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I have in mind the following experiment to run in Matlab and I am asking for an help to implement step (3). Any suggestion would be very appreciated.
(1) Consider the random variables X and Y both uniformly distributed on [0,1]
(2) Draw N realisation from the joint distribution of X and Y assuming that X and Y are independent (meaning that X and Y are uniformly jointly distributed on [0,1]x[0,1]). Each draw will be in [0,1]x[0,1].
(3) Transform each draw in [0,1]x[0,1] in a draw in [0,1] using the Hilbert space filling curve: under the Hilbert curve mapping, the draw in [0,1]x[0,1] should be the image of one (or more because of surjectivity) point(s) in [0,1]. I want pick one of these points. Is there any pre-built package in Matlab doing this?
I found this answer which I don't think does what I want as it explains how to obtain the Hilbert value of the draw (curve length from the start of curve to the picked point)
On wikipedia I found this code in C language (from (x,y) to d) which, again, does not fulfil my question.
EDIT This answer does not address updated version of the question, which explicitly asks about constructing Hilbert curve. Instead, this answer addresses a related question on construction of bijective mapping, and the relation to uniform distribution.
Your problem in not really well defined. If you only need the resulting distribution to be uniform, nothing is stopping you from simply picking f:(X,Y)->X. Result would be uniform regardless of whether X and Y are correlated. From your post I can only presume that what you want, in fact, is for the resulting transformation to be bijective, or as close to it as possible given machine precision limitations.
Worth noting that unless you need the algorithm that is best in preserving locality (which is clearly not required for resulting distribution to be bijective, not to mention uniform), there's no need to bother constructing Hilbert curves that you mention in your question. They have just as much to do with the solution as any other space-filling curve, and are incredibly computationally intensive.
So assuming you're looking for a bijective mapping, your question is equivalent to asking whether the set of points in a [unit] square has the same cardinality as the set of points in a [unit] line segment, and if it is, how to construct that bijection, i.e. 1-to-1 correspondence. The intuition says the square should have a higher cardinality, and Cantor spent 3 years trying to prove that, eventually proving quite the opposite - that these sets are in fact equinumerous. He was so surprised at his discovery that he wrote:
I see it, but I don't believe it!
The most commonly referred to bijection, fulfilling** this criteria, is the following. Represent x and y in their decimal form, i.e. x=0. x1 x2 x3 x4 x5..., and y=0. y1 y2 y3 y4 y5..., and let f:(X,Y)->Z be z=0. x1 y1 x2 y2 x3 y3 x4 y4 x5 y5..., i.e. alternating the decimals of the two numbers. The idea behind the bijection is trivial, though a rigorous proof requires quite a bit of prior knowledge.
** The caveat is that if we take e.g. x = 1/3 = 0.33333... and y = 1/5 = 0.199999... = 0.200000..., we can see there are two sequences corresponding to them: z = 0.313939393939... and z = 0.323030303030.... To overcome this obstacle we have to prove that adding a countable set to an uncountable one does not change the cardinality of the latter.
In reality we have to deal with machine precision and not pure math, which strictly speaking means both sets are actually finite and hence not equinumerous (assuming you store result with the same precision as original numbers). Which means we're simply forced to do some assumptions and loose some information, such as, in this case, the last half of significant digits of x and y. That is, unless we use a different data type that allows to store result with a double precision, compared to that of original variables.
Finally, sample implementation in Matlab:
x = rand();
y = rand();
chars = [num2str(x, '%.17f'); num2str(y, '%.17f')];
z = str2double(['0.' reshape(chars(:,3:end), 1, [])]);
>> cellstr(['x=' num2str(x, '%.17f'); 'y=' num2str(y, '%.17f'); 'z=' num2str(z, '%.17f')])
ans =
'x=0.65549803980353738'
'y=0.10975505072305158'
'z=0.61505947958500362'
Edit This answers the original request for a transformation f(x,y) -> t ~ U[0,1] given x,y ~ U[0,1], and additionally for x and y correlated. The updated question asks specifically for a Hilbert curve, H(x,y) -> t ~ U[0,1] and only for x,y ~ U[0,1] so this answer is no longer relevant.
Consider a random uniform sequence in [0,1] r1, r2, r3, .... You are assigning this sequence to pairs of numbers (x1,y1), (x2,y2), .... What you are asking for is a transformation on pairs (x,y) which yield a uniform random number in [0,1].
Consider the random subsequence r1, r3, ... corresponding to x1, x2, .... If you trust that your number generator is random and uncorrelated in [0,1], then the subsequence x1, x2, ... should also be random and uncorrelated in [0,1]. So the rather simple answer to the first part of your question is a projection onto either the x or y axis. That is, just pick x.
Next consider correlations between x and y. Since you haven't specified the nature of the correlation, let's assume a simple scaling of the axes,
such as x' => [0, 0.5], y' => [0, 3.0], followed by a rotation. The scaling doesn't introduce any correlation since x' and y' are still independent. You can generate it easily enough with a matrix multiply:
M1*p = [x_scale, 0; 0, y_scale] * [x; y]
for matrix M1 and point p. You can introduce a correlation by taking this stretched form and rotating it by theta:
M2*M1*p = [cos(theta), sin(theta); -sin(theta), cos(theta)]*M1*p
Putting it all together with theta = pi/4, or 45 degrees, you can see that larger values of y are correlated with larger values of x:
cos_t = sin_t = cos(pi/4); % at 45 degrees, sin(t) = cos(t) = 1/sqrt(2)
M2 = [cos_t, sin_t; -sin_t, cos_t];
M1 = [0.5, 0.0; 0.0, 3.0];
p = random(2,1000);
p_prime = M2*M1*p;
plot(p_prime(1)', p_prime(2)', '.');
axis('equal');
The resulting plot* shows a band of uniformly distributed numbers at a 45 degree angle:
Further transformations are possible with shear, and if you are clever about it, translation (OpenGL uses 4x4 transformation matrices so that translation can be represented as a linear transform matrix, with an extra dimension added before the transformation steps and removed before they are done).
