I have implemented Harris feature detection algorithm and the results are not accurate as compared to using the inbuilt function of Matlab: corner(I,'Harris'). Any ideas why so? Or am i missing something in it?
Results of my code:
im=imread('D:\lena_256.pgm');
im=im2double(im);
% im = double(im(:,:,1));
sigma = 2;k = 0.04;
% derivative masks
s_D = 0.7*sigma;
x = -round(3*s_D):round(3*s_D);
dx = x .* exp(-x.*x/(2*s_D*s_D)) ./ (s_D*s_D*s_D*sqrt(2*pi));
dy = dx';
% image derivatives
Ix = conv2(im, dx, 'same');
Iy = conv2(im, dy, 'same');
% sum of the Auto-correlation matrix
s_I = sigma;
g = fspecial('gaussian',max(1,fix(6*s_I+1)), s_I);
Ix2 = conv2(Ix.^2, g, 'same'); % Smoothed squared image derivatives
Iy2 = conv2(Iy.^2, g, 'same');
Ixy = conv2(Ix.*Iy, g, 'same');
% interest point response
cim = (Ix2.*Iy2 - Ixy.^2) - k*(Ix2 + Iy2).^2; % Original Harris measure.
%cim=(Ix2.*Iy2 - Ixy.^2)./(Ix2 + Iy2); % harmonic mean
% find local maxima on 3x3 neighborgood
[r,c,max_local] = findLocalMaximum(cim,3*s_I);
% set threshold 1% of the maximum value
t = 0.005*max(max_local(:));
% find local maxima greater than threshold
[r,c] = find(max_local>=t);
% build interest points
points = [r,c];
figure, imshow(im),title('Harris Feature Points');
hold on
plot(points(:,1),points(:,2),'r*');
function [row,col,max_local] = findLocalMaximum(val,radius)
mask = fspecial('disk',radius)>0;
val2 = imdilate(val,mask);
index = val==val2;
[row,col] = find(index==1);
max_local = zeros(size(val));
max_local(index) = val(index);
end
Results using inbuilt corner() function:
Related
I would like to generate surface of revolution from bezier curve. I have made bezier curve in MATLAB but beyond this point I am stuck and do not know how to proceed. Please help.
Below is the code that I have made.
clc
clear
close all
% Name : Savla Jinesh Shantilal
% Bits ID : 2021HT30609
%% Define inout parameters
B = [1,1; 2,3; 4,3; 3,1]; % Creating matrix for polygon vertices
[r,s] = size(B); % getting size of matrix in terms of rows and columns
n = r-1; % n+1 represents number of vertices of the polygon
np = 20; % represents number of equi-distance points on the bezier curve
t = linspace(0,1,np);
%% Plot polygon
for k = 1:n
plot([B(k,1),B(k+1,1)], [B(k,2),B(k+1,2)], 'r', 'LineWidth', 2)
hold on
grid on
end
%% Generate the points on the bezier curve
for j = 1:np
P = [0,0];
for i = 0:n
M(i+1) = (factorial(n)/(factorial(i)*factorial(n-i)))*((t(j))^i)*((1-t(j))^(n-i));
P = P + B(i+1,:)*M(i+1);
end
Q(j,:) = P;
end
%% Plot the bezier curve from the obtained points
for l = 1:np-1
plot([Q(l,1),Q(l+1,1)],[Q(l,2),Q(l+1,2)], '-- b', 'LineWidth', 2);
hold on
end
Usually one can use the built-in cylinder function for monotonically increasing x-values. Here, the bezier curve has non monotonic values from max(x) so we break it to two parts to parameterize it, and then add an angle rotation.
