# Structure from Motion (SfM)¶

Theia has a full Structure-from-Motion pipeline that is extremely efficient. Our overall pipeline consists of several steps. First, we extract features (SIFT is the default). Then, we perform two-view matching and geometric verification to obtain relative poses between image pairs and create a ViewGraph. Next, we perform global pose estimation with global SfM. Global SfM is different from incremental SfM in that it considers the entire view graph at the same time instead of incrementally adding more and more images to the Reconstruction. Global SfM methods have been proven to be very fast with comparable or better accuracy to incremental SfM approaches (See [JiangICCV], [MoulonICCV], [WilsonECCV2014]), and they are much more readily parallelized. After we have obtained camera poses, we perform triangulation and BundleAdjustment to obtain a valid 3D reconstruction consisting of cameras and 3D points.

The first step towards creating a reconstruction is to determine images which view the same objects. To do this, we must create a ViewGraph.

1. Extract features in images.
2. Match features to obtain image correspondences.
3. Estimate camera poses from two-view matches and geometries.

#1. and #2. have been covered in other sections, so we will focus on creating a reconstruction from two-view matches and geometry. First, we will describe the fundamental elements of our reconstruction.

## Reconstruction¶

class Reconstruction

At the core of our SfM pipeline is an SfM Reconstruction. A Reconstruction is the representation of a 3D reconstuction consisting of a set of unique Views and Tracks. More importantly, the Reconstruction class contains visibility information relating all of the Views and Tracks to each other. We identify each View uniquely based on the name (a string). A good name for the view is the filename of the image that corresponds to the View

When creating an SfM reconstruction, you should add each View and Track through the Reconstruction object. This will ensure that visibility information (such as which Tracks are observed a given View and which Views see a given Track) stays accurate. Views and Tracks are given a unique ID when added to the Reconstruction to help make use of these structures lightweight and efficient.

ViewId Reconstruction::AddView(const std::string &view_name)

Adds a view to the reconstruction with the default initialization. The ViewId returned is guaranteed to be unique or will be kInvalidViewId if the method fails. Each view is uniquely identified by the view name (a good view name could be the filename of the image).

bool Reconstruction::RemoveView(const ViewId view_id)

Removes the view from the reconstruction and removes all references to the view in the tracks.

Note

Any tracks that have length 0 after the view is removed will also be removed.

int Reconstruction::NumViews() const
int Reconstruction::NumTracks() const
const View *Reconstruction::View(const ViewId view_id) const
View *Reconstruction::MutableView(const ViewId view_id)

Returns the View or a nullptr if the track does not exist.

std::vector<ViewId> Reconstruction::ViewIds() const

Return all ViewIds in the reconstruction.

ViewId Reconstruction::ViewIdFromName(const std::string &view_name) const

Returns to ViewId of the view name, or kInvalidViewId if the view does not exist.

TrackId Reconstruction::AddTrack(const std::vector<std::pair<ViewId, Feature>> &track)

Add a track to the reconstruction with all of its features across views that observe this track. Each pair contains a feature and the corresponding View name (i.e., the string) that observes the feature. A new View will be created if a View with the view name does not already exist. This method will not estimate the position of the track. The TrackId returned will be unique or will be kInvalidTrackId if the method fails.

bool Reconstruction::RemoveTrack(const TrackId track_id)

Removes the track from the reconstruction and from any Views that observe this track. Returns true on success and false on failure (e.g., the track does not exist).

const Track *Reconstruction::Track(const TrackId track_id) const
Track *Reconstruction::MutableTrack(const TrackId track_id)

Returns the Track or a nullptr if the track does not exist.

std::vector<TrackId> Reconstruction::TrackIds() const

Return all TrackIds in the reconstruction.

## ViewGraph¶

class ViewGraph

A ViewGraph is a basic SfM construct that is created from two-view matching information. Any pair of views that have a view correlation form an edge in the ViewGraph such that the nodes in the graph are View that are connected by TwoViewInfo objects that contain information about the relative pose between the Views as well as matching information.