Given a known affine correlation structure, you can transform back from random points (x',y') to points (x,y) where x and y are independent in [0,1] by solving Mk*...*M1 p = p_prime for p, or equivalently, by setting p = inv(Mk*...*M1) * p_prime, where p=[x;y]. Again, just pick x, which will be uniform in [0,1]. This doesn't work if the transformation matrix is singular, e.g., if you introduce a projection matrix Mj into the mix (though if the projection is the first step you can still recover).
* You may notice that the plot is from python rather than matlab. I don't have matlab or octave sitting in front of me right now, so I hope I got the syntax details right.
You could compute the hilbert curve from f(x,y)=z. Basically it's a hamiltonian path traversal. You can find a good description at Nick's spatial index hilbert curve quadtree blog. Or take a look at monotonic n-ary gray code. I've written an implementation based on Nick's blog in php:http://monstercurves.codeplex.com.
I will focus only on your last point
(3) Transform each draw in [0,1]x[0,1] in a draw in [0,1] using the Hilbert space filling curve: under the Hilbert curve mapping, the draw in [0,1]x[0,1] should be the image of one (or more because of surjectivity) point(s) in [0,1]. I want pick one of these points. Is there any pre-built package in Matlab doing this?
As far as I know, there aren't pre-built packages in Matlab doing this, but the good news is that the code on wikipedia can be called from MATLAB, and it is as simple as putting together the conversion routine with a gateway function in a xy2d.c file:
#include "mex.h"
// source: https://en.wikipedia.org/wiki/Hilbert_curve
// rotate/flip a quadrant appropriately
void rot(int n, int *x, int *y, int rx, int ry) {
if (ry == 0) {
if (rx == 1) {
*x = n-1 - *x;
*y = n-1 - *y;
}
//Swap x and y
int t = *x;
*x = *y;
*y = t;
}
}
// convert (x,y) to d
int xy2d (int n, int x, int y) {
int rx, ry, s, d=0;
for (s=n/2; s>0; s/=2) {
rx = (x & s) > 0;
ry = (y & s) > 0;
d += s * s * ((3 * rx) ^ ry);
rot(s, &x, &y, rx, ry);
}
return d;
}
/* The gateway function */
void mexFunction( int nlhs, mxArray *plhs[],
int nrhs, const mxArray *prhs[])
{
int n; /* input scalar */
int x; /* input scalar */
int y; /* input scalar */
int *d; /* output scalar */
/* check for proper number of arguments */
if(nrhs!=3) {
mexErrMsgIdAndTxt("MyToolbox:arrayProduct:nrhs","Three inputs required.");
}
if(nlhs!=1) {
mexErrMsgIdAndTxt("MyToolbox:arrayProduct:nlhs","One output required.");
}
/* get the value of the scalar inputs */
n = mxGetScalar(prhs[0]);
x = mxGetScalar(prhs[1]);
y = mxGetScalar(prhs[2]);
/* create the output */
plhs[0] = mxCreateDoubleScalar(xy2d(n,x,y));
/* get a pointer to the output scalar */
d = mxGetPr(plhs[0]);
}
and compile it with mex('xy2d.c').
The above implementation
[...] assumes a square divided into n by n cells, for n a power of 2, with integer coordinates, with (0,0) in the lower left corner, (n-1,n-1) in the upper right corner.
In practice, a discretization step is required before applying the mapping. As in every discretization problem, it is crucial to choose the precision wisely. The snippet below puts everything together.
close all; clear; clc;
% number of random samples
NSAMPL = 100;
% unit square divided into n-by-n cells
% has to be a power of 2
n = 2^2;
% quantum
d = 1/n;
N = 0:d:1;
% generate random samples
x = rand(1,NSAMPL);
y = rand(1,NSAMPL);
% discretization
bX = floor(x/d);
bY = floor(y/d);
% 2d to 1d mapping
dd = zeros(1,NSAMPL);
for iid = 1:length(dd)
dd(iid) = xy2d(n, bX(iid), bY(iid));
end
figure;
hold on;
axis equal;
plot(x, y, '.');
plot(repmat([0;1], 1, length(N)), repmat(N, 2, 1), '-r');
plot(repmat(N, 2, 1), repmat([0;1], 1, length(N)), '-r');
figure;
plot(1:NSAMPL, dd);
xlabel('# of sample')
I am using MATLAB for image processing and have images with segmented regions. Here is an example image:
http://www.mathworks.de/help/releases/R2013b/images/examples/ipexroundness_04.png
How can I find the minimum distance from one region to the closest neighbour region? I don't need full implementation but can anyone refer an algorithm which finds the closest neighbour regions and caluclates the minimum distance and the closest points of the region.
I want to use this information to connect the image regions, i.e. build bridges between image regions.
Off the top of my head, the quick-and-dirty nested loop approach:
For region ii=1:n, run the mask of region ii through bwdist to generate a distance transform. For regions jj=1:n, use the mask of region jj to index into that distance transform. That gets you the distances of every pixel in region jj from their closest counterpart in region ii - find the minimum value (and its coordinates) and stuff it in some kind of pairwise distance matrix. Repeat until done, then process the pairwise distance matrix to work out which regions you want to connect.