% first define the x and y coordinate from your Q info:
xx = Q(:,1);
yy = Q(:,2);
N = 1e2;
[~, id] = max(xx); % the position where we split
t = linspace(xx(1),xx(id),N);
% Parameterize the function:
t = linspace(0,2*(xx(id)-xx(1)),N);
x = zeros(1, N);
L = t <= ( xx(id)-xx(1) ); % the "Left" side of the curve
x(L) = t(L)+xx(1);
x(~L) = flip(x(L));
%define the r value
r = x;
r(L) = interp1(xx(1:id) ,yy(1:id) , x(L) ); % Left side
r(~L) = interp1(xx(id:end),yy(id:end), x(~L)); % right side (not left)
% define your x values
x = repmat(x', [1 N]);
% define the theta that will perform the rotation
theta = linspace(0,2*pi, N);
% initialize values for y and z
y = zeros(N);
z = zeros(N);
% calculate the y and z values
for i=1:N
y(i,:) = r(i) *cos(theta);
z(i,:) = r(i) *sin(theta);
end
%plot the surface of revolution and the original curve
s = surf(x,y,z);
alpha 0.4
hold on
plot(xx,yy,'k','LineWidth',3)
I am running simulations of a protein/membrane system, and want to quantify the degree of membrane deformation. I have averaged the membrane surface on a grid during the simulation, which results in a text file with three columns, containing the x, y, and z points of the membrane. I then convert this information to a mesh surface in matlab, which I then use to calculate the gaussian and/or mean curvature. The problem is, I'm getting similar values at the very beginning of the simulation, when the surface (membrane) is very flat, and at the end, when it is completely deformed. Unless I'm misunderstanding what curvature is, this is not correct and I believe I am doing something wrong in the matlab portion of this process. Here's the script where I loop over many frames (each of which is a different file containing x,y,z coordinates of the averaged membrane) and convert it to a mesh:
https://pastebin.com/reqWAz01
for i = 0:37
file = strcat('cg-topmem_pos-', num2str(i), '.out');
A = load(file);
x = A(:,1);
y = A(:,2);
z = A(:,3);
xv = linspace(min(x), max(x), 20);
yv = linspace(min(y), max(y), 20);
[X,Y] = meshgrid(xv, yv);
Z = griddata(x,y,z,X,Y);
[K,H,Pmax,Pmin] = surfature(X,Y,Z);
M = max(max(H))
if i == 0
fileID = fopen('Average2-NoTMHS-5nmns.txt', 'a');
fprintf(fileID, '%f %f\n',M);
fclose(fileID);
else
fileID = fopen('Average2-NoTMHS-5nmns.txt', 'a');
fprintf(fileID, '\n%f %f\n',M);
fclose(fileID);
end
end
Then using the following calculates curvature:
https://pastebin.com/5D21PdBQ
function [K,H,Pmax,Pmin] = surfature(X,Y,Z),
% SURFATURE - COMPUTE GAUSSIAN AND MEAN CURVATURES OF A SURFACE
% [K,H] = SURFATURE(X,Y,Z), WHERE X,Y,Z ARE 2D ARRAYS OF POINTS ON THE
% SURFACE. K AND H ARE THE GAUSSIAN AND MEAN CURVATURES, RESPECTIVELY.
% SURFATURE RETURNS 2 ADDITIONAL ARGUMENTS,
% [K,H,Pmax,Pmin] = SURFATURE(...), WHERE Pmax AND Pmin ARE THE MINIMUM
% AND MAXIMUM CURVATURES AT EACH POINT, RESPECTIVELY.
% First Derivatives
[Xu,Xv] = gradient(X);
[Yu,Yv] = gradient(Y);
[Zu,Zv] = gradient(Z);
% Second Derivatives
[Xuu,Xuv] = gradient(Xu);
[Yuu,Yuv] = gradient(Yu);
[Zuu,Zuv] = gradient(Zu);
[Xuv,Xvv] = gradient(Xv);
[Yuv,Yvv] = gradient(Yv);
[Zuv,Zvv] = gradient(Zv);
% Reshape 2D Arrays into Vectors
Xu = Xu(:); Yu = Yu(:); Zu = Zu(:);
Xv = Xv(:); Yv = Yv(:); Zv = Zv(:);
Xuu = Xuu(:); Yuu = Yuu(:); Zuu = Zuu(:);
Xuv = Xuv(:); Yuv = Yuv(:); Zuv = Zuv(:);
Xvv = Xvv(:); Yvv = Yvv(:); Zvv = Zvv(:);
Xu = [Xu Yu Zu];
Xv = [Xv Yv Zv];
Xuu = [Xuu Yuu Zuu];
Xuv = [Xuv Yuv Zuv];
Xvv = [Xvv Yvv Zvv];
% First fundamental Coeffecients of the surface (E,F,G)
E = dot(Xu,Xu,2);
F = dot(Xu,Xv,2);
G = dot(Xv,Xv,2);
m = cross(Xu,Xv,2);
p = sqrt(dot(m,m,2));
n = m./[p p p];
% Second fundamental Coeffecients of the surface (L,M,N)
L = dot(Xuu,n,2);
M = dot(Xuv,n,2);
N = dot(Xvv,n,2);
[s,t] = size(Z);
% Gaussian Curvature
K = (L.*N - M.^2)./(E.*G - F.^2);
K = reshape(K,s,t);
% Mean Curvature
H = (E.*N + G.*L - 2.*F.*M)./(2*(E.*G - F.^2));
H = reshape(H,s,t);
% Principal Curvatures
Pmax = H + sqrt(H.^2 - K);
Pmin = H - sqrt(H.^2 - K);
Any help would be greatly appreciated. I'm afraid that there is some issue between how the mesh is created and how curvature is calculated, but I am not matlab literate and could use some help. Thanks very much.