Once you have a set of views and match information, you can add them to the view graph:

std::vector<View> views;
// Match all views in the set.
std::vector<ViewIdPair, TwoViewInfo> view_pair_matches;

ViewGraph view_graph;
for (const auto& view_pair : view_pair_matches) {
const ViewIdPair& view_id_pair = view_pair.first;
const TwoViewInfo& two_view_info = view_pair.second;
// Only add view pairs to the view graph if they have strong visual coherence.
if (two_view_info.num_matched_features > min_num_matched_features) {
views[view_id_pair.second],
two_view_info);
}
}

// Process and/or manipulate the view graph.


The edge values are especially useful for one-shot SfM where the relative poses are heavily exploited for computing the final poses. Without a proper ViewGraph, one-shot SfM would not be possible.

## Views and Tracks¶

class View

At the heart of our SfM framework is the View class which represents everything about an image that we want to reconstruct. It contains information about features from the image, camera pose information, and EXIF information. Views make up our basic visiblity constraints and are a fundamental part of the SfM pipeline.

class Track

A Track represents a feature that has been matached over potentially many images. When a feature appears in multiple images it typically means that the features correspond to the same 3D point. These 3D points are useful constraints in SfM reconstruction, as they represent the “structure” in “Structure-from-Motion” and help to build a point cloud for our reconstruction.

## TwoViewInfo¶

class TwoViewInfo

After image matching is performed we can create a ViewGraph that explains the relative pose information between two images that have been matched. The TwoViewInfo struct is specified as:

struct TwoViewInfo {
double focal_length_1;
double focal_length_2;

Eigen::Vector3d position_2;
Eigen::Vector3d rotation_2;

// Number of features that were matched and geometrically verified betwen the
// images.
int num_verified_matches;
};


This information serves the purpose of an edge in the view graph that describes visibility information between all views. The relative poses here are used to estimate global poses for the cameras.

## Camera¶

class Camera

Each View contains a Camera object that contains intrinsic and extrinsic information about the camera that observed the scene. Theia has an efficient, compact Camera class that abstracts away common image operations. This greatly relieves the pain of manually dealing with calibration and geometric transformations of images. We represent camera intrinsics such that the calibration matrix is:

$\begin{split}K = \left[\begin{matrix}f & s & p_x \\ 0 & f * a & p_y \\ 0 & 0 & 1 \end{matrix} \right]\end{split}$

where $$f$$ is the focal length (in pixels), $$s$$ is the skew, $$a$$ is the aspect ratio and $$p$$ is the principle point of the camera. All of these intrinsics may be accessed with getter and setter methods, e.g., double GetFocalLenth() or void SetFocalLength(const double focal_length). Note that we do additionally allow for up to two radial distortion parameters, but these are not part of the calibration matrix so they must be set or retrieved separately from the corresponding getter/setter methods.

We store the camera pose information as the transformation which maps world coordinates into camera coordinates. Our rotation is stored internally as an SO(3) rotation, which makes optimization with BundleAdjustment more effective since the value is always a valid rotation (unlike e.g., Quaternions that must be normalized after each optimization step). However, for convenience we provide an interface to retrieve the rotation as a rotation matrix as well. Further, we store the camera position as opposed to the translation.

The convenience of this camera class is clear with the common example of 3D point reprojection.

// Open an image and obtain camera parameters.
FloatImage image("my_image.jpg");
double focal_length;
CHECK(image.FocalLengthPixels(&focal_length));
const double radial_distortion1 = value obtained elsewhere...
const double radial_distortion2 = value obtained elsewhere...
const Eigen::Matrix3d rotation = value obtained elsewhere...
const Eigen::Vector3d position = value obtained elsewhere...