Edit: Having got that far, I figured I might as well knock something together - here's a rough version that takes a binary image and returns the distances, x and y coordinates of the potential connection points as pairwise matrices:
function [d x y] = regiondist(img)
label = bwlabel(img);
n = max(label(:));
[x y d] = deal(zeros(n));
for ii = 1:n
dt = bwdist(label == ii);
for jj = 1:n
if ii == jj
continue
end
reg = (label == jj);
[mindist idx] = min(dt(reg));
d(ii, jj) = mindist;
[ry rx] = find(reg);
x(ii, jj) = rx(idx);
y(ii, jj) = ry(idx);
end
end
I need to calculate length of the object in a binary image (maximum distance between the pixels inside the object). As it is a binary image, so we might consider it a 2D array with values 0 (white) and 1 (black). The thing I need is a clever (and preferably simple) algorithm to perform this operation. Keep in mind there are many objects in the image.
The image to clarify:
Sample input image:
I think the problem is simple if the boundary of an object is convex and no three vertices are on a line (i.e. no vertex can be removed without changing the polygon): Then you can just pick two points at random and use a simple gradient-descent type search to find the longest line:
Start with random vertices A, B
See if the line A' - B is longer than A - B where A' is the point left of A; if so, replace A with A'
See if the line A' - B is longer than A - B where A' is the point right of A; if so, replace A with A'
Do the same for B
repeat until convergence
So I'd suggest finding the convex hull for each seed blob, removing all "superfluos" vertices (to ensure convergence) and running the algorithm above.
Constructing a convex hull is an O(n log n) operation IIRC, where n is the number of boundary pixels. Should be pretty efficient for small objects like these. EDIT: I just remembered that the O(n log n) for the convex hull algorithm was needed to sort the points. If the boundary points are the result of a connected component analysis, they are already sorted. So the whole algorithm should run in O(n) time, where n is the number of boundary points. (It's a lot of work, though, because you might have to write your own convex-hull algorithm or modify one to skip the sort.)
Add: Response to comment
If you don't need 100% accuracy, you could simply fit an ellipse to each blob and calculate the length of the major axis: This can be computed from central moments (IIRC it's simply the square root if the largest eigenvalue of the covariance matrix), so it's an O(n) operation and can efficiently be calculated in a single sweep over the image. It has the additional advantage that it takes all pixels of a blob into account, not just two extremal points, i.e. it is far less affected by noise.
Find the major-axis length of the ellipse that has the same normalized second central moments as the region. In MATLAB you can use regionprops.
A very crude, brute-force approach would be to first identify all the edge pixels (any black pixel in the object adjacent to a non-black pixel) and calculate the distances between all possible pairs of edge pixels. The longest of these distances will give you the length of the object.
If the objects are always shaped like the ones in your sample, you could speed this up by only evaluating the pixels with the highest and lowest x and y values within the object.
I would suggest trying an "reverse" distance transform. In the magical world of mathematical morphology (sorry couldn't resist the alliteration) the distance transform gives you the closest distance of each pixel to its nearest boundary pixel. In your case, you are interested in the farthest distance to a boundary pixel, hence I have cleverly applied a "reverse" prefix.
You can find information on the distance transform here and here. I believe that matlab implements the distance transform as per here. That would lead me to believe that you can find an open source implementation of the distance transform in octave. Furthermore, it would not surprise me in the least if opencv implemented it.
I haven't given it much thought but its intuitive to me that you should be able to reverse the distance transform and calculate it in roughly the same amount of time as the original distance transform.
I think you could consider using a breadth first search algorithm.
The basic idea is that you loop over each row and column in the image, and if you haven't visited the node (a node is a row and column with a colored pixel) yet, then you would run the breadth first search. You would visit each node you possibly could, and keep track of the max and min points for the object.
Here's some C++ sample code (untested):
#include <vector>
#include <queue>
#include <cmath>
using namespace std;
// used to transition from given row, col to each of the
// 8 different directions
int dr[] = { -1, 0, 1, -1, 1, -1, 0, 1 };
int dc[] = { -1, -1, -1, 0, 0, 1, 1, 1 };
// WHITE or COLORED cells
const int WHITE = 0;
const int COLORED = 1;
// number of rows and columns
int nrows = 2000;
int ncols = 2000;
// assume G is the image
int G[2000][2000];
// the "visited array"
bool vis[2000][2000];
// get distance between 2 points
inline double getdist(double x1, double y1, double x2, double y2) {
double d1 = x1 - x2;
double d2 = y1 - y2;
return sqrt(d1*d1+d2*d2);
}
// this function performs the breadth first search
double bfs(int startRow, int startCol) {
queue< int > q;
q.push(startRow);
q.push(startCol);
vector< pair< int, int > > points;
while(!q.empty()) {
int r = q.front();
q.pop();
int c = q.front();
q.pop();
// already visited?
if (vis[r][c])
continue;
points.push_back(make_pair(r,c));
vis[r][c] = true;
// try all eight directions
for(int i = 0; i < 8; ++i) {
int nr = r + dr[i];
int nc = c + dc[i];
if (nr < 0 || nr >= nrows || nc < 0 || nc >= ncols)
continue; // out of bounds
// push next node on queue
q.push(nr);
q.push(nc);
}
}
// the distance is maximum difference between any 2 points encountered in the BFS
double diff = 0;
for(int i = 0; i < (int)points.size(); ++i) {
for(int j = i+1; j < (int)points.size(); ++j) {
diff = max(diff,getdist(points[i].first,points[i].second,points[j].first,points[j].second));
}
}
return diff;
}
int main() {
vector< double > lengths;
memset(vis,false,sizeof vis);
for(int r = 0; r < nrows; ++r) {
for(int c = 0; c < ncols; ++c) {
if (G[r][c] == WHITE)
continue; // we don't care about cells without objects
if (vis[r][c])
continue; // we've already processed this object
// find the length of this object
double len = bfs(r,c);
lengths.push_back(len);
}
}
return 0;
}
What's the algorithm for computing a least squares plane in (x, y, z) space, given a set of 3D data points? In other words, if I had a bunch of points like (1, 2, 3), (4, 5, 6), (7, 8, 9), etc., how would one go about calculating the best fit plane f(x, y) = ax + by + c? What's the algorithm for getting a, b, and c out of a set of 3D points?
