0dd9a4a576
Is based on Google style which was used in the Libmv project before, but is now consistently applied for the sources of the library itself and to C-API. With some time C-API will likely be removed, and it makes it easier to make it follow Libmv style, hence the diversion from Blender's style. There are quite some exceptions (clang-format off) in the code around Eigen matrix initialization. It is rather annoying, and there could be some neat way to make initialization readable without such exception. Could be some places where loss of readability in matrix initialization got lost as the change is quite big. If this has happened it is easier to address readability once actually working on the code. This change allowed to spot some missing header guards, so that's nice. Doing it in bundled version, as the upstream library needs to have some of the recent development ported over from bundle to upstream. There should be no functional changes.
783 lines
27 KiB
C++
783 lines
27 KiB
C++
// Copyright (c) 2009 libmv authors.
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//
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// Permission is hereby granted, free of charge, to any person obtaining a copy
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// of this software and associated documentation files (the "Software"), to
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// deal in the Software without restriction, including without limitation the
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// rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
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// sell copies of the Software, and to permit persons to whom the Software is
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// furnished to do so, subject to the following conditions:
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//
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// The above copyright notice and this permission notice shall be included in
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// all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
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// IN THE SOFTWARE.
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#include "libmv/multiview/euclidean_resection.h"
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#include <cmath>
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#include <limits>
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#include <Eigen/Geometry>
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#include <Eigen/SVD>
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#include "libmv/base/vector.h"
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#include "libmv/logging/logging.h"
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#include "libmv/multiview/projection.h"
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namespace libmv {
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namespace euclidean_resection {
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typedef unsigned int uint;
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bool EuclideanResection(const Mat2X& x_camera,
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const Mat3X& X_world,
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Mat3* R,
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Vec3* t,
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ResectionMethod method) {
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switch (method) {
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case RESECTION_ANSAR_DANIILIDIS:
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EuclideanResectionAnsarDaniilidis(x_camera, X_world, R, t);
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break;
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case RESECTION_EPNP:
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return EuclideanResectionEPnP(x_camera, X_world, R, t);
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break;
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case RESECTION_PPNP:
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return EuclideanResectionPPnP(x_camera, X_world, R, t);
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break;
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default: LOG(FATAL) << "Unknown resection method.";
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}
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return false;
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}
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bool EuclideanResection(const Mat& x_image,
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const Mat3X& X_world,
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const Mat3& K,
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Mat3* R,
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Vec3* t,
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ResectionMethod method) {
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CHECK(x_image.rows() == 2 || x_image.rows() == 3)
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<< "Invalid size for x_image: " << x_image.rows() << "x"
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<< x_image.cols();
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Mat2X x_camera;
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if (x_image.rows() == 2) {
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EuclideanToNormalizedCamera(x_image, K, &x_camera);
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} else if (x_image.rows() == 3) {
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HomogeneousToNormalizedCamera(x_image, K, &x_camera);
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}
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return EuclideanResection(x_camera, X_world, R, t, method);
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}
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void AbsoluteOrientation(const Mat3X& X, const Mat3X& Xp, Mat3* R, Vec3* t) {
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int num_points = X.cols();
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Vec3 C = X.rowwise().sum() / num_points; // Centroid of X.
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Vec3 Cp = Xp.rowwise().sum() / num_points; // Centroid of Xp.
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// Normalize the two point sets.
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Mat3X Xn(3, num_points), Xpn(3, num_points);
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for (int i = 0; i < num_points; ++i) {
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Xn.col(i) = X.col(i) - C;
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Xpn.col(i) = Xp.col(i) - Cp;
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}
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// Construct the N matrix (pg. 635).
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double Sxx = Xn.row(0).dot(Xpn.row(0));
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double Syy = Xn.row(1).dot(Xpn.row(1));
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double Szz = Xn.row(2).dot(Xpn.row(2));
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double Sxy = Xn.row(0).dot(Xpn.row(1));
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double Syx = Xn.row(1).dot(Xpn.row(0));
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double Sxz = Xn.row(0).dot(Xpn.row(2));
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double Szx = Xn.row(2).dot(Xpn.row(0));
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double Syz = Xn.row(1).dot(Xpn.row(2));
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double Szy = Xn.row(2).dot(Xpn.row(1));
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Mat4 N;
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// clang-format off
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N << Sxx + Syy + Szz, Syz - Szy, Szx - Sxz, Sxy - Syx,
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Syz - Szy, Sxx - Syy - Szz, Sxy + Syx, Szx + Sxz,
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Szx - Sxz, Sxy + Syx, -Sxx + Syy - Szz, Syz + Szy,
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Sxy - Syx, Szx + Sxz, Syz + Szy, -Sxx - Syy + Szz;
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// clang-format on
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// Find the unit quaternion q that maximizes qNq. It is the eigenvector
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// corresponding to the lagest eigenvalue.
