Two Material Circular Mesh - Gmsh - numeric

I have the following .geo for GMSH:
cl__1 = 1.125;
Point(1) = {0, 0, 0, cl__1};
Point(2) = {1, 0, 0, cl__1};
Point(3) = {1, 1, 0, cl__1};
Point(4) = {0, 1, 0, cl__1};
Point(5) = {0.350000,0.100000,cl__1};
Point(6) = {0.350000,0.350000,cl__1};
Point(7) = {0.100000,0.350000,cl__1};
Point(8) = {0.85,0.85,cl__1};
Line(1) = {1, 2};
Line(2) = {2, 3};
Line(3) = {3, 4};
Line(4) = {4, 1};
Circle(5) = {5,6,7};
Line(6) = {7,8};
Line(7) = {8,5};
Line Loop(1) = {1, 2, 3, 4};
Line Loop(2) = {5,6,7};
Plane Surface(1) = {1,2};
Plane Surface(2) = {2};
Circle {5} In Surface {2};
Line {6} In Surface {2};
Line {7} In Surface {2};
I have drawn exactly the geometry I want to build but somehow Gmsh is only meshing inside Plane Surface 2.
Is there anything wrong in this code?

The point definition for points 5 - 8 are not in plane z = 0 they are in plane (z = cl_1). A suggestion, when you write this kind of files try to arrange things in the following way:
cl_1 = 1.125;
Point(1) = {0 , 0 , 0, cl_1};
Point(2) = {1 , 0 , 0, cl_1};
Point(3) = {1 , 1 , 0, cl_1};
Point(4) = {0 , 1 , 0, cl_1};
Point(5) = {0.350000, 0.100000, 0, cl_1};
Point(6) = {0.350000, 0.350000, 0, cl_1};
Point(7) = {0.100000, 0.350000, 0, cl_1};
Point(8) = {0.85 , 0.85 , 0, cl_1};
Line(1) = {1, 2};
Line(2) = {2, 3};
Line(3) = {3, 4};
Line(4) = {4, 1};
Circle(5) = {5,6,7};
Line(6) = {7,8};
Line(7) = {8,5};
Line Loop(1) = {1, 2, 3, 4};
Line Loop(2) = {5,6,7};
Plane Surface(8) = {1, 2};
Plane Surface(9) = {2};
you can save lot of time only by inspection.

Related

plot grid not visible in jupyter using matplotlib

Can someone help me identify why the gridlines are missing even though plt.grid() is already added in the code please. Thank you.
km = KMeans(n_clusters = 3, init = 'k-means++', max_iter = 300, n_init = 10, random_state = 0)
y_means = km.fit_predict(x)
plt.scatter(x[y_means == 0, 0], x[y_means == 0, 1], s = 100, c = 'pink', label = 'Uninterested Customers')
plt.scatter(x[y_means == 1, 0], x[y_means == 1, 1], s = 100, c = 'yellow', label = 'General Customers')
plt.scatter(x[y_means == 2, 0], x[y_means == 2, 1], s = 100, c = 'cyan', label = 'Target Customers')
plt.scatter(km.cluster_centers_[:,0], km.cluster_centers_[:, 1], s = 50, c = 'blue' , label = 'centeroid')
plt.title('Administrative Duration vs Duration', fontsize = 20)
plt.grid()
plt.xlabel('Administrative Duration')
plt.ylabel('Bounce Rates')
plt.legend()
plt.show()

Extract PhysicalFace as 2D mesh in fipy

I created a mesh with Gmsh (a surface with a hole and then extruded it). Now I would like to plot the model in individual slices after a simulation with e.g. the MatplotlibViewer (Mayavi does not work on both my computers). I had hoped that it would be possible to define a new net using mesh.physicalFaces, but if that is possible, I have not figured it out yet. My second attempt was to apply the mesh again with Gmsh up to the Extrude command. But the mesh corresponds not to that of the 3D version. Can somebody give me a clue on this? Also like just another approach to representation.