If you have n data points (x[i], y[i], z[i]), compute the 3x3 symmetric matrix A whose entries are:
sum_i x[i]*x[i], sum_i x[i]*y[i], sum_i x[i]
sum_i x[i]*y[i], sum_i y[i]*y[i], sum_i y[i]
sum_i x[i], sum_i y[i], n
Also compute the 3 element vector b:
{sum_i x[i]*z[i], sum_i y[i]*z[i], sum_i z[i]}
Then solve Ax = b for the given A and b. The three components of the solution vector are the coefficients to the least-square fit plane {a,b,c}.
Note that this is the "ordinary least squares" fit, which is appropriate only when z is expected to be a linear function of x and y. If you are looking more generally for a "best fit plane" in 3-space, you may want to learn about "geometric" least squares.
Note also that this will fail if your points are in a line, as your example points are.
The equation for a plane is: ax + by + c = z. So set up matrices like this with all your data:
x_0 y_0 1
A = x_1 y_1 1
...
x_n y_n 1
And
a
x = b
c
And
z_0
B = z_1
...
z_n
In other words: Ax = B. Now solve for x which are your coefficients. But since (I assume) you have more than 3 points, the system is over-determined so you need to use the left pseudo inverse. So the answer is:
a
b = (A^T A)^-1 A^T B
c
And here is some simple Python code with an example:
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
N_POINTS = 10
TARGET_X_SLOPE = 2
TARGET_y_SLOPE = 3
TARGET_OFFSET = 5
EXTENTS = 5
NOISE = 5
# create random data
xs = [np.random.uniform(2*EXTENTS)-EXTENTS for i in range(N_POINTS)]
ys = [np.random.uniform(2*EXTENTS)-EXTENTS for i in range(N_POINTS)]
zs = []
for i in range(N_POINTS):
zs.append(xs[i]*TARGET_X_SLOPE + \
ys[i]*TARGET_y_SLOPE + \
TARGET_OFFSET + np.random.normal(scale=NOISE))
# plot raw data
plt.figure()
ax = plt.subplot(111, projection='3d')
ax.scatter(xs, ys, zs, color='b')
# do fit
tmp_A = []
tmp_b = []
for i in range(len(xs)):
tmp_A.append([xs[i], ys[i], 1])
tmp_b.append(zs[i])
b = np.matrix(tmp_b).T
A = np.matrix(tmp_A)
fit = (A.T * A).I * A.T * b
errors = b - A * fit
residual = np.linalg.norm(errors)
print("solution:")
print("%f x + %f y + %f = z" % (fit[0], fit[1], fit[2]))
print("errors:")
print(errors)
print("residual:")
print(residual)
# plot plane
xlim = ax.get_xlim()
ylim = ax.get_ylim()
X,Y = np.meshgrid(np.arange(xlim[0], xlim[1]),
np.arange(ylim[0], ylim[1]))
Z = np.zeros(X.shape)
for r in range(X.shape[0]):
for c in range(X.shape[1]):
Z[r,c] = fit[0] * X[r,c] + fit[1] * Y[r,c] + fit[2]
ax.plot_wireframe(X,Y,Z, color='k')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
plt.show()
unless someone tells me how to type equations here, let me just write down the final computations you have to do:
first, given points r_i \n \R, i=1..N, calculate the center of mass of all points:
r_G = \frac{\sum_{i=1}^N r_i}{N}
then, calculate the normal vector n, that together with the base vector r_G defines the plane by calculating the 3x3 matrix A as
A = \sum_{i=1}^N (r_i - r_G)(r_i - r_G)^T
with this matrix, the normal vector n is now given by the eigenvector of A corresponding to the minimal eigenvalue of A.
To find out about the eigenvector/eigenvalue pairs, use any linear algebra library of your choice.
This solution is based on the Rayleight-Ritz Theorem for the Hermitian matrix A.
See 'Least Squares Fitting of Data' by David Eberly for how I came up with this one to minimize the geometric fit (orthogonal distance from points to the plane).
bool Geom_utils::Fit_plane_direct(const arma::mat& pts_in, Plane& plane_out)
{
bool success(false);
int K(pts_in.n_cols);
if(pts_in.n_rows == 3 && K > 2) // check for bad sizing and indeterminate case
{
plane_out._p_3 = (1.0/static_cast<double>(K))*arma::sum(pts_in,1);
arma::mat A(pts_in);
A.each_col() -= plane_out._p_3; //[x1-p, x2-p, ..., xk-p]
arma::mat33 M(A*A.t());
arma::vec3 D;
arma::mat33 V;
if(arma::eig_sym(D,V,M))
{
// diagonalization succeeded
plane_out._n_3 = V.col(0); // in ascending order by default
if(plane_out._n_3(2) < 0)
{
plane_out._n_3 = -plane_out._n_3; // upward pointing
}
success = true;
}
}
return success;
}
Timed at 37 micro seconds fitting a plane to 1000 points (Windows 7, i7, 32bit program)
This reduces to the Total Least Squares problem, that can be solved using SVD decomposition.