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Vec4 q = N.jacobiSvd(Eigen::ComputeFullU).matrixU().col(0);
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// Retrieve the 3x3 rotation matrix.
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Vec4 qq = q.array() * q.array();
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double q0q1 = q(0) * q(1);
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double q0q2 = q(0) * q(2);
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double q0q3 = q(0) * q(3);
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double q1q2 = q(1) * q(2);
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double q1q3 = q(1) * q(3);
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double q2q3 = q(2) * q(3);
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// clang-format off
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(*R) << qq(0) + qq(1) - qq(2) - qq(3),
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2 * (q1q2 - q0q3),
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2 * (q1q3 + q0q2),
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2 * (q1q2+ q0q3),
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qq(0) - qq(1) + qq(2) - qq(3),
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2 * (q2q3 - q0q1),
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2 * (q1q3 - q0q2),
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2 * (q2q3 + q0q1),
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qq(0) - qq(1) - qq(2) + qq(3);
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// clang-format on
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// Fix the handedness of the R matrix.
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if (R->determinant() < 0) {
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R->row(2) = -R->row(2);
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}
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// Compute the final translation.
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*t = Cp - *R * C;
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}
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// Convert i and j indices of the original variables into their quadratic
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// permutation single index. It follows that t_ij = t_ji.
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static int IJToPointIndex(int i, int j, int num_points) {
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// Always make sure that j is bigger than i. This handles t_ij = t_ji.
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if (j < i) {
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std::swap(i, j);
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}
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int idx;
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int num_permutation_rows = num_points * (num_points - 1) / 2;
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// All t_ii's are located at the end of the t vector after all t_ij's.
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if (j == i) {
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idx = num_permutation_rows + i;
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} else {
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int offset = (num_points - i - 1) * (num_points - i) / 2;
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idx = (num_permutation_rows - offset + j - i - 1);
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}
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return idx;
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};
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// Convert i and j indexes of the solution for lambda to their linear indexes.
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static int IJToIndex(int i, int j, int num_lambda) {
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if (j < i) {
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std::swap(i, j);
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}
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int A = num_lambda * (num_lambda + 1) / 2;
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int B = num_lambda - i;
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int C = B * (B + 1) / 2;
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int idx = A - C + j - i;
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return idx;
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};
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static int Sign(double value) {
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return (value < 0) ? -1 : 1;
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};
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// Organizes a square matrix into a single row constraint on the elements of
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// Lambda to create the constraints in equation (5) in "Linear Pose Estimation
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// from Points or Lines", by Ansar, A. and Daniilidis, PAMI 2003. vol. 25, no.
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// 5.
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static Vec MatrixToConstraint(const Mat& A, int num_k_columns, int num_lambda) {
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Vec C(num_k_columns);
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C.setZero();
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int idx = 0;
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for (int i = 0; i < num_lambda; ++i) {
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for (int j = i; j < num_lambda; ++j) {
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C(idx) = A(i, j);
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if (i != j) {
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C(idx) += A(j, i);
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}
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++idx;
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}
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}
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return C;
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}
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// Normalizes the columns of vectors.
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static void NormalizeColumnVectors(Mat3X* vectors) {
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int num_columns = vectors->cols();
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for (int i = 0; i < num_columns; ++i) {
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vectors->col(i).normalize();
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}
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}
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void EuclideanResectionAnsarDaniilidis(const Mat2X& x_camera,
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const Mat3X& X_world,
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Mat3* R,
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Vec3* t) {
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CHECK(x_camera.cols() == X_world.cols());
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CHECK(x_camera.cols() > 3);
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int num_points = x_camera.cols();
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// Copy the normalized camera coords into 3 vectors and normalize them so
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// that they are unit vectors from the camera center.
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Mat3X x_camera_unit(3, num_points);
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x_camera_unit.block(0, 0, 2, num_points) = x_camera;
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x_camera_unit.row(2).setOnes();
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NormalizeColumnVectors(&x_camera_unit);
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int num_m_rows = num_points * (num_points - 1) / 2;
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int num_tt_variables = num_points * (num_points + 1) / 2;
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int num_m_columns = num_tt_variables + 1;
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Mat M(num_m_columns, num_m_columns);
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M.setZero();
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Matu ij_index(num_tt_variables, 2);
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// Create the constraint equations for the t_ij variables (7) and arrange
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// them into the M matrix (8). Also store the initial (i, j) indices.
