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Structure from Motion

Structure from motion (SfM) is the process of estimating the 3-D structure of a scene from a set of 2-D images. SfM is used in many applications, such as 3-D scanning and augmented reality.

SfM can be computed in many different ways. The way in which you approach the problem depends on different factors, such as the number and type of cameras used, and whether the images are ordered. If the images are taken with a single calibrated camera, then the 3-D structure and camera motion can only be recovered up to scale. up to scale means that you can rescale the structure and the magnitude of the camera motion and still maintain observations. For example, if you put a camera close to an object, you can see the same image as when you enlarge the object and move the camera far away. If you want to compute the actual scale of the structure and motion in world units, you need additional information, such as:

• The size of an object in the scene

• Information from another sensor, for example, an odometer.

Structure from Motion from Two Views

For the simple case of structure from two stationary cameras or one moving camera, one view must be considered camera 1 and the other one camera 2. In this scenario, the algorithm assumes that camera 1 is at the origin and its optical axis lies along the z-axis.

1. SfM requires point correspondences between images. Find corresponding points either by matching features or tracking points from image 1 to image 2. Feature tracking techniques, such as Kanade-Lucas-Tomasi (KLT) algorithm, work well when the cameras are close together. As cameras move further apart, the KLT algorithm breaks down, and feature matching can be used instead.

Distance Between Cameras (Baseline)Method for Finding Point CorrespondencesExample
WideMatch features using `matchFeatures`Find Image Rotation and Scale Using Automated Feature Matching
NarrowTrack features using `vision.PointTracker`Face Detection and Tracking Using the KLT Algorithm

2. To find the pose of the second camera relative to the first camera, you must compute the fundamental matrix. Use the corresponding points found in the previous step for the computation. The fundamental matrix describes the epipolar geometry of the two cameras. It relates a point in one camera to an epipolar line in the other camera. Use the `estimateFundamentalMatrix` function to estimate the fundamental matrix.

3. Input the fundamental matrix to the `relativeCameraPose` function. `relativeCameraPose` returns the orientation and the location of the second camera in the coordinate system of the first camera. The location can only be computed up to scale, so the distance between two cameras is set to 1. In other words, the distance between the cameras is defined to be 1 unit.

4. Determine the 3-D locations of the matched points using `triangulate`. Because the pose is up to scale, when you compute the structure, it has the right shape but not the actual size.

The `triangulate` function takes two camera matrices, which you can compute using `cameraMatrix`.

5. Use `pcshow` to display the reconstruction, and use `plotCamera` to visualize the camera poses.

To recover the scale of the reconstruction, you need additional information. One method to recover the scale is to detect an object of a known size in the scene. The Structure From Motion From Two Views example shows how to recover scale by detecting a sphere of a known size in the point cloud of the scene.

Structure from Motion from Multiple Views

For most applications, such as robotics and autonomous driving, SfM uses more than two views.

The approach used for SfM from two views can be extended for multiple views. The set of multiple views used for SfM can be ordered or unordered. The approach taken here assumes an ordered sequence of views. SfM from multiple views requires point correspondences across multiple images, called tracks. A typical approach is to compute the tracks from pairwise point correspondences. You can use `viewSet` to manage the pairwise correspondences and find the tracks. Each track corresponds to a 3-D point in the scene. To compute 3-D points from the tracks, use `triangulateMultiview`.

Using the approach in SfM from two views, you can find the pose of camera 2 relative to camera 1. To extend this approach to the multiple view case, find the pose of camera 3 relative to camera 2, and so on. The relative poses must be transformed into a common coordinate system. Typically, all camera poses are computed relative to camera 1 so that all poses are in the same coordinate system. You can use `viewSet` to manage camera poses. The `viewSet` object stores the views and connections between the views.

Every camera pose estimation from one view to the next contains errors. The errors arise from imprecise point localization in images, and from noisy matches and imprecise calibration. These errors accumulate as the number of views increases, an effect known as drift. One way to reduce the drift, is to refine camera poses and 3-D point locations. The nonlinear optimization algorithm, called bundle adjustment, implemented by the `bundleAdjustment` function, can be used for the refinement.

The Structure From Motion From Multiple Views example shows how to reconstruct a 3-D scene from a sequence of 2-D views. The example uses the Camera Calibrator app to calibrate the camera that takes the views. It uses a `viewSet` object to store and manage the data associated with each view.

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