C++ code using OpenCV:
float fitPlaneToSetOfPoints(const std::vector<cv::Point3f> &pts, cv::Point3f &p0, cv::Vec3f &nml) {
const int SCALAR_TYPE = CV_32F;
typedef float ScalarType;
// Calculate centroid
p0 = cv::Point3f(0,0,0);
for (int i = 0; i < pts.size(); ++i)
p0 = p0 + conv<cv::Vec3f>(pts[i]);
p0 *= 1.0/pts.size();
// Compose data matrix subtracting the centroid from each point
cv::Mat Q(pts.size(), 3, SCALAR_TYPE);
for (int i = 0; i < pts.size(); ++i) {
Q.at<ScalarType>(i,0) = pts[i].x - p0.x;
Q.at<ScalarType>(i,1) = pts[i].y - p0.y;
Q.at<ScalarType>(i,2) = pts[i].z - p0.z;
}
// Compute SVD decomposition and the Total Least Squares solution, which is the eigenvector corresponding to the least eigenvalue
cv::SVD svd(Q, cv::SVD::MODIFY_A|cv::SVD::FULL_UV);
nml = svd.vt.row(2);
// Calculate the actual RMS error
float err = 0;
for (int i = 0; i < pts.size(); ++i)
err += powf(nml.dot(pts[i] - p0), 2);
err = sqrtf(err / pts.size());
return err;
}
As with any least-squares approach, you proceed like this:
Before you start coding
Write down an equation for a plane in some parameterization, say 0 = ax + by + z + d in thee parameters (a, b, d).
Find an expression D(\vec{v};a, b, d) for the distance from an arbitrary point \vec{v}.
Write down the sum S = \sigma_i=0,n D^2(\vec{x}_i), and simplify until it is expressed in terms of simple sums of the components of v like \sigma v_x, \sigma v_y^2, \sigma v_x*v_z ...
Write down the per parameter minimization expressions dS/dx_0 = 0, dS/dy_0 = 0 ... which gives you a set of three equations in three parameters and the sums from the previous step.
Solve this set of equations for the parameters.
(or for simple cases, just look up the form). Using a symbolic algebra package (like Mathematica) could make you life much easier.
The coding
Write code to form the needed sums and find the parameters from the last set above.
Alternatives
Note that if you actually had only three points, you'd be better just finding the plane that goes through them.
Also, if the analytic solution in unfeasible (not the case for a plane, but possible in general) you can do steps 1 and 2, and use a Monte Carlo minimizer on the sum in step 3.
CGAL::linear_least_squares_fitting_3
Function linear_least_squares_fitting_3 computes the best fitting 3D
line or plane (in the least squares sense) of a set of 3D objects such
as points, segments, triangles, spheres, balls, cuboids or tetrahedra.
http://www.cgal.org/Manual/latest/doc_html/cgal_manual/Principal_component_analysis_ref/Function_linear_least_squares_fitting_3.html
It sounds like all you want to do is linear regression with 2 regressors. The wikipedia page on the subject should tell you all you need to know and then some.
All you'll have to do is to solve the system of equations.
If those are your points:
(1, 2, 3), (4, 5, 6), (7, 8, 9)
That gives you the equations:
3=a*1 + b*2 + c
6=a*4 + b*5 + c
9=a*7 + b*8 + c
So your question actually should be: How do I solve a system of equations?
Therefore I recommend reading this SO question.
If I've misunderstood your question let us know.
EDIT:
Ignore my answer as you probably meant something else.
We first present a linear least-squares plane fitting method that minimizes the residuals between the estimated normal vector and provided points.
Recall that the equation for a plane passing through origin is Ax + By + Cz = 0, where (x, y, z) can be any point on the plane and (A, B, C) is the normal vector perpendicular to this plane.
The equation for a general plane (that may or may not pass through origin) is Ax + By + Cz + D = 0, where the additional coefficient D represents how far the plane is away from the origin, along the direction of the normal vector of the plane. [Note that in this equation (A, B, C) forms a unit normal vector.]
Now, we can apply a trick here and fit the plane using only provided point coordinates. Divide both sides by D and rearrange this term to the right-hand side. This leads to A/D x + B/D y + C/D z = -1. [Note that in this equation (A/D, B/D, C/D) forms a normal vector with length 1/D.]
We can set up a system of linear equations accordingly, and then solve it by an Eigen solver in C++ as follows.
// Example for 5 points
Eigen::Matrix<double, 5, 3> matA; // row: 5 points; column: xyz coordinates
Eigen::Matrix<double, 5, 1> matB = -1 * Eigen::Matrix<double, 5, 1>::Ones();
// Find the plane normal
Eigen::Vector3d normal = matA.colPivHouseholderQr().solve(matB);
// Check if the fitting is healthy
double D = 1 / normal.norm();
normal.normalize(); // normal is a unit vector from now on
bool planeValid = true;
for (int i = 0; i < 5; ++i) { // compare Ax + By + Cz + D with 0.2 (ideally Ax + By + Cz + D = 0)
if ( fabs( normal(0)*matA(i, 0) + normal(1)*matA(i, 1) + normal(2)*matA(i, 2) + D) > 0.2) {
planeValid = false; // 0.2 is an experimental threshold; can be tuned
break;
}
}
We then discuss its equivalence to the typical SVD-based method and their comparison.
