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use crate::{CameraPoint, KeyPointWorldMatch, KeyPointsMatch, NormalizedKeyPoint, WorldPoint}; use core::cmp::Ordering; use derive_more::{AsMut, AsRef, Deref, DerefMut, From, Into}; use nalgebra::{ dimension::{U2, U3, U7}, Isometry3, Matrix3, Matrix3x2, MatrixMN, Quaternion, Rotation3, Translation3, UnitQuaternion, Vector2, Vector3, Vector4, VectorN, SVD, }; use sample_consensus::Model; /// This contains a world pose, which is a pose of the world relative to the camera. /// This maps [`WorldPoint`] into [`CameraPoint`], changing an absolute position into /// a vector relative to the camera. #[derive(Debug, Clone, Copy, PartialEq, AsMut, AsRef, Deref, DerefMut, From, Into)] pub struct WorldPose(pub Isometry3<f64>); impl Model<KeyPointWorldMatch> for WorldPose { fn residual(&self, data: &KeyPointWorldMatch) -> f32 { let WorldPose(iso) = *self; let KeyPointWorldMatch(image, world) = *data; let new_bearing = (iso * world.coords).normalize(); let bearing_vector = image.to_homogeneous().normalize(); (1.0 - bearing_vector.dot(&new_bearing)) as f32 } } impl WorldPose { /// Computes difference between the image keypoint and the projected keypoint. pub fn projection_error(&self, correspondence: KeyPointWorldMatch) -> Vector2<f64> { let KeyPointWorldMatch(NormalizedKeyPoint(image), world) = correspondence; let NormalizedKeyPoint(projection) = self.project(world); image - projection } /// Projects the `WorldPoint` onto the camera as a `NormalizedKeyPoint`. pub fn project(&self, point: WorldPoint) -> NormalizedKeyPoint { self.transform(point).into() } /// Projects the [`WorldPoint`] into camera space as a [`CameraPoint`]. pub fn transform(&self, WorldPoint(point): WorldPoint) -> CameraPoint { let WorldPose(iso) = *self; CameraPoint(iso * point) } /// Computes the Jacobian of the projection in respect to the `WorldPose`. /// The Jacobian is in the format: /// ```no_build,no_run /// | dx/dtx dy/dPx | /// | dx/dty dy/dPy | /// | dx/dtz dy/dPz | /// | dx/dqr dy/dqr | /// | dx/dqx dy/dqx | /// | dx/dqy dy/dqy | /// | dx/dqz dy/dqz | /// ``` /// /// Where `t` refers to the translation vector and `q` refers to the unit quaternion. #[rustfmt::skip] pub fn projection_pose_jacobian(&self, point: WorldPoint) -> MatrixMN<f64, U7, U2> { let q = self.0.rotation.quaternion().coords; // World point (input) let p = point.0.coords; // Camera point (intermediate output) let pc = (self.0 * point.0).coords; // dP/dT (Jacobian of camera point in respect to translation component) let dp_dt = Matrix3::identity(); // d/dQv (Qv x (Qv x P)) let qv_qv_p = Matrix3::new( q.y * p.y + q.z * p.z, q.y * p.x - 2.0 * q.x * p.y, q.z * p.x - 2.0 * q.x * p.z, q.x * p.y - 2.0 * q.y * p.x, q.x * p.x + q.z * p.z, q.z * p.y - 2.0 * q.y * p.z, q.x * p.z - 2.0 * q.z * p.x, q.y * p.z - 2.0 * q.z * p.y, q.x * p.x + q.y * p.y ); // d/dQv (Qv x P) let qv_p = Matrix3::new( 0.0, -p.z, p.y, p.z, 0.0, -p.x, -p.y, p.x, 0.0, ); // dP/dQv = d/dQv (2 * Qs * Qv x P + 2 * Qv x (Qv x P)) // Jacobian of camera point in respect to vector component of quaternion let dp_dqv = 2.0 * (q.w * qv_p + qv_qv_p); // dP/Ds = d/Qs (2 * Qs * Qv x P) // Jacobian of camera point in respect to real component of quaternion let dp_ds = 2.0 * q.xyz().cross(&p); // dP/dT,Q (Jacobian of 3d camera point in respect to translation and quaternion) let dp_dtq = MatrixMN::<f64, U7, U3>::from_rows(&[ dp_dt.row(0).into(), dp_dt.row(1).into(), dp_dt.row(2).into(), dp_dqv.row(0).into(), dp_dqv.row(1).into(), dp_dqv.row(2).into(), dp_ds.transpose(), ]); // 1 / pz let pcz = pc.z.recip(); // - 1 / pz^2 let npcz2 = -(pcz * pcz); // dK/dp (Jacobian of normalized image coordinate in respect to 3d camera point) let dk_dp = Matrix3x2::new( pcz, 0.0, 0.0, pcz, npcz2, npcz2, ); dp_dtq * dk_dp } pub fn to_vec(&self) -> VectorN<f64, U7> { let Self(iso) = *self; let t = iso.translation.vector; let rc = iso.rotation.quaternion().coords; t.push(rc.x).push(rc.y).push(rc.z).push(rc.w) } pub fn from_vec(v: VectorN<f64, U7>) -> Self { Self(Isometry3::from_parts( Translation3::from(Vector3::new(v[0], v[1], v[2])), UnitQuaternion::from_quaternion(Quaternion::from(Vector4::new(v[3], v[4], v[5], v[6]))), )) } } impl From<CameraPose> for WorldPose { fn from(camera: CameraPose) -> Self { Self(camera.inverse()) } } /// This contains a camera pose, which is a pose of the camera relative to the world. /// This transforms camera points (with depth as `z`) into world coordinates. /// This also tells you where the camera is located and oriented in the world. #[derive(Debug, Clone, Copy, PartialEq, AsMut, AsRef, Deref, DerefMut, From, Into)] pub struct CameraPose(pub Isometry3<f64>); impl From<WorldPose> for CameraPose { fn from(world: WorldPose) -> Self { Self(world.inverse()) } } /// This contains a relative pose, which is a pose that transforms the [`CameraPoint`] /// of one image into the corresponding [`CameraPoint`] of another image. This transforms /// the point from the camera space of camera `A` to camera `B`. /// /// Camera space for a given camera is defined as thus: /// /// * Origin is the optical center /// * Positive z axis is forwards /// * Positive y axis is up /// * Positive x axis is right /// /// Note that this is a left-handed coordinate space. #[derive(Debug, Clone, Copy, PartialEq, AsMut, AsRef, Deref, DerefMut, From, Into)] pub struct RelativeCameraPose(pub Isometry3<f64>); impl RelativeCameraPose { /// The relative pose transforms the point in camera space from camera `A` to camera `B`. pub fn transform(&self, CameraPoint(point): CameraPoint) -> CameraPoint { let Self(iso) = *self; CameraPoint(iso * point) } /// Generates an essential matrix corresponding to this relative camera pose. /// /// If a point `a` is transformed using [`RelativeCameraPose::transform`] into /// a point `b`, then the essential matrix returned by this method will /// give a residual of approximately `0.0` when you call /// `essential.residual(&KeyPointsMatch(a.into(), b.into()))`. /// /// See the documentation of [`EssentialMatrix`] for more information. /// /// ``` /// # use cv_core::{RelativeCameraPose, CameraPoint, KeyPointsMatch}; /// # use cv_core::sample_consensus::Model; /// # use cv_core::nalgebra::{Vector3, Point3, Isometry3, UnitQuaternion}; /// let pose = RelativeCameraPose(Isometry3::from_parts( /// Vector3::new(0.3, 0.4, 0.5).into(), /// UnitQuaternion::from_euler_angles(0.2, 0.3, 0.4), /// )); /// let a = CameraPoint(Point3::new(0.5, 0.5, 3.0)); /// let b = pose.transform(a); /// assert!(pose.essential_matrix().residual(&KeyPointsMatch(a.into(), b.into())) < 1e-6); /// ``` pub fn essential_matrix(&self) -> EssentialMatrix { EssentialMatrix( self.0.translation.vector.cross_matrix() * *self.0.rotation.to_rotation_matrix().matrix(), ) } } /// This stores an essential matrix, which is satisfied by the following constraint: /// /// transpose(x') * E * x = 0 /// /// Where `x'` and `x` are homogeneous normalized image coordinates. You can get a /// homogeneous normalized image coordinate by appending `1.0` to a `NormalizedKeyPoint`. /// /// The essential matrix embodies the epipolar constraint between two images. Given that light /// travels in a perfectly straight line (it will not, but for short distances it mostly does) /// and assuming a pinhole camera model, for any point on the camera sensor, the light source /// for that point exists somewhere along a line extending out from the bearing (direction /// of travel) of that point. For a normalized image coordinate, that bearing is `(x, y, 1.0)`. /// That is because normalized image coordinates exist on a virtual plane (the sensor) /// a distance `z = 1.0` from the optical center (the location of the focal point) where /// the unit of distance is the focal length. In epipolar geometry, the point on the virtual /// plane is called an epipole. The line through 3d space created by the bearing that travels /// from the optical center through the epipole is called an epipolar line. /// /// If you look at every point along an epipolar line, each one of those points would show /// up as a different point on the camera sensor of another image (if they are in view). /// If you traced every point along this epipolar line to where it would appear on the sensor /// of the camera (projection of the 3d points into normalized image coordinates), then /// the points would form a straight line. This means that you can draw epipolar lines /// that do not pass through the optical center of an image on that image. /// /// The essential matrix makes it possible to create a vector that is perpendicular to all /// bearings that are formed from the epipolar line on the second image's sensor. This is /// done by computing `E * x`, where `x` is a homogeneous normalized image coordinate /// from the first image. The transpose of the resulting vector then has a dot product /// with the transpose of the second image coordinate `x'` which is equal to `0.0`. /// This can be written as: /// /// ```no_build,no_run /// dot(transpose(E * x), x') = 0 /// ``` /// /// This can be re-written into the form given above: /// /// ```no_build,no_run /// transpose(x') * E * x = 0 /// ``` /// /// Where the first operation creates a pependicular vector to the epipoles on the first image /// and the second takes the dot product which should result in 0. /// /// With a `EssentialMatrix`, you can retrieve the rotation and translation given /// one normalized image coordinate and one bearing that is scaled to the depth /// of the point relative to the current reconstruction. This kind of point can be computed /// using [`WorldPose::project_camera`] to convert a [`WorldPoint`] to a [`CameraPoint`]. #[derive(Debug, Clone, Copy, PartialEq, PartialOrd, AsMut, AsRef, Deref, DerefMut, From, Into)] pub struct EssentialMatrix(pub Matrix3<f64>); impl Model<KeyPointsMatch> for EssentialMatrix { fn residual(&self, data: &KeyPointsMatch) -> f32 { let Self(mat) = *self; let KeyPointsMatch(NormalizedKeyPoint(a), NormalizedKeyPoint(b)) = *data; // The result is a 1x1 matrix which we must get element 0 from. (b.to_homogeneous().transpose() * mat * a.to_homogeneous())[0] as f32 } } impl EssentialMatrix { /// Returns two possible rotations for the essential matrix along with a translation /// bearing of arbitrary length. The translation bearing is not yet in the correct /// space and the inverse rotation (transpose) must be multiplied by the translation /// bearing to make the translation bearing be post-rotation. The translation's length /// is unknown and of unknown sign and must be solved for by using a prior. /// /// `epsilon` is the threshold by which the singular value decomposition is considered /// complete. Making this smaller may improve the precision. It is recommended to /// set this to no higher than `1e-6`. /// /// `max_iterations` is the maximum number of iterations that singular value decomposition /// will run on this matrix. Use this in soft realtime systems to cap the execution time. /// A `max_iterations` of `0` may execute indefinitely and is not recommended. /// /// ``` /// # use cv_core::RelativeCameraPose; /// # use cv_core::nalgebra::{Isometry3, UnitQuaternion, Vector3}; /// let