I am working on Win10, Fipy 3.1.3, Python 3.6
import numpy as np
from fipy import *
#%%
def func_mesh():
mesh = Gmsh3D('''
Geometry.OCCAutoFix = 0;
SetFactory("OpenCASCADE");
x = 1.;
bseg = 0.08;
bs= bseg*x;
ls = 2.1;
cl = 0.01;
radius = 0.006;
// Exterior (bounding box) of mesh
Point(1) = {0, 0, 0, cl};
Point(2) = {0, bs, 0, cl};
Point(4) = { bs, 0, 0, cl};
Point(3) = {bs, bs, 0, cl};
Line(1) = {1, 2};
Line(2) = {2, 3};
Line(3) = {3, 4};
Line(4) = {4, 1};
Line Loop (21) = {1,2,3,4};
//Circle
Point(5) = {bseg/2 - radius, bseg/2, 0, cl};
Point(6) = {bseg/2, bseg/2 + radius, 0, cl};
Point(7) = { bseg/2 + radius, bseg/2, 0, cl};
Point(8) = {bseg/2, bseg/2 - radius, 0, cl};
Point(9) = {bseg/2, bseg/2, 0, cl};
Circle(10) = {5,9,6};
Circle(11) = {6,9,7};
Circle(12) = {7,9,8};
Circle(13) = {8,9,5};
Line Loop(22) = {10,11,12,13};
Plane Surface(40) = {22}; //cycl
Plane Surface(15) = {21, 22}; //Surface with a hole
id[] = Extrude {0, 0, ls} {Surface{15}; Layers{210}; Recombine;};
Surface Loop(2) = {46, 45, 48, 47, 49, 41, 44, 43, 42, 15};
Physical Volume("Vol") = {id[]};
Physical Surface("surf_ges") = {41, 42, 43, 44, 49, 47, 45, 48, 46, 15};
Physical Surface("HX") = {45, 46, 48, 47};
Physical Surface("Extr") = {15};
''')
return mesh
mesh = func_mesh()
x,y,z = mesh.cellCenters
X,Y,Z = mesh.faceCenters
tS = CellVariable(name="storage",
mesh=mesh,
value=367.,
hasOld=True)
submesh = mesh.physicalFaces['Extr']
xsub, ysub = submesh.cellCenters
tSslice = CellVariable(name = 'tSsclice',
mesh = submesh,
value = tS[z== z[0]])
viewer = MatplotlibViewer(vars = tSslice)
The error message for this attemp is: AttributeError: 'binOp' object has no attribute 'cellCenters'.
If I redefine the mesh only until the extrude order I get:
"ValueError: too many values to unpack (expected 2)" because of the shape of tSslice.
I am grateful for any help
submesh is not a Mesh; it's a mask identifying which faces of mesh are included in the surface Extr.
FiPy has no facility for extracting a mesh out of another mesh. It should be feasible to create a Mesh2D using the submesh mask, mesh.vertexCoords, and mesh.faceVertexIDs, but it's not trivial.
In theory, you could invoke Gmsh2D with everything up to Plane Surface(15) = {21, 22}; //Surface with a hole, but I find that doesn't generate the same number of elements as your 3D slice at z == z[0].
Ahah, I see the issue. I thought the Extrude operation resulted in prismatic cells, but it does not. The cells are tetrahedral. Since the cells of mesh do not all have the same tetrahedral geometry, the cells that have their bases on Extr are not all guaranteed to have their centers at z == z[0]. A better way is to use FiPy's CellVariable interpolation to extract the values of tS at the coordinates of tSslice:
from fipy import *
geo = '''
Geometry.OCCAutoFix = 0;
SetFactory("OpenCASCADE");
x = 1.;
bseg = 0.08;
bs= bseg*x;
ls = 2.1;
cl = 0.01;
radius = 0.006;
// Exterior (bounding box) of mesh
Point(1) = {0, 0, 0, cl};
Point(2) = {0, bs, 0, cl};
Point(4) = { bs, 0, 0, cl};
Point(3) = {bs, bs, 0, cl};
Line(1) = {1, 2};
Line(2) = {2, 3};
Line(3) = {3, 4};
Line(4) = {4, 1};
Line Loop (21) = {1,2,3,4};
//Circle
Point(5) = {bseg/2 - radius, bseg/2, 0, cl};
Point(6) = {bseg/2, bseg/2 + radius, 0, cl};
Point(7) = { bseg/2 + radius, bseg/2, 0, cl};
Point(8) = {bseg/2, bseg/2 - radius, 0, cl};
Point(9) = {bseg/2, bseg/2, 0, cl};
Circle(10) = {5,9,6};
Circle(11) = {6,9,7};
Circle(12) = {7,9,8};
Circle(13) = {8,9,5};
Line Loop(22) = {10,11,12,13};
Plane Surface(40) = {22}; //cycl
Plane Surface(15) = {21, 22}; //Surface with a hole
id[] = Extrude {0, 0, ls} {Surface{15}; Layers{210}; Recombine;};
Surface Loop(2) = {46, 45, 48, 47, 49, 41, 44, 43, 42, 15};
Physical Volume("Vol") = {id[]};
Physical Surface("Extr") = {15};
'''
mesh = Gmsh3D(geo + '''
Physical Surface("surf_ges") = {41, 42, 43, 44, 49, 47, 45, 48, 46, 15};
Physical Surface("HX") = {45, 46, 48, 47};
'''
)
submesh = Gmsh2D(geo)
x,y,z = mesh.cellCenters
X,Y = submesh.cellCenters[..., submesh.physicalCells['Extr']]
Z = numerix.ones(X.shape) * z[0]
tS = CellVariable(name="storage",
mesh=mesh,
value=mesh.x * mesh.y * mesh.z,
hasOld=True)
tSslice = CellVariable(name = 'tSsclice',
mesh = submesh)
# interpolate values of tS at positions of tSslice
tSslice[..., submesh.physicalCells['Extr']] = tS(numerix.vstack([X, Y, Z]))
viewer = MatplotlibViewer(vars = tSslice)
Here, I use the same .geo script to define both mesh and submesh. I add the surf_ges and HX physical surfaces only to the definition of mesh, because otherwise all of these faces will be imported into submesh as well, although with an effective z value of 0, so they obscure the faces you're interested in.
