Related
I have a set of frequency data with peaks to which I need to fit a Gaussian curve and then get the full width half maximum from. The FWHM part I can do, I already have a code for that but I'm having trouble writing code to fit the Gaussian.
Does anyone know of any functions that'll do this for me or would be able to point me in the right direction? (I can do least squares fitting for lines and polynomials but I can't get it to work for gaussians)
Also it would be helpful if it was compatible with both Octave and Matlab as I have Octave at the moment but don't get access to Matlab until next week.
Any help would be greatly appreciated!
Fitting a single 1D Gaussian directly is a non-linear fitting problem. You'll find ready-made implementations here, or here, or here for 2D, or here (if you have the statistics toolbox) (have you heard of Google? :)
Anyway, there might be a simpler solution. If you know for sure your data y will be well-described by a Gaussian, and is reasonably well-distributed over your entire x-range, you can linearize the problem (these are equations, not statements):
y = 1/(σ·√(2π)) · exp( -½ ( (x-μ)/σ )² )
ln y = ln( 1/(σ·√(2π)) ) - ½ ( (x-μ)/σ )²
= Px² + Qx + R
where the substitutions
P = -1/(2σ²)
Q = +2μ/(2σ²)
R = ln( 1/(σ·√(2π)) ) - ½(μ/σ)²
have been made. Now, solve for the linear system Ax=b with (these are Matlab statements):
% design matrix for least squares fit
xdata = xdata(:);
A = [xdata.^2, xdata, ones(size(xdata))];
% log of your data
b = log(y(:));
% least-squares solution for x
x = A\b;
The vector x you found this way will equal
x == [P Q R]
which you then have to reverse-engineer to find the mean μ and the standard-deviation σ:
mu = -x(2)/x(1)/2;
sigma = sqrt( -1/2/x(1) );
Which you can cross-check with x(3) == R (there should only be small differences).
Perhaps this has the thing you are looking for? Not sure about compatability:
http://www.mathworks.com/matlabcentral/fileexchange/11733-gaussian-curve-fit
From its documentation:
[sigma,mu,A]=mygaussfit(x,y)
[sigma,mu,A]=mygaussfit(x,y,h)
this function is doing fit to the function
y=A * exp( -(x-mu)^2 / (2*sigma^2) )
the fitting is been done by a polyfit
the lan of the data.
h is the threshold which is the fraction
from the maximum y height that the data
is been taken from.
h should be a number between 0-1.
if h have not been taken it is set to be 0.2
as default.
i had similar problem.
this was the first result on google, and some of the scripts linked here made my matlab crash.
finally i found here that matlab has built in fit function, that can fit Gaussians too.
it look like that:
>> v=-30:30;
>> fit(v', exp(-v.^2)', 'gauss1')
ans =
General model Gauss1:
ans(x) = a1*exp(-((x-b1)/c1)^2)
Coefficients (with 95% confidence bounds):
a1 = 1 (1, 1)
b1 = -8.489e-17 (-3.638e-12, 3.638e-12)
c1 = 1 (1, 1)
I found that the MATLAB "fit" function was slow, and used "lsqcurvefit" with an inline Gaussian function. This is for fitting a Gaussian FUNCTION, if you just want to fit data to a Normal distribution, use "normfit."
Check it
% % Generate synthetic data (for example) % % %
nPoints = 200; binSize = 1/nPoints ;
fauxMean = 47 ;fauxStd = 8;
faux = fauxStd.*randn(1,nPoints) + fauxMean; % REPLACE WITH YOUR ACTUAL DATA
xaxis = 1:length(faux) ;fauxData = histc(faux,xaxis);
yourData = fauxData; % replace with your actual distribution
xAxis = 1:length(yourData) ;
gausFun = #(hms,x) hms(1) .* exp (-(x-hms(2)).^2 ./ (2*hms(3)^2)) ; % Gaussian FUNCTION
% % Provide estimates for initial conditions (for lsqcurvefit) % %
height_est = max(fauxData)*rand ; mean_est = fauxMean*rand; std_est=fauxStd*rand;
x0 = [height_est;mean_est; std_est]; % parameters need to be in a single variable
options=optimset('Display','off'); % avoid pesky messages from lsqcurvefit (optional)
[params]=lsqcurvefit(gausFun,x0,xAxis,yourData,[],[],options); % meat and potatoes
lsq_mean = params(2); lsq_std = params(3) ; % what you want
% % % Plot data with fit % % %
myFit = gausFun(params,xAxis);
figure;hold on;plot(xAxis,yourData./sum(yourData),'k');
plot(xAxis,myFit./sum(myFit),'r','linewidth',3) % normalization optional
xlabel('Value');ylabel('Probability');legend('Data','Fit')
I want to plot the area above and below a particular value in x axis.