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int row = 0;
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for (int i = 0; i < num_points; ++i) {
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for (int j = i + 1; j < num_points; ++j) {
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M(row, row) = -2 * x_camera_unit.col(i).dot(x_camera_unit.col(j));
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M(row, num_m_rows + i) = x_camera_unit.col(i).dot(x_camera_unit.col(i));
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M(row, num_m_rows + j) = x_camera_unit.col(j).dot(x_camera_unit.col(j));
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Vec3 Xdiff = X_world.col(i) - X_world.col(j);
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double center_to_point_distance = Xdiff.norm();
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M(row, num_m_columns - 1) =
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-center_to_point_distance * center_to_point_distance;
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ij_index(row, 0) = i;
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ij_index(row, 1) = j;
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++row;
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}
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ij_index(i + num_m_rows, 0) = i;
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ij_index(i + num_m_rows, 1) = i;
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}
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int num_lambda = num_points + 1; // Dimension of the null space of M.
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Mat V = M.jacobiSvd(Eigen::ComputeFullV)
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.matrixV()
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.block(0, num_m_rows, num_m_columns, num_lambda);
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// TODO(vess): The number of constraint equations in K (num_k_rows) must be
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// (num_points + 1) * (num_points + 2)/2. This creates a performance issue
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// for more than 4 points. It is fine for 4 points at the moment with 18
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// instead of 15 equations.
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int num_k_rows =
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num_m_rows +
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num_points * (num_points * (num_points - 1) / 2 - num_points + 1);
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int num_k_columns = num_lambda * (num_lambda + 1) / 2;
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Mat K(num_k_rows, num_k_columns);
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K.setZero();
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// Construct the first part of the K matrix corresponding to (t_ii, t_jk) for
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// i != j.
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int counter_k_row = 0;
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for (int idx1 = num_m_rows; idx1 < num_tt_variables; ++idx1) {
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for (int idx2 = 0; idx2 < num_m_rows; ++idx2) {
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unsigned int i = ij_index(idx1, 0);
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unsigned int j = ij_index(idx2, 0);
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unsigned int k = ij_index(idx2, 1);
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if (i != j && i != k) {
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int idx3 = IJToPointIndex(i, j, num_points);
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int idx4 = IJToPointIndex(i, k, num_points);
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K.row(counter_k_row) =
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MatrixToConstraint(V.row(idx1).transpose() * V.row(idx2) -
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V.row(idx3).transpose() * V.row(idx4),
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num_k_columns,
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num_lambda);
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++counter_k_row;
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}
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}
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}
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// Construct the second part of the K matrix corresponding to (t_ii,t_jk) for
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// j==k.
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for (int idx1 = num_m_rows; idx1 < num_tt_variables; ++idx1) {
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for (int idx2 = idx1 + 1; idx2 < num_tt_variables; ++idx2) {
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unsigned int i = ij_index(idx1, 0);
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unsigned int j = ij_index(idx2, 0);
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unsigned int k = ij_index(idx2, 1);
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int idx3 = IJToPointIndex(i, j, num_points);
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int idx4 = IJToPointIndex(i, k, num_points);
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K.row(counter_k_row) =
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MatrixToConstraint(V.row(idx1).transpose() * V.row(idx2) -
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V.row(idx3).transpose() * V.row(idx4),
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num_k_columns,
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num_lambda);
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++counter_k_row;
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}
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}
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Vec L_sq = K.jacobiSvd(Eigen::ComputeFullV).matrixV().col(num_k_columns - 1);
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// Pivot on the largest element for numerical stability. Afterwards recover
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// the sign of the lambda solution.
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double max_L_sq_value = fabs(L_sq(IJToIndex(0, 0, num_lambda)));
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int max_L_sq_index = 1;
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for (int i = 1; i < num_lambda; ++i) {
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double abs_sq_value = fabs(L_sq(IJToIndex(i, i, num_lambda)));
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if (max_L_sq_value < abs_sq_value) {
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max_L_sq_value = abs_sq_value;
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max_L_sq_index = i;
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}
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}
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// Ensure positiveness of the largest value corresponding to lambda_ii.
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L_sq =
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L_sq * Sign(L_sq(IJToIndex(max_L_sq_index, max_L_sq_index, num_lambda)));
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Vec L(num_lambda);
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L(max_L_sq_index) =
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sqrt(L_sq(IJToIndex(max_L_sq_index, max_L_sq_index, num_lambda)));
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for (int i = 0; i < num_lambda; ++i) {
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if (i != max_L_sq_index) {
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L(i) = L_sq(IJToIndex(max_L_sq_index, i, num_lambda)) / L(max_L_sq_index);
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}
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}
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// Correct the scale using the fact that the last constraint is equal to 1.