Frankly, I think a better way to achieve a slice through 3D data like this is to use either a customized MayaviClient (see the Cahn-Hilliard sphere and sphereDaemon.py for an example) or export with a VTKViewer and then render your data slices with a tool like ParaView, VisIt, or Mayavi.

Nodes inconsistency between 2D and 3D elements in GMSH

I am trying to produce a tetrahedral mesh of a cube. The problem I encounter is that a few nodes that are used to generate the triangular faces are then not used to generate any of the tetrahedral elements. Is there a way to avoid this?
This is the code.
cl__1 = 0.01;
Point(1) = {0, 0, 0, cl__1};
Point(2) = {1, 0, 0, cl__1};
Point(3) = {1, 1, 0, cl__1};
Point(4) = {0, 1, 0, cl__1};
Line(1) = {1, 2};
Line(2) = {2, 3};
Line(3) = {3, 4};
Line(4) = {4, 1};
Line Loop(5) = {4, 1, 2, 3};
Plane Surface(6) = {5};
Extrude {0, 0, 1} {
Surface{6};
}
Coherence;
Transfinite Line {1, 2, 3, 4, 9, 10, 11, 8, 18, 22, 13, 14} = 2 Using Progression 1;
Transfinite Surface {19};
Transfinite Surface {23};
Transfinite Surface {27};
Transfinite Surface {15};
Transfinite Surface {28};
Transfinite Surface {6};
Transfinite Volume{1} = {1, 2, 3, 4, 6, 10, 14, 5};
Physical Surface("top") = {28};
Physical Surface("bottom") = {6};
Physical Surface("x_min") = {15};
Physical Surface("x_max") = {23};
Physical Surface("y_min") = {19};
Physical Surface("y_max") = {27};
Physical Volume("bottom_volume") = {1};
Thanks
Jian
The result of that input gives these nodes:
$Nodes
8
1 0 0 0
2 1 0 0
3 1 1 0
4 0 1 0
5 0 1 1
6 0 0 1
7 1 0 1
8 1 1 1
$EndNodes
with these nodes you can define all the triangular faces and the tetrahedral elements that define that mesh (there is no inconsistency in my opinion).

Image center in the calibration matrix

I have a camera and a 3D object, I have calculated the camera matrix as given in this. The image captured using the camera is having size 1600x1200. But in the camera matrix I am not getting the value of image centers properly. Instead of 800, 600 I am getting some other values. What will be the possible reasons for this.
#include <iostream>
#include "opencv2/imgproc/imgproc.hpp"
#include "opencv2/highgui/highgui.hpp"
#include <math.h>
using namespace std;
using namespace cv;
int main()
{
int numberofpoints=30;
float p2d[30][2] =
{{72, 169}, {72, 184}, {71, 264}, {96, 168}, {94, 261}, {94, 276}, {257, 133},
{254, 322}, {254, 337}, {278, 132}, {278, 146}, {275, 321}, {439, 228},
{439, 243}, {437, 328}, {435, 431}, {461, 226}, {459, 326}, {459, 342},
{457, 427}, {457, 444}, {612, 196}, {609, 291}, {605, 392}, {605, 406},
{634, 191}, {633, 206}, {630, 288}, {630, 303}, {627, 390}};
float p3d[30][3] =
{{0, 0, 98}, {0, 0, 94}, {0, 0, 75}, {4, 4, 98}, {4, 4, 75}, {4, 4, 71},
{0, 96, 75}, {0, 96, 25}, {0, 96, 21}, {4, 100, 75},
{4, 100, 71}, {4, 100, 25}, {96, 0, 98}, {96, 0, 94},
{96, 0, 75}, {96, 0, 50}, {100, 4, 98}, {100, 4, 75},
{100, 4, 71}, {100, 4, 50}, {100, 4, 46}, {96, 96, 75},
{96, 96, 50}, {96, 96, 25}, {96, 96, 21}, {100, 100, 75},
{100, 100, 71}, {100, 100, 50}, {100, 100, 46}, {100, 100, 25}};
Mat Matrix2D= Mat(30, 2, CV_64FC1, &p2d);
Mat Matrix3D= Mat(30, 3, CV_64FC1, &p3d);
Mat X_3D=Matrix3D.col(0);
Mat Y_3D=Matrix3D.col(1);
Mat Z_3D=Matrix3D.col(2);