pose = RelativeCameraPose(Isometry3::from_parts( /// Vector3::new(-0.8, 0.4, 0.5).into(), /// UnitQuaternion::from_euler_angles(0.2, 0.3, 0.4), /// )); /// // Get the possible poses for the essential matrix created from `pose`. /// let (rot_a, rot_b, t) = pose.essential_matrix().possible_poses_unfixed_bearing(1e-6, 50).unwrap(); /// // The translation must be processed through the reverse rotation. /// let t_a = t; /// let t_b = t; /// // Extract vector from quaternion. /// let qcoord = |uquat: UnitQuaternion<f64>| uquat.quaternion().coords; /// // Convert rotations into quaternion form. /// let quat_a = UnitQuaternion::from(rot_a); /// let quat_b = UnitQuaternion::from(rot_b); /// // Compute residual via cosine distance of quaternions (guaranteed positive w). /// let a_res = quat_a.rotation_to(&pose.rotation).angle(); /// let b_res = quat_b.rotation_to(&pose.rotation).angle(); /// let a_close = a_res < 1e-4; /// let b_close = b_res < 1e-4; /// // At least one rotation is correct. /// assert!(a_close || b_close); /// // The translation points in the same (or reverse) direction /// let a_res = 1.0 - t_a.normalize().dot(&pose.translation.vector.normalize()).abs(); /// let b_res = 1.0 - t_b.normalize().dot(&pose.translation.vector.normalize()).abs(); /// let a_close = a_res < 1e-4; /// let b_close = b_res < 1e-4; /// assert!(a_close || b_close); /// ``` pub fn possible_poses_unfixed_bearing( &self, epsilon: f64, max_iterations: usize, ) -> Option<(Rotation3<f64>, Rotation3<f64>, Vector3<f64>)> { let Self(essential) = *self; let essential = essential; // `W` from https://en.wikipedia.org/wiki/Essential_matrix#Finding_one_solution. let w = Matrix3::new(0.0, -1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0); // Transpose of `W` from https://en.wikipedia.org/wiki/Essential_matrix#Finding_one_solution. let wt = w.transpose(); // Perform SVD. let svd = SVD::try_new(essential, true, true, epsilon, max_iterations); // Extract only the U and V matrix from the SVD. let u_v_t = svd.map(|svd| { if let SVD { u: Some(u), v_t: Some(v_t), singular_values, } = svd { // Sort the singular vectors in U and V*. let mut sources: [usize; 3] = [0, 1, 2]; sources.sort_unstable_by_key(|&ix| float_ord::FloatOrd(-singular_values[ix])); let mut sorted_u = Matrix3::zeros(); let mut sorted_v_t = Matrix3::zeros(); for (&ix, mut column) in sources.iter().zip(sorted_u.column_iter_mut()) { column.copy_from(&u.column(ix)); } for (&ix, mut row) in sources.iter().zip(sorted_v_t.row_iter_mut()) { row.copy_from(&v_t.row(ix)); } (sorted_u, sorted_v_t) } else { panic!("Didn't get U and V matrix in SVD"); } }); // Force the determinants to be positive. I do not know precisely // why this is done since it isn't apparent from the Wikipedia page // on this subject, but this is what TheiaSfM does in essential_matrix_utils.cc. let u_v_t = u_v_t.map(|(mut u, mut v_t)| { // Last column of U is undetermined since d = (a a 0). if u.determinant() < 0.0 { for n in u.column_mut(2).iter_mut() { *n *= -1.0; } } // Last row of Vt is undetermined since d = (a a 0). if v_t.determinant() < 0.0 { for n in v_t.row_mut(2).iter_mut() { *n *= -1.0; } } // Return positive determinant U and V*. (u, v_t) }); // Compute the possible rotations and the bearing with no normalization. u_v_t.map(|(u, v_t)| { ( Rotation3::from_matrix_unchecked(u * w * v_t), Rotation3::from_matrix_unchecked(u * wt * v_t), u.column(2).into_owned(), ) }) } /// See [`EssentialMatrix::possible_poses_unfixed_bearing`]. /// /// This returns only the two rotations that are possible. /// /// ``` /// # use cv_core::RelativeCameraPose; /// # use cv_core::nalgebra::{Isometry3, UnitQuaternion, Vector3}; /// let pose = RelativeCameraPose(Isometry3::from_parts( /// Vector3::new(-0.8, 0.4, 0.5).into(), /// UnitQuaternion::from_euler_angles(0.2, 0.3, 0.4), /// )); /// // Get the possible rotations for the essential matrix created from `pose`. /// let rbs = pose.essential_matrix().possible_rotations(1e-6, 50).unwrap(); /// let one_correct = rbs.iter().any(|&rot| { /// let angle_residual = /// UnitQuaternion::from(rot).rotation_to(&pose.rotation).angle(); /// angle_residual < 1e-4 /// }); /// assert!