I'm using random points to determine the area below a curve (Monte-Carlo):
X: 1xn vector of x values for the function
Y: 1xn vector of y = f(x)
RP: mxn matrix of m random y for each x
I would like to split RP into RPA and RPB depending on it being above or below the curve. The idea is then to plot RPA and RPB against X, in different colors. This code doesn't work because RPA and RPB number of columns is not the same than X:
clf
f = #(x) sin(x/10) + cos(x/60); % Function
xMin = 1; xMax = 100; % x interval
X = [xMin:xMax];
Y = f(X);
plot(X,Y), hold on % Plot function
yMin = min(Y); yMax = max(Y); % Axes limits
set(gca, 'xlim', [xMin, xMax], 'ylim', [yMin, yMax])
m = 20; % Random points per x value
RP = rand(m, columns(X)) .* (yMax-yMin) .+ yMin;
% Split points (doesn't work)
RPA = RP(RP>Y);
RPB = RP(RP<=Y);
br = size(RPB) / size(RP) % Ratio of points below
a = (xMax - xMin) * (yMax - yMin) * br % Area below
% Plot points
plot(X, RPA, 'r.') % Above
plot(X, RPB, 'g.') % Below
Is there a possibility to build RPA and RPB so that they are the same size than RP, with the excluded points y being NaN or something similar, which can be counted and plotted?
You gave a good answer yourself. You can build RPA and RPB with strategic NaNs:
% Split points (works!)
RPA = RP;
RPA(RP<=Y) = NaN;
RPB = RP;
RPB(RPB > Y) = NaN;
And than calculating the ration as the not-NaN:
br = sum(~isnan(RPB)) / sum(~isnan(RP)) % Ratio of points below
I get this nice image:
I am coding a Gaussian Process regression algorithm. Here is the code:
% Data generating function
fh = #(x)(2*cos(2*pi*x/10).*x);
% range
x = -5:0.01:5;
N = length(x);
% Sampled data points from the generating function
M = 50;
selection = boolean(zeros(N,1));
j = randsample(N, M);
% mark them
selection(j) = 1;
Xa = x(j);
% compute the function and extract mean
f = fh(Xa) - mean(fh(Xa));
sigma2 = 1;
% computing the interpolation using all x's
% It is expected that for points used to build the GP cov. matrix, the
% uncertainty is reduced...
K = squareform(pdist(x'));
K = exp(-(0.5*K.^2)/sigma2);
% upper left corner of K
Kaa = K(selection,selection);
% lower right corner of K
Kbb = K(~selection,~selection);
% upper right corner of K
Kab = K(selection,~selection);
% mean of posterior
m = Kab'*inv(Kaa+0.001*eye(M))*f';
% cov. matrix of posterior
D = Kbb - Kab'*inv(Kaa + 0.001*eye(M))*Kab;
% sampling M functions from from GP
[A,B,C] = svd(Kaa);
F0 = A*sqrt(B)*randn(M,M);
% mean from GP using sampled points
F0m = mean(F0,2);
F0d = std(F0,0,2);
%%
% put together data and estimation
F = zeros(N,1);
S = zeros(N,1);
F(selection) = f' + F0m;
S(selection) = F0d;
% sampling M function from posterior
[A,B,C] = svd(D);
a = A*sqrt(B)*randn(N-M,M);
% mean from posterior GPs
Fm = m + mean(a,2);
Fmd = std(a,0,2);
F(~selection) = Fm;
S(~selection) = Fmd;
%%
figure;
% show what we got...
plot(x, F, ':r', x, F-2*S, ':b', x, F+2*S, ':b'), grid on;
hold on;
% show points we got
plot(Xa, f, 'Ok');
% show the whole curve
plot(x, fh(x)-mean(fh(x)), 'k');
grid on;
I expect to get some nice figure where the uncertainty of unknown data points would be big and around sampled data points small. I got an odd figure and even odder is that the uncertainty around sampled data points is bigger than on the rest. Can someone explain to me what I am doing wrong? Thanks!!