Mat X_2D=Matrix2D.col(0);
Mat Y_2D=Matrix2D.col(1);
Mat z = Mat::zeros(numberofpoints,4, CV_64FC1);
Mat o = Mat::ones(numberofpoints,1, CV_64FC1);
Mat temp,temp1,C,D,AoddRows;
hconcat( X_3D, Y_3D,temp);
hconcat(Z_3D ,o,temp1);
hconcat(z, -X_2D.mul(X_3D),C);
hconcat(-X_2D.mul(Y_3D),-X_2D.mul(Z_3D),D);
hconcat(temp,temp1,AoddRows);
hconcat(AoddRows,C,AoddRows);
hconcat(AoddRows,D,AoddRows);
hconcat(AoddRows,-X_2D,AoddRows);
Mat A1,B1,C1,AevenRows;
hconcat(z,Matrix3D,A1);
hconcat(o,-Y_2D.mul(X_3D) ,B1);
hconcat(-Y_2D.mul(Y_3D),-Y_2D.mul(Z_3D),C1);
hconcat(A1,B1,AevenRows);
hconcat(AevenRows,C1,AevenRows);
hconcat(AevenRows,-Y_2D,AevenRows);
Mat A;
vconcat(AoddRows,AevenRows,A);
Mat w, u, EigenVectors;
SVD::compute(A, w, u, EigenVectors);
Mat EigenVector_SmallestEigenvalue=EigenVectors.row(EigenVectors.rows-1);
Mat P=Mat::zeros(3,4, CV_64FC1);
int k=0;
for(int i=0;i<P.rows;i++)
{
for(int j=0;j<P.cols;j++)
{
P.at<double>(i, j) = EigenVector_SmallestEigenvalue.at<double>(0,k);
k++;
}
}
double abs_lambda = sqrt(P.at<double>(2,0) * P.at<double>(2,0) + P.at<double>(2,1) * P.at<double>(2,1) + P.at<double>(2,2) * P.at<double>(2,2));
P = P / abs_lambda;
Mat ProjectionMatrix = P;
Mat T= Mat::zeros(3,1, CV_64FC1);
Mat R = Mat::zeros(3,3, CV_64FC1);
Mat B = Mat::zeros(3,3, CV_64FC1);
Mat b = Mat::zeros(3,1,CV_64FC1);
P.row(0).colRange(0,3).copyTo(B.row(0));
P.row(1).colRange(0,3).copyTo(B.row(1));
P.row(2).colRange(0,3).copyTo(B.row(2));
P.col(3).copyTo(b.col(0));
Mat K=B*B.t();
double Center_x = K.at<double>(0,2);
double Center_y = K.at<double>(1,2);
double beta = sqrt(K.at<double>(1,1)-Center_y*Center_y);
double gemma = (K.at<double>(0,1)-Center_x*Center_y)/beta;
double alpha = sqrt(K.at<double>(0,0)-Center_x*Center_x-gemma*gemma);
Mat CameraMatrix=Mat::zeros(3,3, CV_64FC1);
CameraMatrix.at<double>(0,0)=alpha;
CameraMatrix.at<double>(0,1)=gemma;
CameraMatrix.at<double>(0,2)=Center_x;
CameraMatrix.at<double>(1,1)=beta;
CameraMatrix.at<double>(1,2)=Center_y;
CameraMatrix.at<double>(2,2)=1;
R = CameraMatrix.inv()*B;
T = CameraMatrix.inv()*b;
Mat newMatrix2D=Mat::zeros(numberofpoints,2, CV_64FC1);
for(int i=0;i<numberofpoints;i++)
{
double num_u = P.at<double>(0,0) * Matrix3D.at<double>(i,0)
+ P.at<double>(0,1) * Matrix3D.at<double>(i,1)
+ P.at<double>(0,2) * Matrix3D.at<double>(i,2)
+ P.at<double>(0,3);
double num_v = P.at<double>(1,0) * Matrix3D.at<double>(i,0)
+ P.at<double>(1,1) * Matrix3D.at<double>(i,1)
+ P.at<double>(1,2) * Matrix3D.at<double>(i,2)
+ P.at<double>(1,3);
double den = P.at<double>(2,0) * Matrix3D.at<double>(i,0)
+ P.at<double>(2,1) * Matrix3D.at<double>(i,1)
+ P.at<double>(2,2) * Matrix3D.at<double>(i,2)
+ P.at<double>(2,3);
newMatrix2D.at<double>(i,0)=num_u/den;
newMatrix2D.at<double>(i,1)=num_v/den;
}
Mat reprojMatrix=newMatrix2D;
Mat errorDiff=reprojMatrix-Matrix2D;
int size=errorDiff.rows;
double sum=0;
for(int i=0;i<errorDiff.rows;i++)
{
sum=sum + sqrt(errorDiff.at<double>(i,0) * errorDiff.at<double>(i,0)
+ errorDiff.at<double>(i,1) * errorDiff.at<double>(i,1));
}
sum=sum/size;
cout<<"Average Error="<<sum<<endl;
cout<<"Translation Matrix="<<T<<endl<<endl;
cout<<"Rotational Matrix="<<R<<endl<<endl;
cout<<"Camera Matrix="<<CameraMatrix<<endl;
return 0;
}
SAMPLE 1:
const int numberofpoints = 30;