(one_correct); /// ``` pub fn possible_rotations( &self, epsilon: f64, max_iterations: usize, ) -> Option<[Rotation3<f64>; 2]> { self.possible_poses_unfixed_bearing(epsilon, max_iterations) .map(|(rot_a, rot_b, _)| [rot_a, rot_b]) } /// See [`EssentialMatrix::possible_poses_unfixed_bearing`]. /// /// This returns the rotations and their corresponding post-rotation translation bearing. /// /// ``` /// # use cv_core::RelativeCameraPose; /// # use cv_core::nalgebra::{Isometry3, UnitQuaternion, Vector3}; /// let pose = RelativeCameraPose(Isometry3::from_parts( /// Vector3::new(-0.8, 0.4, 0.5).into(), /// UnitQuaternion::from_euler_angles(0.2, 0.3, 0.4), /// )); /// // Get the possible poses for the essential matrix created from `pose`. /// let rbs = pose.essential_matrix().possible_rotations_and_bearings(1e-6, 50).unwrap(); /// let one_correct = rbs.iter().any(|&(rot, bearing)| { /// let angle_residual = /// UnitQuaternion::from(rot).rotation_to(&pose.rotation).angle(); /// let translation_residual = /// 1.0 - bearing.normalize().dot(&pose.translation.vector.normalize()).abs(); /// angle_residual < 1e-4 && translation_residual < 1e-4 /// }); /// assert!(one_correct); /// ``` pub fn possible_rotations_and_bearings( &self, epsilon: f64, max_iterations: usize, ) -> Option<[(Rotation3<f64>, Vector3<f64>); 2]> { self.possible_poses_unfixed_bearing(epsilon, max_iterations) .map(|(rot_a, rot_b, t)| [(rot_a, t), (rot_b, t)]) } /// Return the [`RelativeCameraPose`] that transforms a [`CameraPoint`] of image /// `A` (source of `a`) to the corresponding [`CameraPoint`] of image B (source of `b`). /// This determines the average expected translation from the points themselves and /// if the points agree with the rotation (points must be in front of the camera). /// The function takes an iterator containing tuples in the form `(depth, a, b)`: /// /// * `depth` - The actual depth (`z` axis, not distance) of normalized keypoint `a` /// * `a` - A keypoint from image `A` /// * `b` - A keypoint from image `B` /// /// `self` must satisfy the constraint: /// /// ```no_build,no_run /// transpose(homogeneous(a)) * E * homogeneous(b) = 0 /// ``` /// /// Also, `a` and `b` must be a correspondence. /// /// This will take the average translation over the entire iterator. This is done /// to smooth out noise and outliers (if present). /// /// `consensus_ratio` is the ratio of points which must be in front of the camera for the model /// to be accepted and return Some. Otherwise, None is returned. /// /// `max_iterations` is the maximum number of iterations that singular value decomposition /// will run on this matrix. Use this in soft realtime systems to cap the execution time. /// A `max_iterations` of `0` may execute indefinitely and is not recommended. /// /// `bearing_scale` is a function that is passed a translation bearing vector, /// an untranslated (but rotated) camera point, and a normalized key point /// where the actual point exists. It must return the scalar which the /// translation bearing vector must by multiplied by to get the actual translation. /// /// `correspondences` must provide an iterator of tuples containing the depth /// (distance along the positive Z axis) of camera A's point `a`, a point `a` /// from camera A, and