Can ceres-solver get exact solution of linear equations with LU decomposition method？

[Moses]

I am using opencv to do Exposure Compensate with GainCompensator method.  

https://github.com/opencv/opencv/blob/master/modules/stitching/src/exposure_compensate.cpp (line 127~141)

OpenCV use solve(A, b, gains_); with CV_EXPORTS_W bool solve(InputArray src1, InputArray src2, OutputArray dst, int flags = DECOMP_LU);。

error equation:

above equation's derivative：

Set to 0:

I replace OpenCV code (line 127~141) with ceres-solver, like below:

---------------

struct F {

F(int Nij, double Iij, double Iji, double alpha, double beta) :Nij_(Nij), Iij_(Iij), Iji_(Iji), alpha_(alpha), beta_(beta) {}

template <typename T> bool operator()(const T* const gi, const T* const gj, T* residual) const {

residual[0] = T(Nij_) * ( alpha_ * alpha_ *  (gi[0] * Iij_ - gj[0] * Iji_) * (gi[0] * Iij_ - gj[0] * Iji_) + beta_ * beta_ * (T(1) - gi[0]) * (T(1) - gi[0]) );

return true;

}

private:

const int Nij_;

const double Iij_, Iji_, alpha_, beta_;

};

struct E {

E(int Nij, double beta) :Nij_(Nij), beta_(beta) {}

template <typename T> bool operator()(const T* const gi, T* residual) const {

residual[0] = T(Nij_) * (beta_ * beta_ * (T(1) - gi[0]) * (T(1) - gi[0]));

return true;

}

private:

const int Nij_;

const double beta_;

};

---------------

//google::InitGoogleLogging("");

gains_ = Mat_<double>::zeros(num_images, 1);

for (int i = 0; i < num_images; ++i) {

gains_(i, 0) = 0; // if I set to all 1, then it will output all 1.

}

ceres::Problem problem;

for (int i = 0; i < num_images; ++i)

{

for (int j = 0; j < num_images; ++j)

{

if (i != j)

problem.AddResidualBlock(new AutoDiffCostFunction<F, 1, 1, 1>(new F(N(i, j), I(i, j), I(j, i), alpha, beta)), NULL, &gains_(i, 0), &gains_(j, 0));

else

problem.AddResidualBlock(new AutoDiffCostFunction<E, 1, 1>(new E(N(i, j), beta)), NULL, &gains_(i, 0));

}

}

ceres::Solver::Options options;

//options.max_num_iterations = 25;

options.linear_solver_type = ceres::DENSE_QR;

options.num_linear_solver_threads = 4;

options.function_tolerance = 0.1;

options.minimizer_progress_to_stdout = true;

ceres::Solver::Summary summary;

ceres::Solve(options, &problem, &summary);

//std::cout << summary.BriefReport() << "\n";

----------------

I use seven images to test exposure compensate, ceres-solver quick a little, but the answer is not what I want.

I think this is because ceres-solver is getting a local solution, how can I get the same answer the same with Opencv, because ceres-solver is very easy to use.

Does anyone know more about math with this problem, thank you very much!

[Moses]

Update missing images 

在 2018年7月8日星期日 UTC+8下午7:14:35，Moses写道：

I am using opencv to do Exposure Compensate with GainCompensator method.  

https://github.com/opencv/opencv/blob/master/modules/stitching/src/exposure_compensate.cpp (line 127~141)

OpenCV use solve(A, b, gains_); with CV_EXPORTS_W bool solve(InputArray src1, InputArray src2, OutputArray dst, int flags = DECOMP_LU);。

 

error equation:

above equation's derivative：

[Sameer Agarwal]

Moses,

I am confused.

Are you asking if Ceres can do LU factorization? The answer is no.

Are you asking if ceres can get the same solution as an LU factorization? for a linear problem yes.

But looks like your problem is non-linear. In which case all you can get is a local minimum.

Sameer

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