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L = L / (V.row(num_m_columns - 1).dot(L));
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Vec X = V * L;
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// Recover the distances from the camera center to the 3D points Q.
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Vec d(num_points);
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d.setZero();
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for (int c_point = num_m_rows; c_point < num_tt_variables; ++c_point) {
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d(c_point - num_m_rows) = sqrt(X(c_point));
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}
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// Create the 3D points in the camera system.
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Mat X_cam(3, num_points);
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for (int c_point = 0; c_point < num_points; ++c_point) {
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X_cam.col(c_point) = d(c_point) * x_camera_unit.col(c_point);
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}
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// Recover the camera translation and rotation.
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AbsoluteOrientation(X_world, X_cam, R, t);
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}
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// Selects 4 virtual control points using mean and PCA.
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static void SelectControlPoints(const Mat3X& X_world,
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Mat* X_centered,
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Mat34* X_control_points) {
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size_t num_points = X_world.cols();
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// The first virtual control point, C0, is the centroid.
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Vec mean, variance;
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MeanAndVarianceAlongRows(X_world, &mean, &variance);
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X_control_points->col(0) = mean;
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// Computes PCA
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X_centered->resize(3, num_points);
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for (size_t c = 0; c < num_points; c++) {
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X_centered->col(c) = X_world.col(c) - mean;
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}
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Mat3 X_centered_sq = (*X_centered) * X_centered->transpose();
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Eigen::JacobiSVD<Mat3> X_centered_sq_svd(X_centered_sq, Eigen::ComputeFullU);
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Vec3 w = X_centered_sq_svd.singularValues();
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Mat3 u = X_centered_sq_svd.matrixU();
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for (size_t c = 0; c < 3; c++) {
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double k = sqrt(w(c) / num_points);
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X_control_points->col(c + 1) = mean + k * u.col(c);
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}
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}
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// Computes the barycentric coordinates for all real points
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static void ComputeBarycentricCoordinates(const Mat3X& X_world_centered,
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const Mat34& X_control_points,
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Mat4X* alphas) {
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size_t num_points = X_world_centered.cols();
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Mat3 C2;
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for (size_t c = 1; c < 4; c++) {
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C2.col(c - 1) = X_control_points.col(c) - X_control_points.col(0);
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}
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Mat3 C2inv = C2.inverse();
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Mat3X a = C2inv * X_world_centered;
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alphas->resize(4, num_points);
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alphas->setZero();
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alphas->block(1, 0, 3, num_points) = a;
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for (size_t c = 0; c < num_points; c++) {
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(*alphas)(0, c) = 1.0 - alphas->col(c).sum();
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}
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}
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// Estimates the coordinates of all real points in the camera coordinate frame
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static void ComputePointsCoordinatesInCameraFrame(
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const Mat4X& alphas,
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const Vec4& betas,
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const Eigen::Matrix<double, 12, 12>& U,
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Mat3X* X_camera) {
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size_t num_points = alphas.cols();
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// Estimates the control points in the camera reference frame.
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Mat34 C2b;
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C2b.setZero();
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for (size_t cu = 0; cu < 4; cu++) {
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for (size_t c = 0; c < 4; c++) {
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C2b.col(c) += betas(cu) * U.block(11 - cu, c * 3, 1, 3).transpose();
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}
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}
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// Estimates the 3D points in the camera reference frame
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X_camera->resize(3, num_points);
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for (size_t c = 0; c < num_points; c++) {
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X_camera->col(c) = C2b * alphas.col(c);
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}
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// Check the sign of the z coordinate of the points (should be positive)
|
||
uint num_z_neg = 0;
|
||
for (size_t i = 0; i < X_camera->cols(); ++i) {
|
||
if ((*X_camera)(2, i) < 0) {
|
||
num_z_neg++;
|
||
}
|
||
}
|
||
|
||
// If more than 50% of z are negative, we change the signs
|
||
if (num_z_neg > 0.5 * X_camera->cols()) {
|
||
C2b = -C2b;
|
||
*X_camera = -(*X_camera);
|
||
}
|
||
}
|
||
|
||
bool EuclideanResectionEPnP(const Mat2X& x_camera,
|
||
const Mat3X& X_world,
|
||
Mat3* R,
|
||
Vec3* t) {
|
||
CHECK(x_camera.cols() == X_world.cols());
|
||
CHECK(x_camera.cols() > 3);
|
||
size_t num_points = X_world.cols();
|
||
|
||
// Select the control points.
|
||
Mat34 X_control_points;
|
||
Mat X_centered;
|
||
SelectControlPoints(X_world, &X_centered, &X_control_points);
|
||
|
||
// Compute the barycentric coordinates.