The aforementioned linear least-squares (LLS) method fits the general plane equation Ax + By + Cz + D = 0, whereas the SVD-based method replaces D with D = - (Ax0 + By0 + Cz0) and fits the plane equation A(x-x0) + B(y-y0) + C(z-z0) = 0, where (x0, y0, z0) is the mean of all points that serves as the origin of the new local coordinate frame.
Comparison between two methods:
The LLS fitting method is much faster than the SVD-based method, and is suitable for use when points are known to be roughly in a plane shape.
The SVD-based method is more numerically stable when the plane is far away from origin, because the LLS method would require more digits after decimal to be stored and processed in such cases.
The LLS method can detect outliers by checking the dot product residual between each point and the estimated normal vector, whereas the SVD-based method can detect outliers by checking if the smallest eigenvalue of the covariance matrix is significantly smaller than the two larger eigenvalues (i.e. checking the shape of the covariance matrix).
We finally provide a test case in C++ and MATLAB.
// Test case in C++ (using LLS fitting method)
matA(0,0) = 5.4637; matA(0,1) = 10.3354; matA(0,2) = 2.7203;
matA(1,0) = 5.8038; matA(1,1) = 10.2393; matA(1,2) = 2.7354;
matA(2,0) = 5.8565; matA(2,1) = 10.2520; matA(2,2) = 2.3138;
matA(3,0) = 6.0405; matA(3,1) = 10.1836; matA(3,2) = 2.3218;
matA(4,0) = 5.5537; matA(4,1) = 10.3349; matA(4,2) = 1.8796;
// With this sample data, LLS fitting method can produce the following result
// fitted normal vector = (-0.0231143, -0.0838307, -0.00266429)
// unit normal vector = (-0.265682, -0.963574, -0.0306241)
// D = 11.4943
% Test case in MATLAB (using SVD-based method)
points = [5.4637 10.3354 2.7203;
5.8038 10.2393 2.7354;
5.8565 10.2520 2.3138;
6.0405 10.1836 2.3218;
5.5537 10.3349 1.8796]
covariance = cov(points)
[V, D] = eig(covariance)
normal = V(:, 1) % pick the eigenvector that corresponds to the smallest eigenvalue
% normal = (0.2655, 0.9636, 0.0306)
From the man page for XFillPolygon:
If shape is Complex, the path may self-intersect. Note that contiguous coincident points in the path are not treated as self-intersection.
If shape is Convex, for every pair of points inside the polygon, the line segment connecting them does not intersect the path. If known by the client, specifying Convex can improve performance. If you specify Convex for a path that is not convex, the graphics results are undefined.
If shape is Nonconvex, the path does not self-intersect, but the shape is not wholly convex. If known by the client, specifying Nonconvex instead of Complex may improve performance. If you specify Nonconvex for a self-intersecting path, the graphics results are undefined.
I am having performance problems with fill XFillPolygon and, as the man page suggests, the first step I want to take is to specify the correct shape of the polygon. I am currently using Complex to be on the safe side.
Is there an efficient algorithm to determine if a polygon (defined by a series of coordinates) is convex, non-convex or complex?
You can make things a lot easier than the Gift-Wrapping Algorithm... that's a good answer when you have a set of points w/o any particular boundary and need to find the convex hull.
In contrast, consider the case where the polygon is not self-intersecting, and it consists of a set of points in a list where the consecutive points form the boundary. In this case it is much easier to figure out whether a polygon is convex or not (and you don't have to calculate any angles, either):
For each consecutive pair of edges of the polygon (each triplet of points), compute the z-component of the cross product of the vectors defined by the edges pointing towards the points in increasing order. Take the cross product of these vectors:
given p[k], p[k+1], p[k+2] each with coordinates x, y:
dx1 = x[k+1]-x[k]
dy1 = y[k+1]-y[k]
dx2 = x[k+2]-x[k+1]
dy2 = y[k+2]-y[k+1]
zcrossproduct = dx1*dy2 - dy1*dx2
The polygon is convex if the z-components of the cross products are either all positive or all negative. Otherwise the polygon is nonconvex.
If there are N points, make sure you calculate N cross products, e.g. be sure to use the triplets (p[N-2],p[N-1],p[0]) and (p[N-1],p[0],p[1]).
If the polygon is self-intersecting, then it fails the technical definition of convexity even if its directed angles are all in the same direction, in which case the above approach would not produce the correct result.
This question is now the first item in either Bing or Google when you search for "determine convex polygon." However, none of the answers are good enough.
The (now deleted) answer by #EugeneYokota works by checking whether an unordered set of points can be made into a convex polygon, but that's not what the OP asked for. He asked for a method to check whether a given polygon is convex or not. (A "polygon" in computer science is usually defined [as in the XFillPolygon documentation] as an ordered array of 2D points, with consecutive points joined with a side as well as the last point to the first.) Also, the gift wrapping algorithm in this case would have the time-complexity of O(n^2) for n points - which is much larger than actually needed to solve this problem, while the question asks for an efficient algorithm.
#JasonS's answer, along with the other answers that follow his idea, accepts star polygons such as a pentagram or the one in #zenna's comment, but star polygons are not considered to be convex. As
#plasmacel notes in a comment, this is a good approach to use if you have prior knowledge that the polygon is not self-intersecting, but it can fail if you do not have that knowledge.
#Sekhat's answer is correct but it also has the time-complexity of O(n^2) and thus is inefficient.
#LorenPechtel's added answer after her edit is the best one here but it is vague.