float p2d[numberofpoints][2] =
{{72, 169}, {72, 184}, {71, 264}, {96, 168}, {94, 261}, {94, 276}, {257, 133},
{254, 322}, {254, 337}, {278, 132}, {278, 146}, {275, 321}, {439, 228},
{439, 243}, {437, 328}, {435, 431}, {461, 226}, {459, 326}, {459, 342},
{457, 427}, {457, 444}, {612, 196}, {609, 291}, {605, 392}, {605, 406},
{634, 191}, {633, 206}, {630, 288}, {630, 303}, {627, 390}};
float p3d[numberofpoints][3] =
{{0, 0, 98}, {0, 0, 94}, {0, 0, 75}, {4, 4, 98}, {4, 4, 75}, {4, 4, 71},
{0, 96, 75}, {0, 96, 25}, {0, 96, 21}, {4, 100, 75},
{4, 100, 71}, {4, 100, 25}, {96, 0, 98}, {96, 0, 94},
{96, 0, 75}, {96, 0, 50}, {100, 4, 98}, {100, 4, 75},
{100, 4, 71}, {100, 4, 50}, {100, 4, 46}, {96, 96, 75},
{96, 96, 50}, {96, 96, 25}, {96, 96, 21}, {100, 100, 75},
{100, 100, 71}, {100, 100, 50}, {100, 100, 46}, {100, 100, 25}};
SAMPLE 2:
const int numberofpoints = 24;
float p2d[24][2] =
{{41, 291}, {41, 494}, {41, 509}, {64, 285}, {64, 303},
{64, 491}, {195, 146}, {216, 144}, {216, 160}, {431, 337},
{431, 441}, {431, 543}, {431, 558}, {452, 336}, {452, 349},
{452, 438}, {452, 457}, {452, 539}, {568, 195}, {566, 291},
{588, 190}, {587, 209}, {586, 289}, {586, 302}};
float p3d[24][3] =
{{0, 0, 75}, {0, 0, 25}, {0, 0, 21}, {4, 4, 75}, {4, 4, 71}, {4, 4, 25},
{0, 96, 75 }, {4, 100, 75}, {4, 100, 71}, {96, 0, 75}, {96, 0, 50}, {96, 0, 25},
{96, 0, 21}, {100, 4, 75}, {100, 4, 71}, {100, 4, 50}, {100, 4, 46}, {100, 4, 25}, {96, 96, 75 }, {96, 96, 50 }, {100, 100, 75}, {100, 100, 71}, {100, 100, 50}, {100, 100, 46}};
SAMPLE 3:
const int numberofpoints = 33;
float p2d[numberofpoints][2] =
{{45, 310}, {43, 412}, {41, 513}, {41, 530}, {68, 305},
{68, 321}, {66, 405}, {66, 423}, {64, 509}, {201, 70}, {201, 84},
{200, 155}, {198, 257}, {218, 259}, {218, 271}, {441, 364},
{439, 464}, {437, 569}, {437, 586}, {462, 361}, {462, 376},
{460, 462}, {460, 479}, {459, 565}, {583, 117}, {583, 131},
{580, 215}, {576, 313}, {603, 113}, {600, 214}, {599, 229},
{596, 311}, {596, 323}};
float p3d[numberofpoints][3] =
{{0, 0, 75}, {0, 0, 50}, {0, 0, 25}, {0, 0, 21}, {4, 4, 75}, {4, 4, 71},
{4, 4, 50}, {4, 4, 46}, {4, 4, 25}, {0, 96, 98}, {0, 96, 94}, {0, 96, 75},
{0, 96, 50}, {4, 100,75}, {4, 100,71}, {96, 0, 75}, {96, 0, 50}, {96, 0, 25},
{96, 0, 21}, {100, 4,75}, {100, 4,71}, {100, 4,50}, {100, 4,46}, {100, 4,25},{96, 96,98}, {96, 96,94}, {96, 96,75}, {96, 96,50}, {100, 100, 98}, {100, 100, 75},{100, 100, 71}, {100, 100, 50}, {100, 100, 46}};
EDIT:
I've taken one of the 3D coordinate point sets you've provided, and then tried to project them on a virtual camera (to prevent any errors with 3d-2d point correspondences). The alpha, beta and gamma are estimated correctly in this case, but u and v (that you are looking for) were for some reason negative in all of my experiments. Perhaps there's a bug in my code, but maybe this is due to inaccuracy of OpenCV solveZ implementation (I've had problems with OpenCV's eigenvalue/vector computation before and try to use OpenBLAS when high accuracy is needed).
Another possible reason - as authors state, this approach performs algebraic minimization, which may produce physically meaningless results. Perhaps you should check remaining part of paper or try a different approach.