|
||
Mat4X alphas(4, num_points);
|
||
ComputeBarycentricCoordinates(X_centered, X_control_points, &alphas);
|
||
|
||
// Estimates the M matrix with the barycentric coordinates
|
||
Mat M(2 * num_points, 12);
|
||
Eigen::Matrix<double, 2, 12> sub_M;
|
||
for (size_t c = 0; c < num_points; c++) {
|
||
double a0 = alphas(0, c);
|
||
double a1 = alphas(1, c);
|
||
double a2 = alphas(2, c);
|
||
double a3 = alphas(3, c);
|
||
double ui = x_camera(0, c);
|
||
double vi = x_camera(1, c);
|
||
// clang-format off
|
||
M.block(2*c, 0, 2, 12) << a0, 0,
|
||
a0*(-ui), a1, 0,
|
||
a1*(-ui), a2, 0,
|
||
a2*(-ui), a3, 0,
|
||
a3*(-ui), 0,
|
||
a0, a0*(-vi), 0,
|
||
a1, a1*(-vi), 0,
|
||
a2, a2*(-vi), 0,
|
||
a3, a3*(-vi);
|
||
// clang-format on
|
||
}
|
||
|
||
// TODO(julien): Avoid the transpose by rewriting the u2.block() calls.
|
||
Eigen::JacobiSVD<Mat> MtMsvd(M.transpose() * M, Eigen::ComputeFullU);
|
||
Eigen::Matrix<double, 12, 12> u2 = MtMsvd.matrixU().transpose();
|
||
|
||
// Estimate the L matrix.
|
||
Eigen::Matrix<double, 6, 3> dv1;
|
||
Eigen::Matrix<double, 6, 3> dv2;
|
||
Eigen::Matrix<double, 6, 3> dv3;
|
||
Eigen::Matrix<double, 6, 3> dv4;
|
||
|
||
dv1.row(0) = u2.block(11, 0, 1, 3) - u2.block(11, 3, 1, 3);
|
||
dv1.row(1) = u2.block(11, 0, 1, 3) - u2.block(11, 6, 1, 3);
|
||
dv1.row(2) = u2.block(11, 0, 1, 3) - u2.block(11, 9, 1, 3);
|
||
dv1.row(3) = u2.block(11, 3, 1, 3) - u2.block(11, 6, 1, 3);
|
||
dv1.row(4) = u2.block(11, 3, 1, 3) - u2.block(11, 9, 1, 3);
|
||
dv1.row(5) = u2.block(11, 6, 1, 3) - u2.block(11, 9, 1, 3);
|
||
dv2.row(0) = u2.block(10, 0, 1, 3) - u2.block(10, 3, 1, 3);
|
||
dv2.row(1) = u2.block(10, 0, 1, 3) - u2.block(10, 6, 1, 3);
|
||
dv2.row(2) = u2.block(10, 0, 1, 3) - u2.block(10, 9, 1, 3);
|
||
dv2.row(3) = u2.block(10, 3, 1, 3) - u2.block(10, 6, 1, 3);
|
||
dv2.row(4) = u2.block(10, 3, 1, 3) - u2.block(10, 9, 1, 3);
|
||
dv2.row(5) = u2.block(10, 6, 1, 3) - u2.block(10, 9, 1, 3);
|
||
dv3.row(0) = u2.block(9, 0, 1, 3) - u2.block(9, 3, 1, 3);
|
||
dv3.row(1) = u2.block(9, 0, 1, 3) - u2.block(9, 6, 1, 3);
|
||
dv3.row(2) = u2.block(9, 0, 1, 3) - u2.block(9, 9, 1, 3);
|
||
dv3.row(3) = u2.block(9, 3, 1, 3) - u2.block(9, 6, 1, 3);
|
||
dv3.row(4) = u2.block(9, 3, 1, 3) - u2.block(9, 9, 1, 3);
|
||
dv3.row(5) = u2.block(9, 6, 1, 3) - u2.block(9, 9, 1, 3);
|
||
dv4.row(0) = u2.block(8, 0, 1, 3) - u2.block(8, 3, 1, 3);
|
||
dv4.row(1) = u2.block(8, 0, 1, 3) - u2.block(8, 6, 1, 3);
|
||
dv4.row(2) = u2.block(8, 0, 1, 3) - u2.block(8, 9, 1, 3);
|
||
dv4.row(3) = u2.block(8, 3, 1, 3) - u2.block(8, 6, 1, 3);
|
||
dv4.row(4) = u2.block(8, 3, 1, 3) - u2.block(8, 9, 1, 3);
|
||
dv4.row(5) = u2.block(8, 6, 1, 3) - u2.block(8, 9, 1, 3);
|
||
|
||
Eigen::Matrix<double, 6, 10> L;
|
||
for (size_t r = 0; r < 6; r++) {
|
||
// clang-format off
|
||
L.row(r) << dv1.row(r).dot(dv1.row(r)),
|
||
2.0 * dv1.row(r).dot(dv2.row(r)),
|
||
dv2.row(r).dot(dv2.row(r)),
|
||
2.0 * dv1.row(r).dot(dv3.row(r)),