// Set up the camera.
Camera camera;
camera.SetOrientationFromRotationMatrix(rotation);
camera.SetPosition(position);
camera.SetFocalLength(focal_length);
camera.SetPrincipalPoint(image.Width() / 2.0, image.Height() / 2.0);

// Obtain a homogeneous 3D point
const Eigen::Vector4d homogeneous_point3d = value obtained elsewhere...

// Reproject the 3D point to a pixel.
Eigen::Vector2d reprojection_pixel;
const double depth = camera.ProjectPoint(homogeneous_point3d, &pixel);
if (depth < 0) {
LOG(INFO) << "Point was behind the camera!";
}

LOG(INFO) << "Homogeneous 3D point: " << homogeneous_point3d
<< " reprojected to the pixel value of " << reprojection_pixel;


Point projection can be a tricky function when considering the camera intrinsics and extrinsics. Theia takes care of this projection (including radial distortion) in a simple and efficient manner.

In addition to typical getter/setter methods for the camera parameters, the Camera class also defines several helper functions:.

bool Camera::InitializeFromProjectionMatrix(const int image_width, const int image_height, const Matrix3x4d projection_matrix)

Initializes the camera intrinsic and extrinsic parameters from the projection matrix by decomposing the matrix with a RQ decomposition.

Note

The projection matrix does not contain information about radial distortion, so those parameters will need to be set separately.

void Camera::GetProjectionMatrix(Matrix3x4d *pmatrix) const

Returns the projection matrix. Does not include radial distortion.

void Camera::GetCalibrationMatrix(Eigen::Matrix3d *kmatrix) const

Returns the calibration matrix in the form specified above.

Eigen::Vector3d Camera::PixelToUnitDepthRay(const Eigen::Vector2d &pixel) const

Converts the pixel point to a ray in 3D space such that the origin of the ray is at the camera center and the direction is the pixel direction rotated according to the camera orientation in 3D space. The returned vector is not unit length.

## Global SfM Pipeline¶

The global SfM pipelines in Theia follow a general procedure of filtering outliers and estimating camera poses or structure. Removing outliers can help increase performance dramatically for global SfM, though robust estimation methods are still required to obtain good results. The general pipeline is as follows:

1. Create the intial view graph from 2-view matches and TwoViewInfo that describes the relative pose between matched images.
2. Filter initial view graph and remove outlier 2-view matches.
3. Calibrate internal parameters of all cameras (either from EXIF or another calibration method).
4. Estimate global orientations of each camera.
5. Filter the view graph: remove any TwoViewInfos where the relative rotation does not agree with the estimated global rotations.
6. Refine the relative translation estimation to account for the estimated global rotations.
7. Filter any bad TwoViewInfo based on the relative translations.
8. Estimate the global positions of all cameras from the estimated rotations and TwoViewInfo.
9. Estimate 3D points.
11. (Optional) Attempt to estimate any remaining 3D points and bundle adjust again.
class ReconstructionEstimator

This is the base class for which all SfM reconstruction pipelines derive from. The reconstruction estimation type can be specified at runtime, though currently only NONLINEAR is implemented.

ReconstructionEstimator::ReconstructionEstimator(const ReconstructorEstimatorOptions &options)
ReconstructionEstimator::ReconstructionEstimatorSummary Estimate(const ViewGraph &view_graph, Reconstruction *reconstruction)

Estimates the cameras poses and 3D points from a view graph. The details of each step in the estimation process are described below.

class ReconstructorEstimatorOptions
ReconstructionEstimatorType ReconstructorEstimatorOptions::reconstruction_estimator_type

DEFAULT: ReconstructionEstimatorType::NONLINEAR

Type of reconstruction estimation to use.

int ReconstructorEstimatorOptions::num_threads

DEFAULT: 1

Number of threads to use during the various stages of reconstruction.

double ReconstructorEstimatorOptions::max_reprojection_error_in_pixels

DEFAULT: 5.0

Maximum reprojection error. This is used to determine inlier correspondences for absolute pose estimation. Additionally, this is the threshold used for filtering outliers after bundle adjustment.