Here's the updated code:
#include <opencv2/opencv.hpp>
#include <iostream>
int main()
{
// generate random 3d points
const int numberofpoints = 33;
float p3d_[numberofpoints][3] =
{{0, 0, 75}, {0, 0, 50}, {0, 0, 25}, {0, 0, 21}, {4, 4, 75}, {4, 4, 71},
{4, 4, 50}, {4, 4, 46}, {4, 4, 25}, {0, 96, 98}, {0, 96, 94}, {0, 96, 75},
{0, 96, 50}, {4, 100,75}, {4, 100,71}, {96, 0, 75}, {96, 0, 50}, {96, 0, 25},
{96, 0, 21}, {100, 4,75}, {100, 4,71}, {100, 4,50}, {100, 4,46}, {100, 4,25},{96, 96,98}, {96, 96,94}, {96, 96,75}, {96, 96,50}, {100, 100, 98}, {100, 100, 75},{100, 100, 71}, {100, 100, 50}, {100, 100, 46}};
std::vector<cv::Point3d> p3d;
for (int i = 0; i < numberofpoints; i++) {
cv::Point3d X(p3d_[i][0], p3d_[i][1], p3d_[i][2]);
p3d.push_back(X);
}
// set up virtual camera
const cv::Size imgSize(1600, 1200);
const double fx = 100;
const double fy = 120;
cv::Mat1d K = (cv::Mat1d(3, 3) << fx, 0, imgSize.width/2,
0, fy, imgSize.height/2,
0, 0, 1);
std::cout << "K = " << std::endl;
std::cout << K << std::endl << std::endl;
cv::Mat1d t = (cv::Mat1d(3, 1) << 10, 20, 20);
cv::Mat1d rvecDeg = (cv::Mat1d(3, 1) << 10, 20, 30);
// project point on camera
std::vector<cv::Point2d> p2d;
cv::projectPoints(p3d,
rvecDeg*CV_PI/180,
t,
K,
cv::Mat(),
p2d);
// estimate projection
cv::Mat1d G = cv::Mat1d::zeros(numberofpoints*2, 12);
for (int i = 0; i < numberofpoints; i++) {
const double X = p3d[i].x;
const double Y = p3d[i].y;
const double Z = p3d[i].z;
G(i*2 + 0, 0) = X;
G(i*2 + 0, 1) = Y;
G(i*2 + 0, 2) = Z;
G(i*2 + 0, 3) = 1;
G(i*2 + 1, 4) = X;
G(i*2 + 1, 5) = Y;
G(i*2 + 1, 6) = Z;
G(i*2 + 1, 7) = 1;
const double u = p2d[i].x;
const double v = p2d[i].y;
G(i*2 + 0, 8) = u*X;
G(i*2 + 0, 9) = u*Y;
G(i*2 + 0, 10) = u*Z;
G(i*2 + 0, 11) = u;
G(i*2 + 1, 8) = v*X;
G(i*2 + 1, 9) = v*Y;
G(i*2 + 1, 10) = v*Z;
G(i*2 + 1, 11) = v;
}
std::cout << "G = " << std::endl;
std::cout << G << std::endl << std::endl;
cv::Mat1d p;
cv::SVD::solveZ(G, p);
cv::Mat1d P = p.reshape(0, 3).clone();
P /= P(2, 3);
std::cout << "p = " << std::endl;
std::cout << p << std::endl << std::endl;
std::cout << "P = " << std::endl;
std::cout << P << std::endl << std::endl;
cv::Mat1d B(3, 3);
cv::Mat1d b(3, 1);
P.colRange(0, 3).copyTo(B);
P.col(3).copyTo(b);
std::cout << "B = " << std::endl;
std::cout << B << std::endl << std::endl;
std::cout << "b = " << std::endl;
std::cout << b << std::endl << std::endl;
cv::Mat1d K_ = B*B.t();
K_ /= K_(2, 2);
std::cout << "K_ = " << std::endl;
std::cout << K_ << std::endl << std::endl;
const double u = K_(0, 2);
const double v = K_(1, 2);
const double ku = K_(0, 0);
const double kv = K_(1, 1);
const double kc = K_(0, 1);
const double beta = sqrt(kv - v*v);
const double gamma = (kc - u*v)/beta;
const double alpha = sqrt(ku - u*u - gamma*gamma);
cv::Mat1d A = (cv::Mat1d(3, 3) << alpha, gamma, u,
0, beta, v,
0, 0, 1);
std::cout << "A = " << std::endl;
std::cout << A << std::endl << std::endl;
return 0;
}
And output:
K =
[100, 0, 800;
0, 120, 600;
0, 0, 1]
G =
[0, 0, 75, 1, 0, 0, 0, 0, 0, 0, 63156.71331987197, 842.0895109316264;
0, 0, 0, 0, 0, 0, 75, 1, 0, 0, 46451.70410142483, 619.3560546856644;
0, 0, 50, 1, 0, 0, 0, 0, 0, 0, 42144.23132613153, 842.8846265226306;
0, 0, 0, 0, 0, 0, 50, 1, 0, 0, 31473.60945095979, 629.4721890191959;
0, 0, 25, 1, 0, 0, 0, 0, 0, 0, 21113.32803626328, 844.5331214505311;
0, 0, 0, 0, 0, 0, 25, 1, 0, 0, 16261.14346086196, 650.4457384344785;
0, 0, 21, 1, 0, 0, 0, 0, 0, 0, 17744.50634900177, 844.9764928096083;
0, 0, 0, 0, 0, 0, 21, 1, 0, 0, 13777.82037617329, 656.0866845796805;
4, 4, 75, 1, 0, 0, 0, 0, 3374.871998456306, 3374.871998456306, 63278.84997105574, 843.7179996140766;
0, 0, 0, 0, 4, 4, 75, 1, 2506.953106676701, 2506.953106676701, 47005.37075018814, 626.7382766691752;