|
||
2.0 * dv2.row(r).dot(dv3.row(r)),
|
||
dv3.row(r).dot(dv3.row(r)),
|
||
2.0 * dv1.row(r).dot(dv4.row(r)),
|
||
2.0 * dv2.row(r).dot(dv4.row(r)),
|
||
2.0 * dv3.row(r).dot(dv4.row(r)),
|
||
dv4.row(r).dot(dv4.row(r));
|
||
// clang-format on
|
||
}
|
||
Vec6 rho;
|
||
// clang-format off
|
||
rho << (X_control_points.col(0) - X_control_points.col(1)).squaredNorm(),
|
||
(X_control_points.col(0) - X_control_points.col(2)).squaredNorm(),
|
||
(X_control_points.col(0) - X_control_points.col(3)).squaredNorm(),
|
||
(X_control_points.col(1) - X_control_points.col(2)).squaredNorm(),
|
||
(X_control_points.col(1) - X_control_points.col(3)).squaredNorm(),
|
||
(X_control_points.col(2) - X_control_points.col(3)).squaredNorm();
|
||
// clang-format on
|
||
|
||
// There are three possible solutions based on the three approximations of L
|
||
// (betas). Below, each one is solved for then the best one is chosen.
|
||
Mat3X X_camera;
|
||
Mat3 K;
|
||
K.setIdentity();
|
||
vector<Mat3> Rs(3);
|
||
vector<Vec3> ts(3);
|
||
Vec rmse(3);
|
||
|
||
// At one point this threshold was 1e-3, and caused no end of problems in
|
||
// Blender by causing frames to not resect when they would have worked fine.
|
||
// When the resect failed, the projective followup is run leading to worse
|
||
// results, and often the dreaded "flipping" where objects get flipped
|
||
// between frames. Instead, disable the check for now, always succeeding. The
|
||
// ultimate check is always reprojection error anyway.
|
||
//
|
||
// TODO(keir): Decide if setting this to infinity, effectively disabling the
|
||
// check, is the right approach. So far this seems the case.
|
||
double kSuccessThreshold = std::numeric_limits<double>::max();
|
||
|
||
// Find the first possible solution for R, t corresponding to:
|
||
// Betas = [b00 b01 b11 b02 b12 b22 b03 b13 b23 b33]
|
||
// Betas_approx_1 = [b00 b01 b02 b03]
|
||
Vec4 betas = Vec4::Zero();
|
||
Eigen::Matrix<double, 6, 4> l_6x4;
|
||
for (size_t r = 0; r < 6; r++) {
|
||
l_6x4.row(r) << L(r, 0), L(r, 1), L(r, 3), L(r, 6);
|
||
}
|
||
Eigen::JacobiSVD<Mat> svd_of_l4(l_6x4,
|
||
Eigen::ComputeFullU | Eigen::ComputeFullV);
|
||
Vec4 b4 = svd_of_l4.solve(rho);
|
||
if ((l_6x4 * b4).isApprox(rho, kSuccessThreshold)) {
|
||
if (b4(0) < 0) {
|
||
b4 = -b4;
|
||
}
|
||
b4(0) = std::sqrt(b4(0));
|
||
betas << b4(0), b4(1) / b4(0), b4(2) / b4(0), b4(3) / b4(0);
|
||
ComputePointsCoordinatesInCameraFrame(alphas, betas, u2, &X_camera);
|
||
AbsoluteOrientation(X_world, X_camera, &Rs[0], &ts[0]);
|
||
rmse(0) = RootMeanSquareError(x_camera, X_world, K, Rs[0], ts[0]);
|
||
} else {
|
||
LOG(ERROR) << "First approximation of beta not good enough.";
|
||
ts[0].setZero();
|
||
rmse(0) = std::numeric_limits<double>::max();
|
||
}
|
||
|
||
// Find the second possible solution for R, t corresponding to:
|
||
// Betas = [b00 b01 b11 b02 b12 b22 b03 b13 b23 b33]
|
||
// Betas_approx_2 = [b00 b01 b11]
|
||