There are a few things wrong with your code. Here are the most important points:
The major mistake that makes everything go wrong is the indexing of f. You are defining Xa = x(j), but you should actually do Xa = x(selection), so that the indexing is consistent with the indexing you use on the kernel matrix K.
Subtracting the sample mean f = fh(Xa) - mean(fh(Xa)) does not serve any purpose, and makes the circles in your plot be off from the actual function. (If you choose to subtract something, it should be a fixed number or function, and not depend on the randomly sampled observations.)
You should compute the posterior mean and variance directly from m and D; no need to sample from the posterior and then obtain sample estimates for those.
Here is a modified version of the script with the above points fixed.
%% Init
% Data generating function
fh = #(x)(2*cos(2*pi*x/10).*x);
% range
x = -5:0.01:5;
N = length(x);
% Sampled data points from the generating function
M = 5;
selection = boolean(zeros(N,1));
j = randsample(N, M);
% mark them
selection(j) = 1;
Xa = x(selection);
%% GP computations
% compute the function and extract mean
f = fh(Xa);
sigma2 = 2;
sigma_noise = 0.01;
var_kernel = 10;
% computing the interpolation using all x's
% It is expected that for points used to build the GP cov. matrix, the
% uncertainty is reduced...
K = squareform(pdist(x'));
K = var_kernel*exp(-(0.5*K.^2)/sigma2);
% upper left corner of K
Kaa = K(selection,selection);
% lower right corner of K
Kbb = K(~selection,~selection);
% upper right corner of K
Kab = K(selection,~selection);
% mean of posterior
m = Kab'/(Kaa + sigma_noise*eye(M))*f';
% cov. matrix of posterior
D = Kbb - Kab'/(Kaa + sigma_noise*eye(M))*Kab;
%% Plot
figure;
grid on;
hold on;
% GP estimates
plot(x(~selection), m);
plot(x(~selection), m + 2*sqrt(diag(D)), 'g-');
plot(x(~selection), m - 2*sqrt(diag(D)), 'g-');
% Observations
plot(Xa, f, 'Ok');
% True function
plot(x, fh(x), 'k');
A resulting plot from this with 5 randomly chosen observations, where the true function is shown in black, the posterior mean in blue, and confidence intervals in green.
So when I try to find a template B in a big image A, I can do it by finding the maximum of cross-correlation, like this in spatial domain:
% Finding maximum of correlation:
phi = normxcorr2(B,A);
[ymax, xmax] = find(phi == max(phi(:)));
% Find position in original image:
ypeak = ymax - size(B,1);
xpeak = xmax - size(B,2);
But when I want to do it in frequency domain, I get wrong results:
% Calculate correlation in frequency domain:
Af = fft2(A);
Bf = fft2(B, size(A,1), size(A,2));
phi2f = conj(Af)'*Bf;
% Inverse fft to get back to spatial domain:
phi2 = real(ifft(fftshift(phi2f)));
% Now we have correlation matrix, excatly the same as calculated in
% the spatial domain.
[ymax2, xmax2] = find(phi2 == max(phi2(:)));
I don't understand what I'm doing wrong in the frequency domain. I've tried it without fftshift, it gives a different result, albeit a wrong one still. How can I do this correctly?
This should do the trick:
t = imread('cameraman.tif');
a = imtranslate(t, [15, 25]);
% Determine padding size in x and y dimension
size_t = size(t);
size_a = size(a);
outsize = size_t + size_a - 1;
% Determine 2D cross correlation in Fourier domain
Ft = fft2(t, outsize(1), outsize(2));
Fa = fft2(a, outsize(1), outsize(2));
c = abs( fftshift( ifft2(Fa .* conj(Ft))) );
% Find peak
[max_c, imax] = max(abs(c(:)));
[ypeak, xpeak] = ind2sub(size(c), imax(1));
% Correct found peak location for image size
corr_offset = round([(ypeak-(size(c, 1)+1)/2) (xpeak-(size(c, 2)+1)/2)]);
% Write out offsets
y_offset = corr_offset(1)
x_offset = corr_offset(2)