The problem i am facing is with discrete values. The code below for instance has an explicit X=10 so i have written it in such a way that i can find the index and calculate the values above and below that particular value but if i want to find the area under the curve above and below 4 this program will now work.
Though in the plot matlab does a spline fitting(or some sort of fitting for connecting discrete values) there is a value for y corresponding to x=4 that matlab computes i cant seem to store or access it.
%Example for Area under the curve and partial area under the curve using Trapezoidal rule of integration
clc;
close all;
clear all;
x=[0,5,10,15,20];% domain
y=[0,25,50,25,0];% Values
LP=log2(y);
plot(x,y);
full = trapz(x,y);% plot of the total area
I=find(x==10);% in our case will be the distance value up to which we want
half = trapz(x(1:I),y(1:I));%Plot of the partial area
How can we find the area under the curve for a value of ie x = 2 or 3 or 4 or 6 or 7 or ...
This is an elaboration of patrik's comment, "first interpolate and then integrate".
For the purpose of this answer I'll assume that the area in question is the area that can be seen in the plot, and since plot connects points by straight lines I assume that linear interpolation is adequate. Moreover, since the trapezoidal rule itself is based on linear interpolation, we only need interpolated values at the beginning and end of the interval.
Starting from the given points
x = [0, 5, 10, 15, 20];
y = [0, 25, 50, 25, 0];
and the integration interval limits, say
xa = 4;
xb = 20;
we first select the data points within the limits
ind = (x > xa) & (x < xb);
xw = x(ind);
yw = y(ind);
and then complete them by interpolation values at the edges:
ya = interp1(x, y, xa);
yb = interp1(x, y, xb);
xw = [xa, xw, xb];
yw = [ya, yw, yb];
Now we can simply apply trapezoidal integration:
area = trapz(xw, yw);
I think that you either need more samples, or to interpolate the data. Another alternative is to use a function handle. Then you need to know the function though. Example using linear interpolation follows.
x0 = [0;5;10;15;20];
y0 = [0,25,50,25,0];
x1 = 0:20;
y1 = interp1(x0,y0,x1,'linear');
xMax = 4;
partInt = trapz(x1(x1<=xMax),y1(x1<=xMax));
Some other kind of interpolation may be suitable, but that is hard to say without more information. Also, this interpolates from the beginning to x. However, I guess figuring out how to change the limits should be easy from here. This solution is different than the former, since it is less depending on the pyramid shape of the data. So to say, it is more general.
I am trying to achieve 3d reconstruction from 2 images. Steps I followed are,
1. Found corresponding points between 2 images using SURF.
2. Implemented eight point algo to find "Fundamental matrix"
3. Then, I implemented triangulation.
I have got Fundamental matrix and results of triangulation till now. How do i proceed further to get 3d reconstruction? I'm confused reading all the material available on internet.
Also, This is code. Let me know if this is correct or not.