int ReconstructorEstimatorOptions::num_retriangulation_iterations

DEFAULT: 1

After computing a model and performing an initial BA, the reconstruction can be further improved (and even densified) if we attempt to retriangulate any tracks that are currently unestimated. For each retriangulation iteration we do the following:

1. Remove features that are above max_reprojection_error_in_pixels.
2. Triangulate all unestimated tracks.
bool ReconstructorEstimatorOptions::initialize_focal_lengths_from_median_estimate

DEFAULT: false

By default, focal lengths for uncalibrated cameras are initialized by setting the focal length to a value that corresponds to a reasonable field of view. If this is true, then we initialize the focal length of all uncalibrated cameras to the median value obtained from decomposing the fundamental matrix of all view pairs connected to that camera. Cameras with calibration or EXIF information are always calibrated using that information regardless of this parameter.

double ReconstructorEstimatorOptions::ransac_confidence

DEFAULT: 0.9999

Confidence using during RANSAC. This determines the quality and termination speed of RANSAC.

int ReconstructorEstimatorOptions::ransac_min_iterations

DEFAULT: 50

Minimum number of iterations for RANSAC.

int ReconstructorEstimatorOptions::ransac_max_iterations

DEFAULT: 1000

Maximum number of iterations for RANSAC.

bool ReconstructorEstimatorOptions::ransac_use_mle

DEFAULT: true

Using the MLE quality assesment (as opposed to simply an inlier count) can improve the quality of a RANSAC estimation with virtually no computational cost.

double ReconstructorEstimatorOptions::max_rotation_error_in_view_graph_cycles

DEFAULT: 3.0

Before orientations are estimated, some “bad” edges may be removed from the view graph by determining the consistency of rotation estimations in loops within the view graph. By examining loops of size 3 (i.e., triplets) the concatenated relative rotations should result in a perfect identity rotation. Any edges that break this consistency may be removed prior to rotation estimation.

double ReconstructorEstimatorOptions::rotation_filtering_max_difference_degrees

DEFAULT: 5.0

After orientations are estimated, view pairs may be filtered/removed if the relative rotation of the view pair differs from the relative rotation formed by the global orientation estimations. That is, measure the angulaar distance between $$R_{i,j}$$ and $$R_j * R_i^T$$ and if it greater than rotation_filtering_max_difference_degrees than we remove that view pair from the graph. Adjust this threshold to control the threshold at which rotations are filtered.

bool ReconstructorEstimatorOptions::refine_relative_translations_after_rotation_estimation

DEFAULT: true

Refine the relative translations based on the epipolar error and known rotation estimations. This can improve the quality of the translation estimation.

int ReconstructorEstimatorOptions::translation_filtering_num_iterations

DEFAULT: 48

double ReconstructorEstimatorOptions::translation_filtering_projection_tolerance

DEFAULT: 0.1

Before the camera positions are estimated, it is wise to remove any relative translations estimates that are low quality. We perform filtering using the 1dSfM technique of [WilsonECCV2014]. See theia/sfm/filter_view_pairs_from_relative_translation.h for more information.

double ReconstructorEstimatorOptions::rotation_estimation_robust_loss_scale

DEFAULT: 0.1

Robust loss function width for nonlinear rotation estimation.

double ReconstructorEstimatorOptions::position_estimation_robust_loss_scale

DEFAULT: 1.0

Robust loss function width to use for the first iteration of nonlinear position estimation.

int ReconstructorEstimatorOptions::position_estimation_min_num_tracks_per_view

DEFAULT: 10

Number of point to camera correspondences used for nonlinear position estimation.

double ReconstructorEstimatorOptions::position_estimation_point_to_camera_weight

DEFAULT: 0.5

Weight of point to camera constraints with respect to camera to camera constraints.

