lib/local/Utilities/include/RotationHelpers.h

///////////////////////////////////////////////////////////////////////////////
// Copyright (C) 2017, Tadas Baltrusaitis all rights reserved.
//
// ACADEMIC OR NON-PROFIT ORGANIZATION NONCOMMERCIAL RESEARCH USE ONLY
//
// BY USING OR DOWNLOADING THE SOFTWARE, YOU ARE AGREEING TO THE TERMS OF THIS LICENSE AGREEMENT.  
// IF YOU DO NOT AGREE WITH THESE TERMS, YOU MAY NOT USE OR DOWNLOAD THE SOFTWARE.
//
// License can be found in OpenFace-license.txt
//
//     * Any publications arising from the use of this software, including but
//       not limited to academic journal and conference publications, technical
//       reports and manuals, must cite at least one of the following works:
//
//       OpenFace 2.0: Facial Behavior Analysis Toolkit
//       Tadas Baltru�aitis, Amir Zadeh, Yao Chong Lim, and Louis-Philippe Morency
//       in IEEE International Conference on Automatic Face and Gesture Recognition, 2018  
//
//       Convolutional experts constrained local model for facial landmark detection.
//       A. Zadeh, T. Baltru�aitis, and Louis-Philippe Morency,
//       in Computer Vision and Pattern Recognition Workshops, 2017.    
//
//       Rendering of Eyes for Eye-Shape Registration and Gaze Estimation
//       Erroll Wood, Tadas Baltru�aitis, Xucong Zhang, Yusuke Sugano, Peter Robinson, and Andreas Bulling 
//       in IEEE International. Conference on Computer Vision (ICCV),  2015 
//
//       Cross-dataset learning and person-specific normalisation for automatic Action Unit detection
//       Tadas Baltru�aitis, Marwa Mahmoud, and Peter Robinson 
//       in Facial Expression Recognition and Analysis Challenge, 
//       IEEE International Conference on Automatic Face and Gesture Recognition, 2015 
//
///////////////////////////////////////////////////////////////////////////////

#ifndef ROTATION_HELPERS_H
#define ROTATION_HELPERS_H

#include <opencv2/core/core.hpp>
#include <opencv2/calib3d.hpp>

namespace Utilities
{
	//===========================================================================
	// Angle representation conversion helpers
	//===========================================================================

	// Using the XYZ convention R = Rx * Ry * Rz, left-handed positive sign
	static cv::Matx33f Euler2RotationMatrix(const cv::Vec3f& eulerAngles)
	{
		cv::Matx33f rotation_matrix;

		float s1 = sin(eulerAngles[0]);
		float s2 = sin(eulerAngles[1]);
		float s3 = sin(eulerAngles[2]);

		float c1 = cos(eulerAngles[0]);
		float c2 = cos(eulerAngles[1]);
		float c3 = cos(eulerAngles[2]);

		rotation_matrix(0, 0) = c2 * c3;
		rotation_matrix(0, 1) = -c2 *s3;
		rotation_matrix(0, 2) = s2;
		rotation_matrix(1, 0) = c1 * s3 + c3 * s1 * s2;
		rotation_matrix(1, 1) = c1 * c3 - s1 * s2 * s3;
		rotation_matrix(1, 2) = -c2 * s1;
		rotation_matrix(2, 0) = s1 * s3 - c1 * c3 * s2;
		rotation_matrix(2, 1) = c3 * s1 + c1 * s2 * s3;
		rotation_matrix(2, 2) = c1 * c2;

		return rotation_matrix;
	}

	// Using the XYZ convention R = Rx * Ry * Rz, left-handed positive sign
	static cv::Vec3f RotationMatrix2Euler(const cv::Matx33f& rotation_matrix)
	{
		float q0 = sqrt(1 + rotation_matrix(0, 0) + rotation_matrix(1, 1) + rotation_matrix(2, 2)) / 2.0f;
		float q1 = (rotation_matrix(2, 1) - rotation_matrix(1, 2)) / (4.0f*q0);
		float q2 = (rotation_matrix(0, 2) - rotation_matrix(2, 0)) / (4.0f*q0);
		float q3 = (rotation_matrix(1, 0) - rotation_matrix(0, 1)) / (4.0f*q0);

		// Slower, but dealing with degenerate cases due to precision
		float t1 = 2.0f * (q0*q2 + q1*q3);
		if (t1 > 1) t1 = 1.0f;
		if (t1 < -1) t1 = -1.0f;

		float yaw = asin(t1);
		float pitch = atan2(2.0f * (q0*q1 - q2*q3), q0*q0 - q1*q1 - q2*q2 + q3*q3);
		float roll = atan2(2.0f * (q0*q3 - q1*q2), q0*q0 + q1*q1 - q2*q2 - q3*q3);

		return cv::Vec3f(pitch, yaw, roll);
	}

	static cv::Vec3f Euler2AxisAngle(const cv::Vec3f& euler)
	{
		cv::Matx33f rotMatrix = Euler2RotationMatrix(euler);
		cv::Vec3f axis_angle;
		cv::Rodrigues(rotMatrix, axis_angle);
		return axis_angle;
	}

