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//! This package provides functions to solve camera pose estimation //! given a set of three 3D points and their corresponding pixel coordinates. //! //! This problem is generally known as ["Perspective-n-Point" (PnP)][pnp]. //! We focus on the P3P case (n = 3) which is the //! minimal amount of points required to solve the problem with a finite number of solutions. //! There are many approaches to solve P3P. In this package, we have implemented: //! //! - Lambda Twist: An Accurate Fast Robust Perspective Three Point (P3P) Solver. //! Mikael Persson, Klas Nordberg. ECCV 2018. ([paper][lambda-twist]) //! //! [pnp]: https://en.wikipedia.org/wiki/Perspective-n-Point //! [lambda-twist]: http://openaccess.thecvf.com/content_ECCV_2018/html/Mikael_Persson_Lambda_Twist_An_ECCV_2018_paper.html //! //! Implementation based on //! "Lambda Twist: An Accurate Fast Robust Perspective Three Point (P3P) Solver" //! Persson, M. and Nordberg, K. ECCV 2018. //! Reference implementation available on the [author github repository][lambda-twist-github]. //! //! [lambda-twist-github]: https://github.com/midjji/lambdatwist-p3p #![no_std] #![warn(missing_docs)] extern crate alloc; use alloc::vec::Vec; use num_traits::Float; // Copyright (c) 2018 Michael Persson // Adapted to openMVG by Romain Janvier and Pierre Moulon // Adapted to Rust by Matthieu Pizenberg // This Source Code Form is subject to the terms of the Mozilla Public // License, v. 2.0. If a copy of the MPL was not distributed with this // file, You can obtain one at http://mozilla.org/MPL/2.0/. use cv_core::nalgebra::{IsometryMatrix3, Matrix3, Rotation3, Translation, Vector3}; use cv_core::sample_consensus::Estimator; use cv_core::{Bearing, FeatureWorldMatch, WorldPoint, WorldPose}; type Iso3 = IsometryMatrix3<f64>; type Mat3 = Matrix3<f64>; type Vec3 = Vector3<f64>; /// The core P3P algorithm. Estimates potential poses of the world relative to the camera. /// /// Returns 0 to 4 potential solutions to the equation: /// /// ![Formula](https://chart.googleapis.com/chart?cht=tx&chl=\lambda_i%20\%20y_i%20=%20R%20x_i%20%2B%20t,\quad%20i%20\in%20\{1,%202,%203\}) /// /// - ![x_i](https://chart.googleapis.com/chart?cht=tx&chl=x_i) are the 3D point world coordinates. /// - ![y_i](https://chart.googleapis.com/chart?cht=tx&chl=y_i) are the image coordinates ![Formula](https://chart.googleapis.com/chart?cht=tx&chl=y_i%20\%20\sim%20\%20(u_i,\%20v_i,\%201)) /// (also sometimes called "bearing vectors"). /// - ![lambda_i](https://chart.googleapis.com/chart?cht=tx&chl=lambda_i) are the signed distances from the camera. /// /// The rotation and pose of the camera itself can be retrieved by converting the [`cv_core::WorldPose`] into a [`cv_core::CameraPose`]. /// /// This struct is marked as `#lnon_exhaustive]` to allow the backwards-compatible addition of new fields. #[derive(Copy, Clone, Debug, PartialEq)] #[non_exhaustive] pub struct LambdaTwist { /// This determines the number of iterations that Gauss-Newton is ran on the poses to optimize them. /// The paper notes that it rarely improves after two iterations. The original implementation uses 5 iterations. pub gauss_newton_iterations: usize, /// This determines the number of iterations to spend converging on a proper rotation matrix. pub