double ReconstructorEstimatorOptions::min_triangulation_angle_degrees

DEFAULT: 3.0

In order to triangulate features accurately, there must be a sufficient baseline between the cameras relative to the depth of the point. Points with a very high depth and small baseline are very inaccurate. We require that at least one pair of cameras has a sufficient viewing angle to the estimated track in order to consider the estimation successful.

bool ReconstructorEstimatorOptions::bundle_adjust_tracks

DEFAULT: true

Bundle adjust a track immediately after estimating it.

double ReconstructorEstimatorOptions::triangulation_max_reprojection_error_in_pixels

DEFAULT: 10.0

The reprojection error to use for determining a valid triangulation. If the reprojection error of any observation is greater than this value then we can consider the triangluation unsuccessful.

int ReconstructorEstimatorOptions::min_cameras_for_iterative_solver

DEFAULT: 1000

Use SPARSE_SCHUR for problems smaller than this size and ITERATIVE_SCHUR for problems larger than this size.

bool ReconstructorEstimatorOptions::constant_camera_intrinsics

DEFAULT: false

If accurate calibration is known ahead of time then it is recommended to set the camera intrinsics constant during bundle adjustment.

## Estimating Global Rotations¶

Theia estimates the global rotations of cameras robustly using a nonlinear optimization. Using the relative rotations obtained from all TwoViewInfo, we enforce the constraint that

(1)$R_{i,j} = R_j * R_i^T$

We use the angle-axis representation of rotations to ensure that proper rotations are formed. All pairwise constraints are put into a nonlinear optimization with a robust loss function and the global orienations are computed. The optimization usually converges within just a few iterations and provides a very accurate result. The nonlinear optimization is initialized by forming a random spanning tree of the view graph and walking along the edges. There are two potential methods that may be used.

bool EstimateRotationsNonlinear(const std::unordered_map<ViewIdPair, Eigen::Vector3d> &relative_rotations, const double robust_loss_width, std::unordered_map<ViewId, Eigen::Vector3d> *global_orientations)

Using the relative rotations and an initial guess for the global rotations, minimize the error between the relative rotations and the global orientations. We use as SoftL1Loss for robustness to outliers.

class RobustRotationEstimator

The nonlinear method above is very sensitive to initialization. We recommend to use the RobustRotationEstimator of [ChatterjeeICCV13]. This rotation estimator is similar in spirit to EstimateRotationsNonlinear(), however, it utilizes L1 minimization to maintain efficiency to outliers. After several iterations of L1 minimization, an iteratively reweighted least squares approach is used to refine the solution.

int RobustRotationEstimator::Options::max_num_l1_iterations

DEFAULT: 5

Maximum number of L1 iterations to perform before performing the reweighted least squares minimization. Typically only a very small number of L1 iterations are needed.

int RobustRotationEstimator::Options::max_num_irls_iterations

DEFAULT: 100

Maximum number of reweighted least squares iterations to perform. These steps are much faster than the L2 iterations.

RobustRotationEstimator::RobustRotationEstimator(const RobustRotationEstimator::Options &options, const std::unordered_map<ViewIdPair, Eigen::Vector3d> &relative_rotations)

The options and the relative rotations remain constant throughout the rotation estimation and are passed in the constructor.

bool RobustRotationEstimator::EstimateRotations(std::unordered_map<ViewId, Eigen::Vector3d> *global_orientations)

Estimates the global orientation using the robust method described above. An initial estimation for the rotations is required. [ChatterjeeICCV13] suggests to use a random spanning tree to initialize the rotations.

