* Fits the shape of a Morphable Model to given 2D landmarks (i.e. estimates the maximum likelihood solution of the shape coefficients) as proposed in [1].
* It's a linear, closed-form solution fitting of the shape, with regularisation (prior towards the mean).
*
* [1] O. Aldrian & W. Smith, Inverse Rendering of Faces with a 3D Morphable Model, PAMI 2013.
*
* Note: Using less than the maximum number of coefficients to fit is not thoroughly tested yet and may contain an error.
* Note: Returns coefficients following standard normal distribution (i.e. all have similar magnitude). Why? Because we fit using the normalised basis?
* Note: The standard deviations given should be a vector, i.e. different for each landmark. This is not implemented yet.
*
* @param[in] morphable_model The Morphable Model whose shape (coefficients) are estimated.
* @param[in] affine_camera_matrix A 3x4 affine camera matrix from model to screen-space (should probably be of type CV_32FC1 as all our calculations are done with float).
* @param[in] landmarks 2D landmarks from an image to fit the model to.
* @param[in] vertex_ids The vertex ids in the model that correspond to the 2D points.
* @param[in] base_face The base or reference face from where the fitting is started. Usually this would be the models mean face, which is what will be used if the parameter is not explicitly specified.
* @param[in] lambda The regularisation parameter (weight of the prior towards the mean). Gets normalized by the number of images given.
* @param[in] num_coefficients_to_fit How many shape-coefficients to fit (all others will stay 0). Should be bigger than zero, or boost::none to fit all coefficients.
* @param[in] detector_standard_deviation The standard deviation of the 2D landmarks given (e.g. of the detector used), in pixels.
* @param[in] model_standard_deviation The standard deviation of the 3D vertex points in the 3D model, projected to 2D (so the value is in pixels).
* @return The estimated shape-coefficients (alphas).
assert(affine_camera_matrix.size()==landmarks.size()&&landmarks.size()==vertex_ids.size());// same number of instances (i.e. images/frames) for each of them
MatrixXfbasis_rows_=morphable_model.get_shape_model().get_rescaled_pca_basis_at_point(vertex_ids[k][i]);// In the paper, the orthonormal basis might be used? I'm not sure, check it. It's even a mess in the paper. PH 26.5.2014: I think the rescaled basis is fine/better.
constMatrixXfrhs=-A.transpose()*Omega.asDiagonal()*b;// It's -A^t*Omega^t*b, but we don't need to transpose Omega, since it's a diagonal matrix, and Omega^t = Omega.
// c_s: The 'x' that we solve for. (The variance-normalised shape parameter vector, $c_s = [a_1/sigma_{s,1} , ..., a_m-1/sigma_{s,m-1}]^t$.)
// We get coefficients ~ N(0, 1), because we're fitting with the rescaled basis. The coefficients are not multiplied with their eigenvalues.
* Fits the shape of a Morphable Model to given 2D landmarks (i.e. estimates the maximum likelihood solution of the shape coefficients) as proposed in [1].
* It's a linear, closed-form solution fitting of the shape, with regularisation (prior towards the mean).