Added linear, closed-form blendshape fitting

Based on the linear landmark fitting.
The regularisation is set conservatively, in the experiments so far even a zero value yielded good results.

CMakeLists.txt

@@ -100,6 +100,7 @@ set(HEADERS
 	include/eos/fitting/detail/nonlinear_camera_estimation_detail.hpp
 	include/eos/fitting/linear_shape_fitting.hpp
 	include/eos/fitting/contour_correspondence.hpp
+	include/eos/fitting/blendshape_fitting.hpp
 	include/eos/render/Mesh.hpp
 	include/eos/render/utils.hpp
 	include/eos/render/render.hpp

include/eos/fitting/blendshape_fitting.hpp

+/*
+ * Eos - A 3D Morphable Model fitting library written in modern C++11/14.
+ *
+ * File: include/eos/fitting/blendshape_fitting.hpp
+ *
+ * Copyright 2015 Patrik Huber
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+#pragma once
+
+#ifndef BLENDSHAPEFITTING_HPP_
+#define BLENDSHAPEFITTING_HPP_
+
+#include "eos/morphablemodel/Blendshape.hpp"
+
+#include "opencv2/core/core.hpp"
+
+#include <vector>
+#include <cassert>
+
+namespace eos {
+	namespace fitting {
+
+/**
+ * Fits blendshape coefficients to given 2D landmarks, given a current face shape instance.
+ * It's a linear, closed-form solution fitting algorithm, with regularisation (constraining the L2-norm of the coefficients).
+ * 
+ * This algorithm is very similar to the shape fitting in fit_shape_to_landmarks_linear. Instead of the PCA basis, the blendshapes are used, and instead of
+ * the mean, a current face instance is used to do the fitting from.
+ *
+ * @param[in] blendshapes A vector with blendshapes to estimate the coefficients for.
+ * @param[in] face_instance A shape instance from which the blendshape coefficients should be estimated (i.e. the current mesh without expressions, e.g. estimated from a previous PCA-model fitting). A 3m x 1 matrix.
+ * @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 blendshapes to.
+ * @param[in] vertex_ids The vertex ids in the model that correspond to the 2D points.
+ * @param[in] lambda A regularisation parameter, constraining the L2-norm of the coefficients.
+ * @return The estimated blendshape-coefficients.
+ */
+inline std::vector<float> fit_blendshapes_to_landmarks_linear(std::vector<eos::morphablemodel::Blendshape> blendshapes, cv::Mat face_instance, cv::Mat affine_camera_matrix, std::vector<cv::Vec2f> landmarks, std::vector<int> vertex_ids, float lambda=500.0f)
+{
+	using cv::Mat;
+	assert(landmarks.size() == vertex_ids.size());
+
+	int num_coeffs_to_fit = blendshapes.size();
+	int num_landmarks = static_cast<int>(landmarks.size());
+
+	// Copy all blendshapes into a "basis" matrix with each blendshape being a column:
+	cv::Mat blendshapes_as_basis(blendshapes[0].deformation.rows, blendshapes.size(), CV_32FC1); // assert blendshapes.size() > 0 and all of them have same number of rows, and 1 col
+	for (int i = 0; i < blendshapes.size(); ++i)
+	{
+		blendshapes[i].deformation.copyTo(blendshapes_as_basis.col(i));
+	}
+
+	// $\hat{V} \in R^{3N\times m-1}$, subselect the rows of the eigenvector matrix $V$ associated with the $N$ feature points
+	// And we insert a row of zeros after every third row, resulting in matrix $\hat{V}_h \in R^{4N\times m-1}$:
+	Mat V_hat_h = Mat::zeros(4 * num_landmarks, num_coeffs_to_fit, CV_32FC1);
+	int row_index = 0;
+	for (int i = 0; i < num_landmarks; ++i) {
+		//Mat basis_rows = morphable_model.get_shape_model().get_normalised_pca_basis(vertex_ids[i]); // In the paper, the not-normalised 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 normalised basis is fine/better.
+		Mat basis_rows = blendshapes_as_basis.rowRange(vertex_ids[i] * 3, (vertex_ids[i] * 3) + 3);
+		//basisRows.copyTo(V_hat_h.rowRange(rowIndex, rowIndex + 3));
+		basis_rows.colRange(0, num_coeffs_to_fit).copyTo(V_hat_h.rowRange(row_index, row_index + 3));
+		row_index += 4; // replace 3 rows and skip the 4th one, it has all zeros
+	}
+	// Form a block diagonal matrix $P \in R^{3N\times 4N}$ in which the camera matrix C (P_Affine, affine_camera_matrix) is placed on the diagonal:
+	Mat P = Mat::zeros(3 * num_landmarks, 4 * num_landmarks, CV_32FC1);
+	for (int i = 0; i < num_landmarks; ++i) {
+		Mat submatrix_to_replace = P.colRange(4 * i, (4 * i) + 4).rowRange(3 * i, (3 * i) + 3);
+		affine_camera_matrix.copyTo(submatrix_to_replace);
+	}
+
+	// The landmarks in matrix notation (in homogeneous coordinates), $3N\times 1$
+	Mat y = Mat::ones(3 * num_landmarks, 1, CV_32FC1);
+	for (int i = 0; i < num_landmarks; ++i) {
+		y.at<float>(3 * i, 0) = landmarks[i][0];
+		y.at<float>((3 * i) + 1, 0) = landmarks[i][1];
+		//y.at<float>((3 * i) + 2, 0) = 1; // already 1, stays (homogeneous coordinate)
+	}
+	// The mean, with an added homogeneous coordinate (x_1, y_1, z_1, 1, x_2, ...)^t
+	Mat v_bar = Mat::ones(4 * num_landmarks, 1, CV_32FC1);
+	for (int i = 0; i < num_landmarks; ++i) {
+		//cv::Vec4f model_mean = morphable_model.get_shape_model().get_mean_at_point(vertex_ids[i]);
+		cv::Vec4f model_mean(face_instance.at<float>(vertex_ids[i]*3), face_instance.at<float>(vertex_ids[i]*3 + 1), face_instance.at<float>(vertex_ids[i]*3 + 2), 1.0f);
+		v_bar.at<float>(4 * i, 0) = model_mean[0];
+		v_bar.at<float>((4 * i) + 1, 0) = model_mean[1];
+		v_bar.at<float>((4 * i) + 2, 0) = model_mean[2];
+		//v_bar.at<float>((4 * i) + 3, 0) = 1; // already 1, stays (homogeneous coordinate)
+		// note: now that a Vec4f is returned, we could use copyTo?
+	}
+
+	// Bring into standard regularised quadratic form:
+	Mat A = P * V_hat_h; // camera matrix times the basis
+	Mat b = P * v_bar - y; // camera matrix times the mean, minus the landmarks.
+	
+	Mat AtA = A.t() * A;
+	Mat AtAReg = AtA + lambda * Mat::eye(num_coeffs_to_fit, num_coeffs_to_fit, CV_32FC1);
+	// Invert using OpenCV:
+	Mat AtARegInv = AtAReg.inv(cv::DECOMP_SVD); // DECOMP_SVD calculates the pseudo-inverse if the matrix is not invertible.
+
+	Mat c_s = -AtARegInv * A.t() * b;
+
+	return std::vector<float>(c_s);
+};
+
+	} /* namespace fitting */
+} /* namespace eos */
+
+#endif /* BLENDSHAPEFITTING_HPP_ */