## Estimating Global Positions¶

Positions of cameras may be estimated simultaneously after the rotations are known. We use a nonlinear optimization to estimate camera positions based. Given pairwise relative translations from TwoViewInfo and the estimated rotation, the constraint

$R_i * (c_j - c_i) = \alpha_{i,j} * t_{i,j}$

Where $$\alpha_{i,j} = ||c_j - c_i||^2$$. This ensures that we optimize for positions that agree with the relative positions computed in two-view estimation.

class NonlinearPositionEstimatorOptions
int NonlinearPositionEstimatorOptions::num_threads

DEFAULT: 1

Number of threads to use with Ceres for nonlinear optimization.

bool NonlinearPositionEstimatorOptions::verbose

DEFAULT: false

Set to true for verbose logging.

int NonlinearPositionEstimatorOptions::max_num_iterations

DEFAULT: 400

The maximum number of iterations for each minimization (i.e., for a single IRLS iteration).

double NonlinearPositionEstimatorOptions::robust_loss_width

DEFAULT: 1.0

The width of the robust Huber loss function used in the first minimization iteration.

int NonlinearPositionEstimatorOptions::min_num_points_per_view

DEFAULT: 0

The number of 3D point-to-camera constraints to use for position recovery. Using points-to-camera constraints can sometimes improve robustness to collinear scenes. Points are taken from tracks in the reconstruction such that the minimum number of points is used such that each view has at least min_num_points_per_view point-to-camera constraints.

double NonlinearPositionEstimatorOptions::point_to_camera_weight

DEFAULT: 0.5

Each point-to-camera constraint (if any) is weighted by point_to_camera_weight compared to the camera-to-camera weights.

class NonlinearPositionEstimator
NonlinearPositionEstimator(const NonlinearPositionEstimatorOptions &options, const Reconstruction &reconstruction, const std::unordered_map<ViewIdPair, TwoViewInfo> &view_pairs)

The constructor takes the options for the nonlinear position estimator, as well as const references to the reconstruction (which contains camera and track information) and the two view geometry information that will be use to recover the positions.

bool EstimatePositions(const std::unordered_map<ViewId, Eigen::Vector3d> &orientation, std::unordered_map<ViewId, Eigen::Vector3d> *positions)

Estimates the positions of cameras given the global orientation estimates by using the nonlinear algorithm described above. Only positions that have an orientation set are estimated. Returns true upons success and false on failure.

## Triangulation¶

Triangulation in structure from motion calculates the 3D position of an image coordinate that has been tracked through two or more images. All cameras with an estimated camera pose are used to estimate the 3D point of a track.

class EstimateTrackOptions
bool EstimateTrackOptions::bundle_adjustment

DEFAULT: true

Bundle adjust the track (holding all camera parameters constant) after initial estimation. This is highly recommended in order to obtain good 3D point estimations.

double EstimateTrackOptions::max_acceptable_reprojection_error_pixels

DEFAULT: 5.0

Track estimation is only considered successful if the reprojection error for all observations is less than this value.

double EstimateTrackOptions::min_triangulation_angle_degrees

DEFAULT: 3.0

In order to triangulate features accurately, there must be a sufficient baseline between the cameras relative to the depth of the point. Points with a very high depth and small baseline are very inaccurate. We require that at least one pair of cameras has a sufficient viewing angle to the estimated track in order to consider the estimation successful.

bool EstimateAllTracks(const EstimateTrackOptions &options, const int num_threads, Reconstruction *reconstruction)

Performs (potentially multithreaded) track estimation. Track estimation is embarassingly parallel so multithreading is recommended.

bool Triangulate(const Matrix3x4d &pose1, const Matrix3x4d &pose2, const Eigen::Vector2d &point1, const Eigen::Vector2d &point2, Eigen::Vector4d *triangulated_point)

2-view triangulation using the method described in [Lindstrom]. This method is optimal in an L2 sense such that the reprojection errors are minimized while enforcing the epipolar constraint between the two cameras. Additionally, it basically the same speed as the TriangulateDLT() method.