rotation_convergence_iterations: usize, } impl LambdaTwist { /// Creates a default lambda twist with 5 iterations of Gauss-Newton. pub fn new() -> Self { Self::default() } /// Sets the [`LambdaTwist::gauss_newton_iterations`]. pub fn gauss_newton_iterations(self, gauss_newton_iterations: usize) -> Self { Self { gauss_newton_iterations, ..self } } /// Sets the [`LambdaTwist::rotation_convergence_iterations`]. pub fn rotation_convergence_iterations(self, rotation_convergence_iterations: usize) -> Self { Self { rotation_convergence_iterations, ..self } } } impl Default for LambdaTwist { fn default() -> Self { Self { gauss_newton_iterations: 5, rotation_convergence_iterations: 100, } } } impl<P> Estimator<FeatureWorldMatch<P>> for LambdaTwist where P: Bearing, { type Model = WorldPose; type ModelIter = Vec<WorldPose>; const MIN_SAMPLES: usize = 3; fn estimate<I>(&self, mut data: I) -> Self::ModelIter where I: Iterator<Item = FeatureWorldMatch<P>> + Clone, { compute_poses_nordberg( self.gauss_newton_iterations, self.rotation_convergence_iterations, [ data.next() .expect("must provide 3 samples at minimum to LambdaTwist"), data.next() .expect("must provide 3 samples at minimum to LambdaTwist"), data.next() .expect("must provide 3 samples at minimum to LambdaTwist"), ], ) } } /// Refine a valid solution with a Gauss-Newton Solver. /// `refined_lambda = gauss_newton_refine_lambda(lambda, a12, a13, a23, b12, b13, b23);` /// lambda: Vec3, the solution to refine. /// a12: f64, the squared distance between 3D point 1 and 3D point 2. /// a13: f64, the squared distance between 3D point 1 and 3D point 3. /// a23: f64, the squared distance between 3D point 2 and 3D point 3. /// b12: f64, -2.0 * cosine of the angle between bearing vector 1 and bearing vector 2. /// b13: f64, -2.0 * cosine of the angle between bearing vector 1 and bearing vector 3. /// b23: f64, -2.0 * cosine of the angle between bearing vector 2 and bearing vector 3. /// The paper note it rarely improve after two iterations. The original implementation use 5 iterations. #[allow(clippy::similar_names)] #[allow(clippy::too_many_arguments)] fn gauss_newton_refine_lambda( lambda: Vec3, iterations: usize, a12: f64, a13: f64, a23: f64, b12: f64, b13: f64, b23: f64, ) -> Vec3 { let compute_residual = |l: &Vec3| { let l1 = l.x; let l2 = l.y; let l3 = l.z; let r1 = l1 * l1 + l2 * l2 + b12 * l1 * l2 - a12; let r2 = l1 * l1 + l3 * l3 + b13 * l1 * l3 - a13; let r3 = l2 * l2 + l3 * l3 + b23 * l2 * l3 - a23; (l1, l2, l3, Vec3::new(r1, r2, r3)) }; let (mut l1, mut l2, mut l3, mut res) = compute_residual(&lambda); for _ in 0..iterations { if l1_norm(res) < 1e-10 { break; } let dr1dl1 = 2.0 * l1 + b12 * l2; let dr1dl2 = 2.0 * l2 + b12 * l1; let dr2dl1 = 2.0 * l1 + b13 * l3; let dr2dl3 = 2.0 * l3 + b13 * l1; let dr3dl2 = 2.0 * l2 + b23 * l3; let dr3dl3 = 2.0 * l3 + b23 * l2; let det = 1.0 / (-dr1dl1 * dr2dl3 * dr3dl2 - dr1dl2 * dr2dl1 * dr3dl3); #[rustfmt::skip] let jacobian = Mat3::new( -dr2dl3 * dr3dl2, -dr1dl2 * dr3dl3, dr1dl2 * dr2dl3, -dr2dl1 * dr3dl3, dr1dl1 * dr3dl3, -dr1dl1 * dr2dl3, dr2dl1 * dr3dl2, -dr1dl1 * dr3dl2, -dr1dl2 * dr2dl1, ); let lambda_new = Vec3::new(l1, l2, l3) - det * (jacobian * res); let (l1_new, l2_new, l3_new, res_new) = compute_residual(&lambda_new); if l1_norm(res_new) > l1_norm(res) { break; } else { l1 = l1_new; l2 = l2_new; l3 = l3_new; res = res_new; } } Vec3::new(l1, l2, l3) } /// Compute L1 norm of a vector. /// L1 norm is the sum of magnitudes. #[inline] fn l1_norm(v: Vec3) -> f64 { v.x.abs() + v.y.abs() + v.z.abs() } /// Compute the real roots of "h(r) = r^2 + b*r + c = 0". /// `let (roots_are_real, r1, r2) = root2real(b, c);` fn root2real(b: f64, c: f64) -> (bool, f64, f64) { let discriminant = b * b - 4.0 * c; if discriminant < 0.0 { let root = 0.5 * b; (false, root, root) } else if b < 0.0 { let y = Float::sqrt(discriminant); (true, 0.5 * (-b + y), 0.5 * (-b - y)) } else { let y = Float::sqrt(discriminant); (true, 2.0 * c / (-b + y), 2.0 * c / (-b - y)) } } /// Compute a single root of the cubic polynomial equation /// "h(r) = r^3 + b*r^2 + c*r + d = 0". /// `let root = cubick(b, c, d);` /// /// The return root is as stable as possible in the sense that it has as high /// derivative as possible. The solution is found by simple Newton-Raphson iterations, /// and the trick is to choose the initial solution r0 in a clever way. /// /// The intial solution is found by considering 5 cases: /// /// Cases I and II: h has no stationary points. In this case its derivative /// is positive. The inital solution to the NR-iteration is r0 here h has /// minimal derivative. /// /// Case III, IV, and V: has two stationary points, t1 < t2. /// In this case, h has negative derivative between t1 and t2. /// In these cases, we can make a second order approximation of h /// around each of t1 and t2, and choose r0 /// as the leftmost or rightmost root of these approximations, /// depending on whether zero, one, or both of h(t1) and h(t2) are > 0. #[allow(clippy::many_single_char_names)] fn cube_root(b: f64, c: f64, d: f64) -> f64 { // Choose an initial solution. let mut r0; // Not monotonic. if b * b >= 3.0 * c { // h has two stationary points, compute them. // double t1 = t - sqrt(diff); let v = (b * b - 3.0 * c).sqrt(); let t1 = (-b - v) / 3.0; // Check if h(t1) > 0, in this case make a 2-order approx of h around t1. let mut k = ((t1 + b) * t1 + c) * t1 + d; if k > 0.0 { // Find leftmost root of 0.5 * (r0 - t1)^2 * (6 * t1 + 2 * b) + k = 0. r0 = t1 - (-k / (3.0 * t1 + b)).sqrt(); } else { let t2 = (-b + v) / 3.0; k = ((t2 + b) * t2 + c) * t2 + d; // Find rightmost root of 0.5 * (r0 - t2)^2 * (6 * t2 + 2 * b) + k1 = 0. r0 = t2 + (-k / (3.0 * t2 + b)).sqrt(); } } else { // r0 = 1.0 / cubick_inv(c/d, b/d, 1.0/d); r0 = -b / 3.0; if ((3.0 * r0 + 2.0 * b) * r0 + c).abs() < 1e-4 { r0 += 1.0; } } // Newton-Raphson iterations (hardcoded, perform at least 7 iterations, at most 50). // Break if position of root changes less than 1e-13. // According to the author, increasing it could lead to a better solution (more robust). for _ in 0..7 { let fx = ((r0 + b) * r0 + c) * r0 + d; let fpx = (3.0 * r0 + 2.0 * b) * r0 + c; r0 -= fx / fpx; } for _ in 0..43 { let fx = ((r0 + b) * r0 + c) * r0 + d; if fx.abs() > 1e-13 { let fpx = (3.0 * r0 + 2.0 * b) * r0 + c; r0 -= fx / fpx; } else { break; } } r0 } /// Eigen decomposition of a singular matrix (which has a 0 eigen value). /// `let (eigenvectors, eigenvalues) = eigen_decomposition_singular(x);` fn eigen_decomposition_singular(x: Mat3) -> (Mat3, Vec3) { let mut eigenvalues = Vec3::zeros(); #[rustfmt::skip] let mut v3 = Vec3::new( x[1] * x[5] - x[2] * x[4], x[2] * x[3] - x[5] * x[0], x[4] * x[0] - x[1] * x[3], ); v3.normalize_mut(); let