Ia=imread('1.jpg');
Ib=imread('2.jpg');
Ia=rgb2gray(Ia);
Ib=rgb2gray(Ib);
% My surf addition
% collect Interest Points from Each Image
blobs1 = detectSURFFeatures(Ia);
blobs2 = detectSURFFeatures(Ib);
figure;
imshow(Ia);
hold on;
plot(selectStrongest(blobs1, 36));
figure;
imshow(Ib);
hold on;
plot(selectStrongest(blobs2, 36));
title('Thirty strongest SURF features in I2');
[features1, validBlobs1] = extractFeatures(Ia, blobs1);
[features2, validBlobs2] = extractFeatures(Ib, blobs2);
indexPairs = matchFeatures(features1, features2);
matchedPoints1 = validBlobs1(indexPairs(:,1),:);
matchedPoints2 = validBlobs2(indexPairs(:,2),:);
figure;
showMatchedFeatures(Ia, Ib, matchedPoints1, matchedPoints2);
legend('Putatively matched points in I1', 'Putatively matched points in I2');
for i=1:matchedPoints1.Count
xa(i,:)=matchedPoints1.Location(i);
ya(i,:)=matchedPoints1.Location(i,2);
xb(i,:)=matchedPoints2.Location(i);
yb(i,:)=matchedPoints2.Location(i,2);
end
matchedPoints1.Count
figure(1) ; clf ;
imshow(cat(2, Ia, Ib)) ;
axis image off ;
hold on ;
xbb=xb+size(Ia,2);
set=[1:matchedPoints1.Count];
h = line([xa(set)' ; xbb(set)'], [ya(set)' ; yb(set)']) ;
pts1=[xa,ya];
pts2=[xb,yb];
pts11=pts1;pts11(:,3)=1;
pts11=pts11';
pts22=pts2;pts22(:,3)=1;pts22=pts22';
width=size(Ia,2);
height=size(Ib,1);
F=eightpoint(pts1,pts2,width,height);
[P1new,P2new]=compute2Pmatrix(F);
XP = triangulate(pts11, pts22,P2new);
eightpoint()
function [ F ] = eightpoint( pts1, pts2,width,height)
X = 1:width;
Y = 1:height;
[X, Y] = meshgrid(X, Y);
x0 = [mean(X(:)); mean(Y(:))];
X = X - x0(1);
Y = Y - x0(2);
denom = sqrt(mean(mean(X.^2+Y.^2)));
N = size(pts1, 1);
%Normalized data
T = sqrt(2)/denom*[1 0 -x0(1); 0 1 -x0(2); 0 0 denom/sqrt(2)];
norm_x = T*[pts1(:,1)'; pts1(:,2)'; ones(1, N)];
norm_x_ = T*[pts2(:,1)';pts2(:,2)'; ones(1, N)];
x1 = norm_x(1, :)';
y1= norm_x(2, :)';
x2 = norm_x_(1, :)';
y2 = norm_x_(2, :)';
A = [x1.*x2, y1.*x2, x2, ...
x1.*y2, y1.*y2, y2, ...
x1, y1, ones(N,1)];
% compute the SVD
[~, ~, V] = svd(A);
F = reshape(V(:,9), 3, 3)';
[FU, FS, FV] = svd(F);
FS(3,3) = 0; %rank 2 constrains
F = FU*FS*FV';
% rescale fundamental matrix
F = T' * F * T;
end
triangulate()
function [ XP ] = triangulate( pts1,pts2,P2 )
n=size(pts1,2);
X=zeros(4,n);
for i=1:n
A=[-1,0,pts1(1,i),0;
0,-1,pts1(2,i),0;
pts2(1,i)*P2(3,:)-P2(1,:);
pts2(2,i)*P2(3,:)-P2(2,:)];
[~,~,va] = svd(A);
X(:,i) = va(:,4);
end
XP(:,:,1) = [X(1,:)./X(4,:);X(2,:)./X(4,:);X(3,:)./X(4,:); X(4,:)./X(4,:)];
end
function [ P1,P2 ] = compute2Pmatrix( F )
P1=[1,0,0,0;0,1,0,0;0,0,1,0];
[~, ~, V] = svd(F');
ep = V(:,3)/V(3,3);
P2 = [skew(ep)*F,ep];
end
From a quick look, it looks correct. Some notes are as follows:
You normalized code in eightpoint() is no ideal.
It is best done on the points involved. Each set of points will have its scaling matrix. That is:
[pts1_n, T1] = normalize_pts(pts1);
[pts2_n, T2] = normalize-pts(pts2);
% ... code
% solution
F = T2' * F * T
As a side note (for efficiency) you should do
[~,~,V] = svd(A, 0);
You also want to enforce the constraint that the fundamental matrix has rank-2. After you compute F, you can do:
[U,D,v] = svd(F);
F = U * diag([D(1,1),D(2,2), 0]) * V';
In either case, normalization is not the only key to make the algorithm work. You'll want to wrap the estimation of the fundamental matrix in a robust estimation scheme like RANSAC.