The poses are the (potentially calibrated) poses of the two cameras, and the points are the 2D image points of the matched features that will be used to triangulate the 3D point. On successful triangulation, true is returned. The homogeneous 3d point is output so that it may be known if the point is at infinity.

bool TriangulateDLT(const Matrix3x4d &pose1, const Matrix3x4d &pose2, const Eigen::Vector2d &point1, const Eigen::Vector2d &point2, Eigen::Vector4d *triangulated_point)

The DLT triangulation method of [HartleyZisserman].

bool TriangulateMidpoint(const Eigen::Vector3d &origin1, const Eigen::Vector3d &ray_direction1, const Eigen::Vector3d &origin2, const Eigen::Vector3d &ray_direction2, Eigen::Vector4d *triangulated_point)

Perform triangulation by determining the closest point between the two rays. In this case, the ray origins are the camera positions and the directions are the (unit-norm) ray directions of the features in 3D space. This method is known to be suboptimal at minimizing the reprojection error, but is approximately 10x faster than the other 2-view triangulation methods.

bool TriangulateNViewSVD(const std::vector<Matrix3x4d> &poses, const std::vector<Eigen::Vector2d> &points, Eigen::Vector3d *triangulated_point)
bool TriangulateNView(const std::vector<Matrix3x4d> &poses, const std::vector<Eigen::Vector2d> &points, Eigen::Vector3d *triangulated_point)

We provide two N-view triangluation methods that minimizes an algebraic approximation of the geometric error. The first is the classic SVD method presented in [HartleyZisserman]. The second is a custom algebraic minimization. Note that we can derive an algebraic constraint where we note that the unit ray of an image observation can be stretched by depth $$\alpha$$ to meet the world point $$X$$ for each of the $$n$$ observations:

$\alpha_i \bar{x_i} = P_i X,$

for images $$i=1,\ldots,n$$. This equation can be effectively rewritten as:

$\alpha_i = \bar{x_i}^\top P_i X,$

which can be substituted into our original constraint such that:

$\bar{x_i} \bar{x_i}^\top P_i X = P_i X$
$0 = (P_i - \bar{x_i} \bar{x_i}^\top P_i) X$

We can then stack this constraint for each observation, leading to the linear least squares problem:

$\begin{split}\begin{bmatrix} (P_1 - \bar{x_1} \bar{x_1}^\top P_1) \\ \vdots \\ (P_n - \bar{x_n} \bar{x_n}^\top P_n) \end{bmatrix} X = \textbf{0}\end{split}$

This system of equations is of the form $$AX=0$$ which can be solved by extracting the right nullspace of $$A$$. The right nullspace of $$A$$ can be extracted efficiently by noting that it is equivalent to the nullspace of $$A^\top A$$, which is a 4x4 matrix.

We perform bundle adjustment using Ceres Solver for nonlinear optimization. Given a Reconstruction, we optimize over all cameras and 3D points to minimize the reprojection error.

class BundleAdjustmentOptions
ceres::LinearSolverType BundleAdjustmentOptions::linear_solver_type

DEFAULT: ceres::SPARSE_SCHUR

ceres::DENSE_SCHUR is recommended for problems with fewer than 100 cameras, ceres::SPARSE_SCHUR for up to 1000 cameras, and ceres::ITERATIVE_SCHUR for larger problems.

ceres::PreconditionerType BundleAdjustmentOptions::preconditioner_type

DEFAULT: ceres::SCHUR_JACOBI

If ceres::ITERATIVE_SCHUR is used, then this preconditioner will be used.

ceres::VisibilityClusteringType BundleAdjustmentOptions::visibility_clustering_type

DEFAULT: ceres::SINGLE_LINKAGE

bool BundleAdjustmentOptions::verbose

DEFAULT: false

Set to true for verbose logging.

bool BundleAdjustmentOptions::constant_camera_intrinsics

DEFAULT: false

If set to true, the camera intrinsics are held constant during optimization. This is useful if the calibration is precisely known ahead of time.

int BundleAdjustmentOptions::num_threads

DEFAULT: 1

The number of threads used by Ceres for optimization. More threads means faster solving time.