betas.setZero();
|
||
Eigen::Matrix<double, 6, 3> l_6x3;
|
||
l_6x3 = L.block(0, 0, 6, 3);
|
||
Eigen::JacobiSVD<Mat> svdOfL3(l_6x3,
|
||
Eigen::ComputeFullU | Eigen::ComputeFullV);
|
||
Vec3 b3 = svdOfL3.solve(rho);
|
||
VLOG(2) << " rho = " << rho;
|
||
VLOG(2) << " l_6x3 * b3 = " << l_6x3 * b3;
|
||
if ((l_6x3 * b3).isApprox(rho, kSuccessThreshold)) {
|
||
if (b3(0) < 0) {
|
||
betas(0) = std::sqrt(-b3(0));
|
||
betas(1) = (b3(2) < 0) ? std::sqrt(-b3(2)) : 0;
|
||
} else {
|
||
betas(0) = std::sqrt(b3(0));
|
||
betas(1) = (b3(2) > 0) ? std::sqrt(b3(2)) : 0;
|
||
}
|
||
if (b3(1) < 0) {
|
||
betas(0) = -betas(0);
|
||
}
|
||
betas(2) = 0;
|
||
betas(3) = 0;
|
||
ComputePointsCoordinatesInCameraFrame(alphas, betas, u2, &X_camera);
|
||
AbsoluteOrientation(X_world, X_camera, &Rs[1], &ts[1]);
|
||
rmse(1) = RootMeanSquareError(x_camera, X_world, K, Rs[1], ts[1]);
|
||
} else {
|
||
LOG(ERROR) << "Second approximation of beta not good enough.";
|
||
ts[1].setZero();
|
||
rmse(1) = std::numeric_limits<double>::max();
|
||
}
|
||
|
||
// Find the third possible solution for R, t corresponding to:
|
||
// Betas = [b00 b01 b11 b02 b12 b22 b03 b13 b23 b33]
|
||
// Betas_approx_3 = [b00 b01 b11 b02 b12]
|
||
betas.setZero();
|
||
Eigen::Matrix<double, 6, 5> l_6x5;
|
||
l_6x5 = L.block(0, 0, 6, 5);
|
||
Eigen::JacobiSVD<Mat> svdOfL5(l_6x5,
|
||
Eigen::ComputeFullU | Eigen::ComputeFullV);
|
||
Vec5 b5 = svdOfL5.solve(rho);
|
||
if ((l_6x5 * b5).isApprox(rho, kSuccessThreshold)) {
|
||
if (b5(0) < 0) {
|
||
betas(0) = std::sqrt(-b5(0));
|
||
if (b5(2) < 0) {
|
||
betas(1) = std::sqrt(-b5(2));
|
||
} else {
|
||
b5(2) = 0;
|
||
}
|
||
} else {
|
||
betas(0) = std::sqrt(b5(0));
|
||
if (b5(2) > 0) {
|
||
betas(1) = std::sqrt(b5(2));
|
||
} else {
|
||
b5(2) = 0;
|
||
}
|
||
}
|
||
if (b5(1) < 0) {
|
||
betas(0) = -betas(0);
|
||
}
|
||
betas(2) = b5(3) / betas(0);
|
||
betas(3) = 0;
|
||
ComputePointsCoordinatesInCameraFrame(alphas, betas, u2, &X_camera);
|
||
AbsoluteOrientation(X_world, X_camera, &Rs[2], &ts[2]);
|
||
rmse(2) = RootMeanSquareError(x_camera, X_world, K, Rs[2], ts[2]);
|
||
} else {
|
||
LOG(ERROR) << "Third approximation of beta not good enough.";
|
||
ts[2].setZero();
|
||
rmse(2) = std::numeric_limits<double>::max();
|
||
}
|
||
|
||
// Finally, with all three solutions, select the (R, t) with the best RMSE.
|
||
VLOG(2) << "RMSE for solution 0: " << rmse(0);
|
||
VLOG(2) << "RMSE for solution 1: " << rmse(1);
|
||
VLOG(2) << "RMSE for solution 2: " << rmse(2);
|
||
size_t n = 0;
|
||
if (rmse(1) < rmse(0)) {
|
||
n = 1;
|
||
}
|
||
if (rmse(2) < rmse(n)) {
|
||
n = 2;
|
||
}
|
||
if (rmse(n) == std::numeric_limits<double>::max()) {
|
||
LOG(ERROR) << "All three possibilities failed. Reporting failure.";
|
||
return false;
|
||
}
|
||
|
||
VLOG(1) << "RMSE for best solution #" << n << ": " << rmse(n);
|
||
*R = Rs[n];
|
||
*t = ts[n];
|
||
|
||
// TODO(julien): Improve the solutions with non-linear refinement.
