf-z;
sse = sum(zerr.^ 2);
end

[victoriap...@gmail.com]

I am using this function with some variations, because I need to estimate two gaussian together

The problem appears in some cases when the residual, and the resnorm are empty, i.e residual=[], so the nlparci can't be calculated and in the command windows appears: "Error using nlparci (line 66)
Requires BETA, RESID, and either J or SIGMA."

I'm looking for a reason of why the residual could be empty, but I don't find anything...Do you have any idea of this?

Thank you! Your fmgaussfit function is really great!!

Victoria