4, 4, 71, 1, 0, 0, 0, 0, 3375.547822963469, 3375.547822963469, 59915.97385760157, 843.8869557408673;
0, 0, 0, 0, 4, 4, 71, 1, 2513.243523511581, 2513.243523511581, 44610.07254233056, 628.3108808778952;
4, 4, 50, 1, 0, 0, 0, 0, 3380.337247599766, 3380.337247599766, 42254.21559499707, 845.0843118999414;
0, 0, 0, 0, 4, 4, 50, 1, 2557.822370597248, 2557.822370597248, 31972.7796324656, 639.4555926493119;
4, 4, 46, 1, 0, 0, 0, 0, 3381.587616222724, 3381.587616222724, 38888.25758656133, 845.396904055681;
0, 0, 0, 0, 4, 4, 46, 1, 2569.460510014068, 2569.460510014068, 29548.79586516178, 642.365127503517;
4, 4, 25, 1, 0, 0, 0, 0, 3391.684639092943, 3391.684639092943, 21198.02899433089, 847.9211597732357;
0, 0, 0, 0, 4, 4, 25, 1, 2663.441243136111, 2663.441243136111, 16646.50776960069, 665.8603107840277;
0, 96, 98, 1, 0, 0, 0, 0, 0, 76956.82972653268, 78560.09701250211, 801.6336429847155;
0, 0, 0, 0, 0, 96, 98, 1, 0, 65682.8899983677, 67051.28354000035, 684.1967708163302;
0, 96, 94, 1, 0, 0, 0, 0, 0, 76853.25603860538, 75252.14653780111, 800.5547504021395;
0, 0, 0, 0, 0, 96, 94, 1, 0, 65937.37528926283, 64563.67997073651, 686.8476592631544;
0, 96, 75, 1, 0, 0, 0, 0, 0, 76268.94727818707, 59585.11506108365, 794.4682008144487;
0, 0, 0, 0, 0, 96, 75, 1, 0, 67373.04865228917, 52635.19425960092, 701.8025901280123;
0, 96, 50, 1, 0, 0, 0, 0, 0, 75153.33695072016, 39142.36299516675, 782.8472599033349;
0, 0, 0, 0, 0, 96, 50, 1, 0, 70114.15427855898, 36517.78868674947, 730.3557737349894;
4, 100, 75, 1, 0, 0, 0, 0, 3182.802966382433, 79570.07415956081, 59677.55561967062, 795.7007415956082;
0, 0, 0, 0, 4, 100, 75, 1, 2830.826944396773, 70770.67360991932, 53078.00520743949, 707.7067360991932;
4, 100, 71, 1, 0, 0, 0, 0, 3176.842851499092, 79421.07128747732, 56388.96061410889, 794.2107128747731;
0, 0, 0, 0, 4, 100, 71, 1, 2846.678125949429, 71166.95314873573, 50528.53673560237, 711.6695314873573;
96, 0, 75, 1, 0, 0, 0, 0, 94501.57967461165, 0, 73829.35912079035, 984.3914549438714;
0, 0, 0, 0, 96, 0, 75, 1, 69392.97525695754, 0, 54213.26191949808, 722.8434922599744;
96, 0, 50, 1, 0, 0, 0, 0, 102665.427189569, 0, 53471.57666123386, 1069.431533224677;
0, 0, 0, 0, 96, 0, 50, 1, 76872.21110081286, 0, 40037.60994834002, 800.7521989668005;
96, 0, 25, 1, 0, 0, 0, 0, 134150.4060603281, 0, 34935.00157821043, 1397.400063128417;
0, 0, 0, 0, 96, 0, 25, 1, 105716.8927391909, 0, 27530.44081749762, 1101.217632699905;
96, 0, 21, 1, 0, 0, 0, 0, 150007.0124893856, 0, 32814.03398205309, 1562.573046764433;
0, 0, 0, 0, 96, 0, 21, 1, 120243.7808209231, 0, 26303.32705457694, 1252.539383551283;
100, 4, 75, 1, 0, 0, 0, 0, 98699.75231231008, 3947.990092492403, 74024.81423423256, 986.9975231231008;
0, 0, 0, 0, 100, 4, 75, 1, 73361.21263523313, 2934.448505409325, 55020.90947642484, 733.6121263523312;
100, 4, 71, 1, 0, 0, 0, 0, 99628.75712124966, 3985.150284849986, 70736.41755608725, 996.2875712124966;
0, 0, 0, 0, 100, 4, 71, 1, 74265.21217550985, 2970.608487020394, 52728.30064461199, 742.6521217550985;
100, 4, 50, 1, 0, 0, 0, 0, 107383.6098967354, 4295.344395869415, 53691.80494836769, 1073.836098967354;
0, 0, 0, 0, 100, 4, 50, 1, 81811.33387099921, 3272.453354839968, 40905.6669354996, 818.113338709992;
100, 4, 46, 1, 0, 0, 0, 0, 109823.0621321851, 4392.922485287403, 50518.60858080514, 1098.230621321851;
0, 0, 0, 0, 100, 4, 46, 1, 84185.12534995917, 3367.405013998367, 38725.15766098122, 841.8512534995917;
100, 4, 25, 1, 0, 0, 0, 0, 141059.7182762867, 5642.388731051467, 35264.92956907167, 1410.597182762867;
0, 0, 0, 0, 100, 4, 25, 1, 114581.0097655446, 4583.240390621784, 28645.25244138615, 1145.810097655446;
96, 96, 98, 1, 0, 0, 0, 0, 83904.83905262669, 83904.83905262669, 85652.85653288974, 874.0087401315279;
0, 0, 0, 0, 96, 96, 98, 1, 72995.44277461393, 72995.44277461393, 74516.18116575171, 760.3691955688951;