|
||
return true;
|
||
}
|
||
|
||
/*
|
||
|
||
Straight from the paper:
|
||
http://www.diegm.uniud.it/fusiello/papers/3dimpvt12-b.pdf
|
||
|
||
function [R T] = ppnp(P,S,tol)
|
||
% input
|
||
% P : matrix (nx3) image coordinates in camera reference [u v 1]
|
||
% S : matrix (nx3) coordinates in world reference [X Y Z]
|
||
% tol: exit threshold
|
||
%
|
||
% output
|
||
% R : matrix (3x3) rotation (world-to-camera)
|
||
% T : vector (3x1) translation (world-to-camera)
|
||
%
|
||
n = size(P,1);
|
||
Z = zeros(n);
|
||
e = ones(n,1);
|
||
A = eye(n)-((e*e’)./n);
|
||
II = e./n;
|
||
err = +Inf;
|
||
E_old = 1000*ones(n,3);
|
||
while err>tol
|
||
[U,˜,V] = svd(P’*Z*A*S);
|
||
VT = V’;
|
||
R=U*[1 0 0; 0 1 0; 0 0 det(U*VT)]*VT;
|
||
PR = P*R;
|
||
c = (S-Z*PR)’*II;
|
||
Y = S-e*c’;
|
||
Zmindiag = diag(PR*Y’)./(sum(P.*P,2));
|
||
Zmindiag(Zmindiag<0)=0;
|
||
Z = diag(Zmindiag);
|
||
E = Y-Z*PR;
|
||
err = norm(E-E_old,’fro’);
|
||
E_old = E;
|
||
end
|
||
T = -R*c;
|
||
end
|
||
|
||
*/
|
||
// TODO(keir): Re-do all the variable names and add comments matching the paper.
|
||
// This implementation has too much of the terseness of the original. On the
|
||
// other hand, it did work on the first try.
|
||
bool EuclideanResectionPPnP(const Mat2X& x_camera,
|
||
const Mat3X& X_world,
|
||
Mat3* R,
|
||
Vec3* t) {
|
||
int n = x_camera.cols();
|
||
Mat Z = Mat::Zero(n, n);
|
||
Vec e = Vec::Ones(n);
|
||
Mat A = Mat::Identity(n, n) - (e * e.transpose() / n);
|
||
Vec II = e / n;
|
||
|
||
Mat P(n, 3);
|
||
P.col(0) = x_camera.row(0);
|
||
P.col(1) = x_camera.row(1);
|
||
P.col(2).setConstant(1.0);
|
||
|
||
Mat S = X_world.transpose();
|
||
|
||
double error = std::numeric_limits<double>::infinity();
|
||
Mat E_old = 1000 * Mat::Ones(n, 3);
|
||
|
||
Vec3 c;
|
||
Mat E(n, 3);
|
||
|
||
int iteration = 0;
|
||
double tolerance = 1e-5;
|
||
// TODO(keir): The limit of 100 can probably be reduced, but this will require
|
||
// some investigation.
|
||
while (error > tolerance && iteration < 100) {
|
||
Mat3 tmp = P.transpose() * Z * A * S;
|
||
Eigen::JacobiSVD<Mat3> svd(tmp, Eigen::ComputeFullU | Eigen::ComputeFullV);
|
||
Mat3 U = svd.matrixU();
|
||
Mat3 VT = svd.matrixV().transpose();
|
||
Vec3 s;
|
||
s << 1, 1, (U * VT).determinant();
|
||
*R = U * s.asDiagonal() * VT;
|
||
Mat PR = P * *R; // n x 3
|
||
c = (S - Z * PR).transpose() * II;
|
||
Mat Y = S - e * c.transpose(); // n x 3
|
||
Vec Zmindiag = (PR * Y.transpose())
|
||
.diagonal()
|
||
.cwiseQuotient(P.rowwise().squaredNorm());
|
||
for (int i = 0; i < n; ++i) {
|
||
Zmindiag[i] = std::max(Zmindiag[i], 0.0);
|
||
}
|
||
Z = Zmindiag.asDiagonal();
|
||
E = Y - Z * PR;
|
||
error = (E - E_old).norm();
|
||
LG << "PPnP error(" << (iteration++) << "): " << error;
|
||
E_old = E;
|
||
}
|
||
*t = -*R * c;
|
||
|
||
// TODO(keir): Figure out what the failure cases are. Is it too many
|
||
// iterations? Spend some time going through the math figuring out if there
|
||
// is some way to detect that the algorithm is going crazy, and return false.
|
||
return true;
|
||
}
|
||
|
||
} // namespace euclidean_resection
|
||
} // namespace libmv
|