	static cv::Vec3f AxisAngle2Euler(const cv::Vec3f& axis_angle)
	{
		cv::Matx33f rotation_matrix;
		cv::Rodrigues(axis_angle, rotation_matrix);
		return RotationMatrix2Euler(rotation_matrix);
	}

	static cv::Matx33f AxisAngle2RotationMatrix(const cv::Vec3f& axis_angle)
	{
		cv::Matx33f rotation_matrix;
		cv::Rodrigues(axis_angle, rotation_matrix);
		return rotation_matrix;
	}

	static cv::Vec3f RotationMatrix2AxisAngle(const cv::Matx33f& rotation_matrix)
	{
		cv::Vec3f axis_angle;
		cv::Rodrigues(rotation_matrix, axis_angle);
		return axis_angle;
	}

	// Generally useful 3D functions
	static void Project(cv::Mat_<float>& dest, const cv::Mat_<float>& mesh, float fx, float fy, float cx, float cy)
	{
		dest = cv::Mat_<float>(mesh.rows, 2, 0.0);

		int num_points = mesh.rows;

		float X, Y, Z;


		cv::Mat_<float>::const_iterator mData = mesh.begin();
		cv::Mat_<float>::iterator projected = dest.begin();

		for (int i = 0; i < num_points; i++)
		{
			// Get the points
			X = *(mData++);
			Y = *(mData++);
			Z = *(mData++);

			float x;
			float y;

			// if depth is 0 the projection is different
			if (Z != 0)
			{
				x = ((X * fx / Z) + cx);
				y = ((Y * fy / Z) + cy);
			}
			else
			{
				x = X;
				y = Y;
			}

			// Project and store in dest matrix
			(*projected++) = x;
			(*projected++) = y;
		}

	}

	//===========================================================================
	// Point set and landmark manipulation functions
	//===========================================================================
	// Using Kabsch's algorithm for aligning shapes
	//This assumes that align_from and align_to are already mean normalised
	static cv::Matx22f AlignShapesKabsch2D(const cv::Mat_<float>& align_from, const cv::Mat_<float>& align_to)
	{

		cv::SVD svd(align_from.t() * align_to);

		// make sure no reflection is there
		// corr ensures that we do only rotaitons and not reflections
		double d = cv::determinant(svd.vt.t() * svd.u.t());

		cv::Matx22f corr = cv::Matx22f::eye();
		if (d > 0)
		{
			corr(1, 1) = 1;
		}
		else
		{
			corr(1, 1) = -1;
		}

		cv::Matx22f R;
		cv::Mat(svd.vt.t()*cv::Mat(corr)*svd.u.t()).copyTo(R);

		return R;
	}

	//=============================================================================
	// Basically Kabsch's algorithm but also allows the collection of points to be different in scale from each other
	static cv::Matx22f AlignShapesWithScale(cv::Mat_<float>& src, cv::Mat_<float> dst)
	{
		int n = src.rows;

		// First we mean normalise both src and dst
		float mean_src_x = (float)cv::mean(src.col(0))[0];
		float mean_src_y = (float)cv::mean(src.col(1))[0];

		float mean_dst_x = (float)cv::mean(dst.col(0))[0];
		float mean_dst_y = (float)cv::mean(dst.col(1))[0];

		cv::Mat_<float> src_mean_normed = src.clone();
		src_mean_normed.col(0) = src_mean_normed.col(0) - mean_src_x;
		src_mean_normed.col(1) = src_mean_normed.col(1) - mean_src_y;

		cv::Mat_<float> dst_mean_normed = dst.clone();
		dst_mean_normed.col(0) = dst_mean_normed.col(0) - mean_dst_x;
		dst_mean_normed.col(1) = dst_mean_normed.col(1) - mean_dst_y;

		// Find the scaling factor of each
		cv::Mat src_sq;
		cv::pow(src_mean_normed, 2, src_sq);

		cv::Mat dst_sq;
		cv::pow(dst_mean_normed, 2, dst_sq);

		float s_src = (float)sqrt(cv::sum(src_sq)[0] / n);
		float s_dst = (float)sqrt(cv::sum(dst_sq)[0] / n);

		src_mean_normed = src_mean_normed / s_src;
		dst_mean_normed = dst_mean_normed / s_dst;

		float s = s_dst / s_src;

		// Get the rotation
		cv::Matx22f R = AlignShapesKabsch2D(src_mean_normed, dst_mean_normed);

		cv::Matx22f	A;
		cv::Mat(s * R).copyTo(A);

		//cv::Mat_<float> aligned = (cv::Mat(cv::Mat(A) * src.t())).t();
		//cv::Mat_<float> offset = dst - aligned;

		//float t_x = cv::mean(offset.col(0))[0];
		//float t_y = cv::mean(offset.col(1))[0];

		return A;

	}

}
#endif // ROTATION_HELPERS_H