96, 96, 94, 1, 0, 0, 0, 0, 84021.59669072446, 84021.59669072446, 82271.1467596677, 875.2249655283798;
0, 0, 0, 0, 96, 96, 94, 1, 73575.81721996517, 73575.81721996517, 72042.98769454923, 766.4147627079706;
96, 96, 75, 1, 0, 0, 0, 0, 84712.67045349024, 84712.67045349024, 66181.77379178925, 882.4236505571899;
0, 0, 0, 0, 96, 96, 75, 1, 77010.98050603706, 77010.98050603706, 60164.82852034145, 802.1977136045526;
96, 96, 50, 1, 0, 0, 0, 0, 86206.34878960505, 86206.34878960505, 44899.13999458597, 897.9827998917193;
0, 0, 0, 0, 96, 96, 50, 1, 84435.70020728504, 84435.70020728504, 43976.92719129429, 879.5385438258859;
100, 100, 98, 1, 0, 0, 0, 0, 87539.46210370047, 87539.46210370047, 85788.67286162646, 875.3946210370046;
0, 0, 0, 0, 100, 100, 98, 1, 76664.75192364832, 76664.75192364832, 75131.45688517536, 766.6475192364833;
100, 100, 75, 1, 0, 0, 0, 0, 88416.27638616599, 88416.27638616599, 66312.20728962449, 884.16276386166;
0, 0, 0, 0, 100, 100, 75, 1, 81008.14162095707, 81008.14162095707, 60756.10621571781, 810.0814162095708;
100, 100, 71, 1, 0, 0, 0, 0, 88614.85267838663, 88614.85267838663, 62916.5454016545, 886.1485267838663;
0, 0, 0, 0, 100, 100, 71, 1, 81991.80970097007, 81991.80970097007, 58214.18488768875, 819.9180970097007;
100, 100, 50, 1, 0, 0, 0, 0, 90038.77512167525, 90038.77512167525, 45019.38756083763, 900.3877512167526;
0, 0, 0, 0, 100, 100, 50, 1, 89045.35592350175, 89045.35592350175, 44522.67796175087, 890.4535592350175;
100, 100, 46, 1, 0, 0, 0, 0, 90415.3917433265, 90415.3917433265, 41591.08020193018, 904.153917433265;
0, 0, 0, 0, 100, 100, 46, 1, 90910.96508045508, 90910.96508045508, 41819.04393700934, 909.1096508045508]
p =
[0.006441365659365744;
-0.006935698812720837;
-0.03488896901596035;
-0.7622591852949104;
0.004771957238156334;
-0.01133107207303337;
-0.02452697177582621;
-0.6456783687203944;
-1.258567018901194e-05;
1.123537587555102e-05;
4.154374841821949e-05;
0.0008967755121116598]
P =
[7.182807260423624, -7.734041261217285, -38.90490824599617, -849.9999999999995;
5.321239455925471, -12.63534956072988, -27.35018011148848, -719.9999999999993;
-0.0140343597913107, 0.01252863813051139, 0.04632569451009583, 1]
B =
[7.182807260423624, -7.734041261217285, -38.90490824599617;
5.321239455925471, -12.63534956072988, -27.35018011148848;
-0.0140343597913107, 0.01252863813051139, 0.04632569451009583]
b =
[-849.9999999999995;
-719.9999999999993;
1]
K_ =
[649999.9999999985, 479999.9999999995, -799.9999999999992;
479999.9999999995, 374400.0000000002, -600.0000000000002;
-799.9999999999992, -600.0000000000002, 1]
A =
[99.99999999999883, -1.940255363782255e-12, -799.9999999999992;
0, 119.9999999999995, -600.0000000000002;
0, 0, 1]
What other values are you getting? Generally, the camera center is never exactly at the center of the image, but it should be reasonably close.

OpenGLES adding a projection

I'm started to learn a OpenGLES and currently I'm reading this TUTORIAL
I have reached paragraph Adding a Projection and I'm stuck there:
// Add to render, right before the call to glViewport
CC3GLMatrix *projection = [CC3GLMatrix matrix];
float h = 4.0f * self.frame.size.height / self.frame.size.width;
[projection populateFromFrustumLeft:-2 andRight:2 andBottom:-h/2 andTop:h/2 andNear:4 andFar:10];
glUniformMatrix4fv(_projectionUniform, 1, 0, projection.glMatrix);
// Modify vertices so they are within projection near/far planes
const Vertex Vertices[] = {
{{1, -1, -7}, {1, 0, 0, 1}},
{{1, 1, -7}, {0, 1, 0, 1}},
{{-1, 1, -7}, {0, 0, 1, 1}},
{{-1, -1, -7}, {0, 0, 0, 1}}
};
The author uses some variables in populateFromFrustumLeft... and doesn't explain them. I want to understand the logic of variable selection to be able to use this function in future.
Help me plz to understan the logic!