Estimation problems like this are very sensitive to non Gaussian noise and outliers. If you have a small number of wrong correspondence, or points with high error, the algorithm will break.
Finally, In 'triangulate' you want to make sure that the points are not at infinity prior to the homogeneous division.
I'd recommend testing the code with 'synthetic' data. That is, generate your own camera matrices and correspondences. Feed them to the estimate routine with varying levels of noise. With zero noise, you should get an exact solution up to floating point accuracy. As you increase the noise, your estimation error increases.
In its current form, running this on real data will probably not do well unless you 'robustify' the algorithm with RANSAC, or some other robust estimator.
Good luck.
Good luck.
Which version of MATLAB do you have?
There is a function called estimateFundamentalMatrix in the Computer Vision System Toolbox, which will give you the fundamental matrix. It may give you better results than your code, because it is using RANSAC under the hood, which makes it robust to spurious matches. There is also a triangulate function, as of version R2014b.
What you are getting is sparse 3D reconstruction. You can plot the resulting 3D points, and you can map the color of the corresponding pixel to each one. However, for what you want, you would have to fit a surface or a triangular mesh to the points. Unfortunately, I can't help you there.
If what you're asking is how to I proceed from fundamental Matrix + corresponding points to a dense model then you still have a lot of work ahead of you.
relative camera locations (R,T) can be calculated from a fundamental matrix assuming you know the internal camera params (up to scale, rotation, translation). To get a full dense matrix there are a few ways to go. you can try using an existing library (PMVS for example). I'd look into OpenMVG but I'm not sure about matlab interface.
Another way to go, you can compute a dense optical flow (many available for matlab). Look for a epipolar OF (It takes a fundamental matrix and restricts the solution to lie on the epipolar lines). Then you can triangulate every pixel to get a depthmap.
Finally you will have to play with format conversions to get from a depthmap to VRML (You can look at meshlab)
Sorry my answer isn't more Matlab oriented.
I have discrete regular grid of a,b points and their corresponding c values and I interpolate it further to get a smooth curve. Now from interpolation data, I further want to create a polynomial equation for curve fitting. How to fit 3D plot in polynomial?
I try to do this in MATLAB. I used Surface fitting toolbox in MATLAB (r2010a) to curve fit 3-dimensional data. But, how does one find a formula that fits a set of data to the best advantage in MATLAB/MAPLE or any other software. Any advice? Also most useful would be some real code examples to look at, PDF files, on the web etc.
This is just a small portion of my data.
a = [ 0.001 .. 0.011];
b = [1, .. 10];
c = [ -.304860225, .. .379710865];
Thanks in advance.
To fit a curve onto a set of points, we can use ordinary least-squares regression. There is a solution page by MathWorks describing the process.
As an example, let's start with some random data:
% some 3d points
data = mvnrnd([0 0 0], [1 -0.5 0.8; -0.5 1.1 0; 0.8 0 1], 50);
As #BasSwinckels showed, by constructing the desired design matrix, you can use mldivide or pinv to solve the overdetermined system expressed as Ax=b:
% best-fit plane
C = [data(:,1) data(:,2) ones(size(data,1),1)] \ data(:,3); % coefficients
% evaluate it on a regular grid covering the domain of the data
[xx,yy] = meshgrid(-3:.5:3, -3:.5:3);
zz = C(1)*xx + C(2)*yy + C(3);
% or expressed using matrix/vector product
%zz = reshape([xx(:) yy(:) ones(numel(xx),1)] * C, size(xx));
Next we visualize the result:
% plot points and surface
figure('Renderer','opengl')
line(data(:,1), data(:,2), data(:,3), 'LineStyle','none', ...
'Marker','.', 'MarkerSize',25, 'Color','r')
surface(xx, yy, zz, ...
'FaceColor','interp', 'EdgeColor','b', 'FaceAlpha',0.2)
grid on; axis tight equal;
view(9,9);
xlabel x; ylabel y; zlabel z;
colormap(cool(64))
As was mentioned, we can get higher-order polynomial fitting by adding more terms to the independent variables matrix (the A in Ax=b).