A correct algorithm with optimal complexity
The algorithm I present here has the time-complexity of O(n), correctly tests whether a polygon is convex or not, and passes all the tests I have thrown at it. The idea is to traverse the sides of the polygon, noting the direction of each side and the signed change of direction between consecutive sides. "Signed" here means left-ward is positive and right-ward is negative (or the reverse) and straight-ahead is zero. Those angles are normalized to be between minus-pi (exclusive) and pi (inclusive). Summing all these direction-change angles (a.k.a the deflection angles) together will result in plus-or-minus one turn (i.e. 360 degrees) for a convex polygon, while a star-like polygon (or a self-intersecting loop) will have a different sum ( n * 360 degrees, for n turns overall, for polygons where all the deflection angles are of the same sign). So we must check that the sum of the direction-change angles is plus-or-minus one turn. We also check that the direction-change angles are all positive or all negative and not reverses (pi radians), all points are actual 2D points, and that no consecutive vertices are identical. (That last point is debatable--you may want to allow repeated vertices but I prefer to prohibit them.) The combination of those checks catches all convex and non-convex polygons.
Here is code for Python 3 that implements the algorithm and includes some minor efficiencies. The code looks longer than it really is due to the the comment lines and the bookkeeping involved in avoiding repeated point accesses.
TWO_PI = 2 * pi
def is_convex_polygon(polygon):
"""Return True if the polynomial defined by the sequence of 2D
points is 'strictly convex': points are valid, side lengths non-
zero, interior angles are strictly between zero and a straight
angle, and the polygon does not intersect itself.
NOTES: 1. Algorithm: the signed changes of the direction angles
from one side to the next side must be all positive or
all negative, and their sum must equal plus-or-minus
one full turn (2 pi radians). Also check for too few,
invalid, or repeated points.
2. No check is explicitly done for zero internal angles
(180 degree direction-change angle) as this is covered
in other ways, including the `n < 3` check.
"""
try: # needed for any bad points or direction changes
# Check for too few points
if len(polygon) < 3:
return False
# Get starting information
old_x, old_y = polygon[-2]
new_x, new_y = polygon[-1]
new_direction = atan2(new_y - old_y, new_x - old_x)
angle_sum = 0.0
# Check each point (the side ending there, its angle) and accum. angles
for ndx, newpoint in enumerate(polygon):
# Update point coordinates and side directions, check side length
old_x, old_y, old_direction = new_x, new_y, new_direction
new_x, new_y = newpoint
new_direction = atan2(new_y - old_y, new_x - old_x)
if old_x == new_x and old_y == new_y:
return False # repeated consecutive points
# Calculate & check the normalized direction-change angle
angle = new_direction - old_direction
if angle <= -pi:
angle += TWO_PI # make it in half-open interval (-Pi, Pi]
elif angle > pi:
angle -= TWO_PI
if ndx == 0: # if first time through loop, initialize orientation
if angle == 0.0:
return False
orientation = 1.0 if angle > 0.0 else -1.0
else: # if other time through loop, check orientation is stable
if orientation * angle <= 0.0: # not both pos. or both neg.
return False
# Accumulate the direction-change angle
angle_sum += angle
# Check that the total number of full turns is plus-or-minus 1
return abs(round(angle_sum / TWO_PI)) == 1
except (ArithmeticError, TypeError, ValueError):
return False # any exception means not a proper convex polygon
The following Java function/method is an implementation of the algorithm described in this answer.
public boolean isConvex()
{
if (_vertices.size() < 4)
return true;
boolean sign = false;
int n = _vertices.size();
for(int i = 0; i < n; i++)
{
double dx1 = _vertices.get((i + 2) % n).X - _vertices.get((i + 1) % n).X;
double dy1 = _vertices.get((i + 2) % n).Y - _vertices.get((i + 1) % n).Y;
double dx2 = _vertices.get(i).X - _vertices.get((i + 1) % n).X;
double dy2 = _vertices.get(i).Y - _vertices.get((i + 1) % n).Y;
double zcrossproduct = dx1 * dy2 - dy1 * dx2;
if (i == 0)
sign = zcrossproduct > 0;
else if (sign != (zcrossproduct > 0))
return false;
}
return true;
}
The algorithm is guaranteed to work as long as the vertices are ordered (either clockwise or counter-clockwise), and you don't have self-intersecting edges (i.e. it only works for simple polygons).
Here's a test to check if a polygon is convex.
Consider each set of three points along the polygon--a vertex, the vertex before, the vertex after. If every angle is 180 degrees or less you have a convex polygon. When you figure out each angle, also keep a running total of (180 - angle). For a convex polygon, this will total 360.
This test runs in O(n) time.
Note, also, that in most cases this calculation is something you can do once and save — most of the time you have a set of polygons to work with that don't go changing all the time.
To test if a polygon is convex, every point of the polygon should be level with or behind each line.
Here's an example picture:
The answer by #RoryDaulton
seems the best to me, but what if one of the angles is exactly 0?
Some may want such an edge case to return True, in which case, change "<=" to "<" in the line :
if orientation * angle < 0.0: # not both pos. or both neg.
Here are my test cases which highlight the issue :
# A square
assert is_convex_polygon( ((0,0), (1,0), (1,1), (0,1)) )
# This LOOKS like a square, but it has an extra point on one of the edges.
assert is_convex_polygon( ((0,0), (0.5,0), (1,0), (1,1), (0,1)) )
The 2nd assert fails in the original answer. Should it?
For my use case, I would prefer it didn't.