a point `b` from camera B. /// /// This does not communicate which points were outliers. /// /// ``` /// # use cv_core::{RelativeCameraPose, CameraPoint}; /// # use cv_core::nalgebra::{Isometry3, UnitQuaternion, Vector3, Point3}; /// let pose = RelativeCameraPose(Isometry3::from_parts( /// Vector3::new(-0.8, 0.4, 0.5).into(), /// UnitQuaternion::from_euler_angles(0.2, 0.3, 0.4), /// )); /// let a_point = CameraPoint(Point3::new(-1.0, 2.0, 3.0)); /// let b_point = pose.transform(a_point); /// // Get the possible poses for the essential matrix created from `pose`. /// let estimate_pose = pose.essential_matrix().solve_pose(0.5, 1e-6, 50, /// cv_core::geom::triangulate_bearing_reproject, /// std::iter::once((a_point, b_point.into())), /// ).unwrap(); /// /// let angle_residual = /// estimate_pose.rotation.rotation_to(&pose.rotation).angle(); /// let translation_residual = (pose.translation.vector - estimate_pose.translation.vector).norm(); /// assert!(angle_residual < 1e-4 && translation_residual < 1e-4, "{}, {}, {:?}, {:?}", angle_residual, translation_residual, pose.translation.vector, estimate_pose.translation.vector); /// ``` pub fn solve_pose( &self, consensus_ratio: f64, epsilon: f64, max_iterations: usize, bearing_scale: impl Fn(Vector3<f64>, CameraPoint, NormalizedKeyPoint) -> f64, correspondences: impl Iterator<Item = (CameraPoint, NormalizedKeyPoint)>, ) -> Option<RelativeCameraPose> { // Get the possible rotations and the translation self.possible_rotations_and_bearings(epsilon, max_iterations) .and_then(|poses| { // Get the net translation scale of points that agree with a and b // in addition to the number of points that agree with a and b. let (ts, total) = correspondences.fold( ([(0.0, 0usize); 2], 0usize), |(mut ts, total), (a, b)| { let trans_and_agree = |(rot, bearing)| { // Triangulate the position of the CameraPoint of b. // We know the precise 3d position of the a point relative // to camera A, but we do not know the // 3d point in relation to camera B since the translation of // the point is unknown. We do know the direction of translation // of the point. We know only the rotation of the camera B // relative to camera A and the epipolar point on camera B. // What we will need to do is start by rotating the point in space. // After rotating the point, we then need to solve for the translation // that minimizes the reprojection error of the untranslated point as much // as possible. See the documentation for reproject_along_translation // to get more details on the process. let untranslated = CameraPoint(rot * a.0); let translation_scale = bearing_scale(bearing, untranslated, b); // Now that we have the translation, we can just verify that the point // is in front (z > 1.0) of the camera to see if it agrees with the model. ( translation_scale, translation_scale * bearing.z + untranslated.z > 1.0, ) }; // Do it for both poses. for (tn, &pose) in ts.iter_mut().zip(&poses) { if let (scale, true) = trans_and_agree(pose) { tn.0 += scale; tn.1 += 1; } } (ts, total + 1) }, ); // Ensure that there is at least one point. if total == 0 { return None; } // Ensure that the best one exceeds the consensus ratio. let best = core::cmp::max(ts[0].1, ts[1].1); if (best as f64 / total as f64) < consensus_ratio && best != 0 { return None; } // TODO: Perhaps if its closer than this we should assume the frame itself is an outlier. let (rot, trans) = match ts[0].1.cmp(&ts[1].1) { Ordering::Equal => return None, Ordering::Greater => (poses[0].0, poses[0].1 * ts[0].0 / ts[0].1 as f64), Ordering::Less => (poses[1].0, poses[1].1 * ts[1].0 / ts[1].1 as f64), }; Some(RelativeCameraPose(Isometry3::from_parts( trans.into(), UnitQuaternion::from_rotation_matrix(&rot), ))) }) } }