int BundleAdjustmentOptions::max_num_iterations

DEFAULT: 500

Maximum number of iterations for Ceres to perform before exiting.

double BundleAdjustmentOptions::max_solver_time_in_seconds

DEFAULT: 3600.0

Maximum solver time is set to 1 hour.

bool BundleAdjustmentOptions::use_inner_iterations

DEFAULT: true

Inner iterations can help improve the quality of the optimization.

double BundleAdjustmentOptions::function_tolerance

DEFAULT: 1e-6

Ceres parameter for determining convergence.

double BundleAdjustmentOptions::gradient_tolerance

DEFAULT: 1e-10

Ceres parameter for determining convergence.

double BundleAdjustmentOptions::parameter_tolerance

DEFAULT: 1e-8

Ceres parameter for determining convergence.

double BundleAdjustmentOptions::max_trust_region_radius

DEFAULT: 1e12

Maximum size that the trust region radius can grow during optimization. By default, we use a value lower than the Ceres default (1e16) to improve solution quality.

BundleAdjustmentSummary BundleAdjustReconstruction(const BundleAdjustmentOptions &options, Reconstruction *reconstruction)

Performs full bundle adjustment on a reconstruction to optimize the camera reprojection error. The BundleAdjustmentSummary returned contains information about the success of the optimization, the initial and final costs, and the time required for various steps of bundle adjustment.

## Similarity Transformation¶

void AlignPointCloudsICP(const int num_points, const double left[], const double right[], double rotation[3 * 3], double translation[3])

We implement ICP for point clouds. We use Besl-McKay registration to align point clouds. We use SVD decomposition to find the rotation, as this is much more likely to find the global minimum as compared to traditional ICP, which is only guaranteed to find a local minimum. Our goal is to find the transformation from the left to the right coordinate system. We assume that the left and right reconstructions have the same number of points, and that the points are aligned by correspondence (i.e. left[i] corresponds to right[i]).

void AlignPointCloudsUmeyama(const int num_points, const double left[], const double right[], double rotation[3 * 3], double translation[3], double *scale)

This function estimates the 3D similiarty transformation using the least squares method of [Umeyama]. The returned rotation, translation, and scale align the left points to the right such that $$Right = s * R * Left + t$$.

void GdlsSimilarityTransform(const std::vector<Eigen::Vector3d> &ray_origin, const std::vector<Eigen::Vector3d> &ray_direction, const std::vector<Eigen::Vector3d> &world_point, std::vector<Eigen::Quaterniond> *solution_rotation, std::vector<Eigen::Vector3d> *solution_translation, std::vector<double> *solution_scale)

Computes the solution to the generalized pose and scale problem based on the paper “gDLS: A Scalable Solution to the Generalized Pose and Scale Problem” by Sweeney et. al. [SweeneyGDLS]. Given image rays from one coordinate system that correspond to 3D points in another coordinate system, this function computes the rotation, translation, and scale that will align the rays with the 3D points. This is used for applications such as loop closure in SLAM and SfM. This method is extremely scalable and highly accurate because the cost function that is minimized is independent of the number of points. Theoretically, up to 27 solutions may be returned, but in practice only 4 real solutions arise and in almost all cases where n >= 6 there is only one solution which places the observed points in front of the camera. The rotation, translation, and scale are defined such that: $$sp_i + \alpha_i d_i = RX_i + t$$ where the observed image ray has an origin at $$p_i$$ in the unit direction $$d_i$$ corresponding to 3D point $$X_i$$.

ray_origin: the origin (i.e., camera center) of the image ray used in the 2D-3D correspondence.

ray_direction: Normalized image rays corresponding to reconstruction points. Must contain at least 4 points.

world_point: 3D location of features. Must correspond to the image_ray of the same index. Must contain the same number of points as image_ray, and at least 4.

solution_rotation: the rotation quaternion of the candidate solutions

solution_translation: the translation of the candidate solutions

solution_scale`: the scale of the candidate solutions