Minimize matrix in Equation using OpenCV

Question

I need to minimize H in following equation:

Where H is 3x3 Matrix.
Pn is 3x1 matrix (point).
Euclidean() gives distance between 2 points.
Dn is the actual distance.

I have one initial estimate of H and m points(P0 to Pm)
I need optimize value of H such that for all m points error is minimized. (All the values in the expression are known) How can I implement this using opencv or dlib (or using boost/NLopt).

Prominence

The question is incomplete and unclear. First, adding scalar (1) to vector isn't generally defined. Second, you have to have something that you optimize in respect to. What is given? And finally, the whole expression will give an object of matrix type (if I am reading it right, that is the initial matrix H is multiplied by the sum) and I don't know what matrix minimization is. – Ron

Its not scalar (1) its next point (n+1) everything is in subscript. I have points P(o) to P(m). I have initial value of Matrix H. – Prominence

For dlib, you can start from this page dlib.net/optimization.html – Deni

@Ron minimize the sum vrt H. It's just poorly formatted. – Deni

@Ron I have formatted my equation properly. – Prominence

@Deni documentation for dlib is very bad. not able to understand anything from that. – Prominence

@user_12 documentation for dlib is good enough, with commented examples. You should start somewhere and ask specific questions where you stumble. – Deni

can you provide the link with sample code where similar expressions are minimized. – Prominence

Answer 1

Although the documentation of find_optimal_parameters function of dlib library was really not enough, there is a unit test that you can find on github which shows how to use the function.

I saw the other question you've asked and seems that the solution was something different than in this question. However, here is an example, how to use the library (this is the first time I'm hearing about it) to calculate what you need or something very close to that. Probably you will need to change the DistanceQuality() function (by replacing the existing loop with two nested ones) and I'll let you do it yourself.

Please note, that everything all over the code is hardcoded, no error handling is done and the testing is done right in the main() function. There's lots of work to be done, though you can find the code working for illustration purposes.

Here we go:

#include <iostream>
#include <dlib/optimization.h>
#include <dlib/optimization/find_optimal_parameters.h>

using namespace dlib;

typedef matrix<double, 3, 1> MyPoint;

std::vector<MyPoint> points;
std::vector<double> distances;

double MyDistance(MyPoint point1, MyPoint point2)
{
   double sum = 0;
   for (int i = 0; i < 3; i++)
   {
      sum += (point1(i, 0) - point2(i, 0)) * (point1(i, 0) - point2(i, 0));
   }
   return sqrt(sum);
}

double DistanceQuality(const matrix<double, 3, 3>& H)
{
   double sum = 0;

   for (int i = 0; i < points.size() - 1; i++)
   {
      auto proj1 = H*points[i];
      auto proj2 = H*points[i+1];
      sum += abs(MyDistance(proj1, proj2) - distances[i]);
   }
   return sum;
}

matrix<double, 3, 3> VecToMatrix(matrix<double, 0, 1> vec)
{
   matrix<double, 3, 3> matrix;
   for (int i = 0; i < 9; i++)
   {
      matrix(i / 3, i % 3) = vec(i);
   }
   return matrix;
}

double test_function(matrix<double, 0, 1> H)
{
   matrix<double, 3, 3> newH = VecToMatrix(H);
   auto result = DistanceQuality(newH);
   return result;
}

int main()
{
   matrix<double, 3, 1> p1;
   matrix<double, 3, 1> p2;
   matrix<double, 3, 1> p3;

   p1 = { 1, 1, 1 };
   p2 = { 2, 2, 3 };
   p3 = { 3, 1.6, 7};

   points.push_back(p1);
   points.push_back(p2);
   points.push_back(p3);

   double d1 = 2.44949;
   double d2 = 4.142463;

   distances.push_back(d1);
   distances.push_back(d2);

   matrix<double, 0, 1> H;
   H = { 3, 1, 1,
         1, 1, 6,
         1, 4, 1 };

   matrix<double, 0, 1> H_min;
   matrix<double, 0, 1> H_max;

   H_min = { 0.5, 0.6, 0.5,
             0.5, 0.7, 0.5,
             0.8, 0.3, 0.5, };

   H_max = { 10, 10, 10,
             10, 10, 10,
             10, 10, 10, };

   dlib::find_optimal_parameters(4, 0.001, 1000, H, H_min, H_max, test_function);
   std::cout << "new H: " << std::endl << VecToMatrix(H) << std::endl;

   return 0;
}

Hope you can adapt the parameters for you specific case.

Ron

I have dlib-18.18 and I am not able to find dlib/optimization/find_optimal_parameters.h – Prominence

According to the repo history, it has been added since Tuesday, April 19, 2016 02:11:38 and was release with version 19 (right after the 18.18). – Ron