This method would work on simple polygons (no self intersecting edges) assuming that the vertices are ordered (either clockwise or counter)
For an array of vertices:
vertices = [(0,0),(1,0),(1,1),(0,1)]
The following python implementation checks whether the z component of all the cross products have the same sign
def zCrossProduct(a,b,c):
return (a[0]-b[0])*(b[1]-c[1])-(a[1]-b[1])*(b[0]-c[0])
def isConvex(vertices):
if len(vertices)<4:
return True
signs= [zCrossProduct(a,b,c)>0 for a,b,c in zip(vertices[2:],vertices[1:],vertices)]
return all(signs) or not any(signs)
I implemented both algorithms: the one posted by #UriGoren (with a small improvement - only integer math) and the one from #RoryDaulton, in Java. I had some problems because my polygon is closed, so both algorithms were considering the second as concave, when it was convex. So i changed it to prevent such situation. My methods also uses a base index (which can be or not 0).
These are my test vertices:
// concave
int []x = {0,100,200,200,100,0,0};
int []y = {50,0,50,200,50,200,50};
// convex
int []x = {0,100,200,100,0,0};
int []y = {50,0,50,200,200,50};
And now the algorithms:
private boolean isConvex1(int[] x, int[] y, int base, int n) // Rory Daulton
{
final double TWO_PI = 2 * Math.PI;
// points is 'strictly convex': points are valid, side lengths non-zero, interior angles are strictly between zero and a straight
// angle, and the polygon does not intersect itself.
// NOTES: 1. Algorithm: the signed changes of the direction angles from one side to the next side must be all positive or
// all negative, and their sum must equal plus-or-minus one full turn (2 pi radians). Also check for too few,
// invalid, or repeated points.
// 2. No check is explicitly done for zero internal angles(180 degree direction-change angle) as this is covered
// in other ways, including the `n < 3` check.
// needed for any bad points or direction changes
// Check for too few points
if (n <= 3) return true;
if (x[base] == x[n-1] && y[base] == y[n-1]) // if its a closed polygon, ignore last vertex
n--;
// Get starting information
int old_x = x[n-2], old_y = y[n-2];
int new_x = x[n-1], new_y = y[n-1];
double new_direction = Math.atan2(new_y - old_y, new_x - old_x), old_direction;
double angle_sum = 0.0, orientation=0;
// Check each point (the side ending there, its angle) and accum. angles for ndx, newpoint in enumerate(polygon):
for (int i = 0; i < n; i++)
{
// Update point coordinates and side directions, check side length
old_x = new_x; old_y = new_y; old_direction = new_direction;
int p = base++;
new_x = x[p]; new_y = y[p];
new_direction = Math.atan2(new_y - old_y, new_x - old_x);
if (old_x == new_x && old_y == new_y)
return false; // repeated consecutive points
// Calculate & check the normalized direction-change angle
double angle = new_direction - old_direction;
if (angle <= -Math.PI)
angle += TWO_PI; // make it in half-open interval (-Pi, Pi]
else if (angle > Math.PI)
angle -= TWO_PI;
if (i == 0) // if first time through loop, initialize orientation
{
if (angle == 0.0) return false;
orientation = angle > 0 ? 1 : -1;
}
else // if other time through loop, check orientation is stable
if (orientation * angle <= 0) // not both pos. or both neg.
return false;
// Accumulate the direction-change angle
angle_sum += angle;
// Check that the total number of full turns is plus-or-minus 1
}
return Math.abs(Math.round(angle_sum / TWO_PI)) == 1;
}
And now from Uri Goren
private boolean isConvex2(int[] x, int[] y, int base, int n)
{
if (n < 4)
return true;
boolean sign = false;
if (x[base] == x[n-1] && y[base] == y[n-1]) // if its a closed polygon, ignore last vertex
n--;
for(int p=0; p < n; p++)
{
int i = base++;
int i1 = i+1; if (i1 >= n) i1 = base + i1-n;
int i2 = i+2; if (i2 >= n) i2 = base + i2-n;
int dx1 = x[i1] - x[i];
int dy1 = y[i1] - y[i];
int dx2 = x[i2] - x[i1];
int dy2 = y[i2] - y[i1];
int crossproduct = dx1*dy2 - dy1*dx2;
if (i == base)
sign = crossproduct > 0;
else
if (sign != (crossproduct > 0))
return false;
}
return true;
}
For a non complex (intersecting) polygon to be convex, vector frames obtained from any two connected linearly independent lines a,b must be point-convex otherwise the polygon is concave.
For example the lines a,b are convex to the point p and concave to it below for each case i.e. above: p exists inside a,b and below: p exists outside a,b
Similarly for each polygon below, if each line pair making up a sharp edge is point-convex to the centroid c then the polygon is convex otherwise it’s concave.
blunt edges (wronged green) are to be ignored
N.B
This approach would require you compute the centroid of your polygon beforehand since it doesn’t employ angles but vector algebra/transformations
Adapted Uri's code into matlab. Hope this may help.
Be aware that Uri's algorithm only works for simple polygons! So, be sure to test if the polygon is simple first!
% M [ x1 x2 x3 ...
% y1 y2 y3 ...]
% test if a polygon is convex
function ret = isConvex(M)
N = size(M,2);
if (N<4)
ret = 1;
return;
end
x0 = M(1, 1:end);
x1 = [x0(2:end), x0(1)];
x2 = [x0(3:end), x0(1:2)];
y0 = M(2, 1:end);
y1 = [y0(2:end), y0(1)];
y2 = [y0(3:end), y0(1:2)];
dx1 = x2 - x1;
dy1 = y2 - y1;
dx2 = x0 - x1;
dy2 = y0 - y1;
zcrossproduct = dx1 .* dy2 - dy1 .* dx2;
% equality allows two consecutive edges to be parallel
t1 = sum(zcrossproduct >= 0);
t2 = sum(zcrossproduct <= 0);
ret = t1 == N || t2 == N;
end