Say we want to fit a quadratic model with constant, linear, interaction, and squared terms (1, x, y, xy, x^2, y^2). We can do this manually:
% best-fit quadratic curve
C = [ones(50,1) data(:,1:2) prod(data(:,1:2),2) data(:,1:2).^2] \ data(:,3);
zz = [ones(numel(xx),1) xx(:) yy(:) xx(:).*yy(:) xx(:).^2 yy(:).^2] * C;
zz = reshape(zz, size(xx));
There is also a helper function x2fx in the Statistics Toolbox that helps in building the design matrix for a couple of model orders:
C = x2fx(data(:,1:2), 'quadratic') \ data(:,3);
zz = x2fx([xx(:) yy(:)], 'quadratic') * C;
zz = reshape(zz, size(xx));
Finally there is an excellent function polyfitn on the File Exchange by John D'Errico that allows you to specify all kinds of polynomial orders and terms involved:
model = polyfitn(data(:,1:2), data(:,3), 2);
zz = polyvaln(model, [xx(:) yy(:)]);
zz = reshape(zz, size(xx));
There might be some better functions on the file-exchange, but one way to do it by hand is this:
x = a(:); %make column vectors
y = b(:);
z = c(:);
%first order fit
M = [ones(size(x)), x, y];
k1 = M\z;
%least square solution of z = M * k1, so z = k1(1) + k1(2) * x + k1(3) * y
Similarly, you can do a second order fit:
%second order fit
M = [ones(size(x)), x, y, x.^2, x.*y, y.^2];
k2 = M\z;
which seems to have numerical problems for the limited dataset you gave. Type help mldivide for more details.
To make an interpolation over some regular grid, you can do like so:
ngrid = 20;
[A,B] = meshgrid(linspace(min(a), max(a), ngrid), ...
linspace(min(b), max(b), ngrid));
M = [ones(numel(A),1), A(:), B(:), A(:).^2, A(:).*B(:), B(:).^2];
C2_fit = reshape(M * k2, size(A)); % = k2(1) + k2(2)*A + k2(3)*B + k2(4)*A.^2 + ...
%plot to compare fit with original data
surfl(A,B,C2_fit);shading flat;colormap gray
hold on
plot3(a,b,c, '.r')
A 3rd-order fit can be done using the formula given by TryHard below, but the formulas quickly become tedious when the order increases. Better write a function that can construct M given x, y and order if you have to do that more than once.
This sounds like more of a philosophical question than specific implementation, specifically to bit - "how does one find a formula that fits a set of data to the best advantage?" In my experience that is a choice you have to make depending on what you're trying to achieve.
What defines "best" for you? For a data fitting problem you can keep adding more and more polynomial coefficients and making a better R^2 value... but will eventually "over fit" the data. A downside of high order polynomials is behavior outside the bounds of the sample data which you've used to fit your response surface - it can quickly go off in some wild direction which may not be appropriate for whatever it is you're trying to model.
Do you have insight into the physical behavior of the system / data you're fitting? That can be used as a basis for what set of equations to use to create a math model. My recommendation would be to use the most economical (simple) model you can get away with.
I have a simple loglog curve as above. Is there some function in Matlab which can fit this curve by segmented lines and show the starting and end points of these line segments ? I have checked the curve fitting toolbox in matlab. They seems to do curve fitting by either one line or some functions. I do not want to curve fitting by one line only.
If there is no direct function, any alternative to achieve the same goal is fine with me. My goal is to fit the curve by segmented lines and get locations of the end points of these segments .
First of all, your problem is not called curve fitting. Curve fitting is when you have data, and you find the best function that describes it, in some sense. You, on the other hand, want to create a piecewise linear approximation of your function.
I suggest the following strategy:
Split manually into sections. The section size should depend on the derivative, large derivative -> small section
Sample the function at the nodes between the sections
Find a linear interpolation that passes through the points mentioned above.