x12_sqr = x.m12 * x.m12; let b = -x.m11 - x.m22 - x.m33; let c = -x12_sqr - x.m13 * x.m13 - x.m23 * x.m23 + x.m11 * (x.m22 + x.m33) + x.m22 * x.m33; let (_, mut e1, mut e2) = root2real(b, c); if e1.abs() < e2.abs() { core::mem::swap(&mut e1, &mut e2); } eigenvalues[0] = e1; eigenvalues[1] = e2; let mx0011 = -x.m11 * x.m22; let prec_0 = x.m12 * x.m23 - x.m13 * x.m22; let prec_1 = x.m12 * x.m13 - x.m11 * x.m23; let compute_eigen_vector = |e: f64| { let tmp = 1.0 / (e * (x.m11 + x.m22) + mx0011 - e * e + x12_sqr); let mut a1 = -(e * x.m13 + prec_0) * tmp; let mut a2 = -(e * x.m23 + prec_1) * tmp; let rnorm = 1.0 / (a1 * a1 + a2 * a2 + 1.0).sqrt(); a1 *= rnorm; a2 *= rnorm; Vec3::new(a1, a2, rnorm) }; let v1 = compute_eigen_vector(e1); let v2 = compute_eigen_vector(e2); #[rustfmt::skip] let eigenvectors = Mat3::new( v1[0], v2[0], v3[0], v1[1], v2[1], v3[1], v1[2], v2[2], v3[2], ); (eigenvectors, eigenvalues) } /// Compute the pose of a camera using three 3D-to-2D correspondences. /// Implementation of the paper /// "Lambda Twist: An Accurate Fast Robust Perspective Three Point (P3P) Solver". /// Persson, M. and Nordberg, K. ECCV 2018. /// /// `let poses = compute_poses_nordberg(world_3d_points, bearing_vectors);` /// /// The 3x3 matrix `world_3d_points` contains one 3D point per column. /// The 3x3 matrix `bearing_vectors` contains one homogeneous image coordinate per column. fn compute_poses_nordberg<P: Bearing>( iterations: usize, rotation_convergence_iterations: usize, samples: [FeatureWorldMatch<P>; 3], ) -> Vec<WorldPose> { // Extraction of 3D points vectors let to_wp = |&FeatureWorldMatch(_, WorldPoint(point))| point; let wps = [to_wp(&samples[0]), to_wp(&samples[1]), to_wp(&samples[2])]; let to_bearing = |FeatureWorldMatch(point, _): &FeatureWorldMatch<P>| point.bearing(); let bearings = [ to_bearing(&samples[0]), to_bearing(&samples[1]), to_bearing(&samples[2]), ]; // Compute vectors between 3D points. let d12 = wps[0] - wps[1]; let d13 = wps[0] - wps[2]; let d23 = wps[1] - wps[2]; let d12xd13 = d12.cross(&d13); // "a12" is the squared distance between 3D points 1 and 2. let a12 = d12.norm_squared(); let a13 = d13.norm_squared(); let a23 = d23.norm_squared(); // "c31" is the cosine between bearing vectors 3 and 1. let c12 = bearings[0].dot(&bearings[1]); let c23 = bearings[1].dot(&bearings[2]); let c31 = bearings[2].dot(&bearings[0]); let blob = c12 * c23 * c31 - 1.0; // "s31" is the sine between bearing vectors 3 and 1. let s12_sqr = 1.0 - c12 * c12; let s23_sqr = 1.0 - c23 * c23; let s31_sqr = 1.0 - c31 * c31; // Other useful constants. let b12 = -2.0 * c12; let b13 = -2.0 * c31; let b23 = -2.0 * c23; // "p[0-3]" here are the four coefficients of the cubic polynomial. // They are refered to as "c[0-3]" in equation (10) of the paper. let p3 = a13 * (a23 * s31_sqr - a13 * s23_sqr); let p2 = 2.0 * blob * a23 * a13 + a13 * (2.0 * a12 + a13) * s23_sqr + a23 * (a23 - a12) * s31_sqr; let p1 = a23 * (a13 - a23) * s12_sqr - a12 * a12 * s23_sqr - 2.0 * a12 * (blob * a23 + a13 * s23_sqr); let p0 = a12 * (a12 * s23_sqr - a23 * s12_sqr); // Get sharpest real root of above. let g = cube_root(p2 / p3, p1 / p3, p0 / p3); // Build the matrix called D0 in the paper. let d0_00 = a23 * (1.0 - g); let d0_01 = -(a23 * c12); let d0_02 = a23 * c31 * g; let d0_11 = a23 - a12 + a13 * g; let d0_12 = -c23 * (a13 * g - a12); let d0_22 = g * (a13 - a23) - a12; #[rustfmt::skip] let d0_mat = Mat3::new( d0_00, d0_01, d0_02, d0_01, d0_11, d0_12, d0_02, d0_12, d0_22, ); // Get sorted eigenvectors and eigenvalues of the singular matrix D0. let (eig_vectors, eig_values) = eigen_decomposition_singular(d0_mat); // Initialize the possible depths triplets for the three image points. // There might be between 0 and 4 possible solutions. let mut lambdas = Vec::with_capacity(4); // Solve the four possible solutions for the depths values. let eigen_ratio = (0.0_f64.max(-eig_values[1] / eig_values[0])).sqrt(); // Helper closure to compute quadratic coefficients. // CF equation (15) in paper. let quadratic_coefficients = |ratio: f64| { let w2 = 1.0 / (ratio * eig_vectors.m12 - eig_vectors.m11); let w0 = w2 * (eig_vectors.m21 - ratio * eig_vectors.m22); let w1 = w2 * (eig_vectors.m31 - ratio * eig_vectors.m32); let a = 1.0 / ((a13 - a12) * w1 * w1 - a12 * b13 * w1 - a12); let b = a * (a13 * b12 * w1 - a12 * b13 * w0 - 2.0 * w0 * w1 * (a12 - a13)); let c = a * ((a13 - a12) * w0 * w0 + a13 * b12 * w0 + a13); (w0, w1, b, c) }; // Helper closure to estimate possible depths values. // CF equation (16) in paper. let possible_depths = |tau: f64, w0: f64, w1: f64| { let d = a23 / (tau * (b23 + tau) + 1.0); if d > 0.0 { let l2 = d.sqrt(); let l3 = tau * l2; let l1 = w0 * l2 + w1 * l3; (true, l1, l2, l3) } else { (false, 0.0, 0.0, 0.0) } }; // Helper closure pushing one potential solution. let mut push_solution = |tau: f64, w0: f64, w1: f64| { if tau > 0.0 { let (valid, l1, l2, l3) = possible_depths(tau, w0, w1); if valid && l1 >= 0.0 { lambdas.push(Vec3::new(l1, l2, l3)); } } }; // Helper closure pushing two potential solutions // corresponding to a given eigen value ratio. let mut push_solutions_to_lambdas = |ratio: f64| { let (w0, w1, b, c) = quadratic_coefficients(ratio); if b * b - 4.0 * c >= 0.0 { let (_, tau1, tau2) = root2real(b, c); push_solution(tau1, w0, w1); push_solution(tau2, w0, w1); } }; push_solutions_to_lambdas(eigen_ratio); push_solutions_to_lambdas(-eigen_ratio); // Recover the rotation R and translation t such that // lambda_i * y_i = R * x_i + t // TODO: replace by the quaternion version. // According to the author, it works slightly better. #[rustfmt::skip] let x_mat = Mat3::new( d12[0], d13[0], d12xd13[0], d12[1], d13[1], d12xd13[1], d12[2], d13[2], d12xd13[2], ); let x_mat = if let Some(x_mat) = x_mat.try_inverse() { x_mat } else { return Vec::new(); }; lambdas .iter() .map(|&lambda| { // Refine estimated depth values. let lambda_refined = gauss_newton_refine_lambda(lambda, iterations, a12, a13, a23, b12, b13, b23); let ry1 = lambda_refined[0] * *bearings[0]; let ry2 = lambda_refined[1] * *bearings[1]; let ry3 = lambda_refined[2] * *bearings[2]; let yd1 = ry1 - ry2; let yd2 = ry1 - ry3; let yd1xd2 = yd1.cross(&yd2); #[rustfmt::skip] let y_mat = Mat3::new( yd1[0], yd2[0], yd1xd2[0], yd1[1], yd2[1], yd1xd2[1], yd1[2], yd2[2], yd1xd2[2], ); let rot = y_mat * x_mat; (rot, ry1 - rot * wps[0].coords) }) .map(|(rot, trans)| { let translation = Translation::from(trans); WorldPose(Iso3::from_parts( translation, Rotation3::from_matrix_eps( &rot, 1e-12, rotation_convergence_iterations, Rotation3::identity(), ), )) }) .collect() }