Here is an example of a code that does that. You can see that the red line (interpolation) is very close to the original function, despite the small amount of sections. This happens due to the adaptive section size.
function fitLogLog()
x = 2:1000;
y = log(log(x));
%# Find section sizes, by using an inverse of the approximation of the derivative
numOfSections = 20;
indexes = round(linspace(1,numel(y),numOfSections));
derivativeApprox = diff(y(indexes));
inverseDerivative = 1./derivativeApprox;
weightOfSection = inverseDerivative/sum(inverseDerivative);
totalRange = max(x(:))-min(x(:));
sectionSize = weightOfSection.* totalRange;
%# The relevant nodes
xNodes = x(1) + [ 0 cumsum(sectionSize)];
yNodes = log(log(xNodes));
figure;plot(x,y);
hold on;
plot (xNodes,yNodes,'r');
scatter (xNodes,yNodes,'r');
legend('log(log(x))','adaptive linear interpolation');
end
Andrey's adaptive solution provides a more accurate overall fit. If what you want is segments of a fixed length, however, then here is something that should work, using a method that also returns a complete set of all the fitted values. Could be vectorized if speed is needed.
Nsamp = 1000; %number of data samples on x-axis
x = [1:Nsamp]; %this is your x-axis
Nlines = 5; %number of lines to fit
fx = exp(-10*x/Nsamp); %generate something like your current data, f(x)
gx = NaN(size(fx)); %this will hold your fitted lines, g(x)
joins = round(linspace(1, Nsamp, Nlines+1)); %define equally spaced breaks along the x-axis
dx = diff(x(joins)); %x-change
df = diff(fx(joins)); %f(x)-change
m = df./dx; %gradient for each section
for i = 1:Nlines
x1 = joins(i); %start point
x2 = joins(i+1); %end point
gx(x1:x2) = fx(x1) + m(i)*(0:dx(i)); %compute line segment
end
subplot(2,1,1)
h(1,:) = plot(x, fx, 'b', x, gx, 'k', joins, gx(joins), 'ro');
title('Normal Plot')
subplot(2,1,2)
h(2,:) = loglog(x, fx, 'b', x, gx, 'k', joins, gx(joins), 'ro');
title('Log Log Plot')
for ip = 1:2
subplot(2,1,ip)
set(h(ip,:), 'LineWidth', 2)
legend('Data', 'Piecewise Linear', 'Location', 'NorthEastOutside')
legend boxoff
end
This is not an exact answer to this question, but since I arrived here based on a search, I'd like to answer the related question of how to create (not fit) a piecewise linear function that is intended to represent the mean (or median, or some other other function) of interval data in a scatter plot.
First, a related but more sophisticated alternative using regression, which apparently has some MATLAB code listed on the wikipedia page, is Multivariate adaptive regression splines.
The solution here is to just calculate the mean on overlapping intervals to get points
function [x, y] = intervalAggregate(Xdata, Ydata, aggFun, intStep, intOverlap)
% intOverlap in [0, 1); 0 for no overlap of intervals, etc.
% intStep this is the size of the interval being aggregated.
minX = min(Xdata);
maxX = max(Xdata);
minY = min(Ydata);
maxY = max(Ydata);
intInc = intOverlap*intStep; %How far we advance each iteraction.
if intOverlap <= 0
intInc = intStep;
end
nInt = ceil((maxX-minX)/intInc); %Number of aggregations
parfor i = 1:nInt
xStart = minX + (i-1)*intInc;
xEnd = xStart + intStep;
intervalIndices = find((Xdata >= xStart) & (Xdata <= xEnd));
x(i) = aggFun(Xdata(intervalIndices));
y(i) = aggFun(Ydata(intervalIndices));
end
For instance, to calculate the mean over some paired X and Y data I had handy with intervals of length 0.1 having roughly 1/3 overlap with each other (see scatter image):
[x,y] = intervalAggregate(Xdat, Ydat, #mean, 0.1, 0.333)
x =
Columns 1 through 8
0.0552 0.0868 0.1170 0.1475 0.1844 0.2173 0.2498 0.2834
Columns 9 through 15
0.3182 0.3561 0.3875 0.4178 0.4494 0.4671 0.4822
y =
Columns 1 through 8
0.9992 0.9983 0.9971 0.9955 0.9927 0.9905 0.9876 0.9846
Columns 9 through 15
0.9803 0.9750 0.9707 0.9653 0.9598 0.9560 0.9537
We see that as x increases, y tends to decrease slightly. From there, it is easy enough to draw line segments and/or perform some other kind of smoothing.
(Note that I did not attempt to vectorize this solution; a much faster version could be assumed if Xdata is sorted.)