Paper Title - Overview of Structure from Motion

Advanced Computing: An International Journal (ACIJ), Vol.15, No.5, September 2024
DOI:10.5121/acij.2024.15502 25
OVERVIEW OF STRUCTURE FROM MOTION
Svetlana Mijakovska
Faculty of Technical Sciences, University St. Klement of Ohrid, Bitola, Macedonia
ABSTRACT
In this paper is given an overview of tools and techniques for obtain information about the geometry of 3D
scenes from 2D images. The generating three-dimensional structure from a series of 2D images or video of
scenes is known as Structure from Motion (SfM). A central tenet of structure from motion is that, given the
position of a feature in one image, it is possible to find the corresponding position of the same feature in
successive images. We described methods for the simultaneous recovery of 3D points and camera
projection matrices using corresponding image points in multiple views. Structure from Motion (SfM) is a
fascinating field within computer vision that seeks to reconstruct a three-dimensional structure of the
environment from a sequence of two-dimensional images. At the heart of SfM lies a set of sophisticated
algorithms that enable the extraction of spatial information from a series of images. Two public
repositories that may be of use are Open SfM and Colmap.
KEYWORDS
Image processing, image projection, epipolar geometry, camera calibration, projection matrix,
triangulation.
1. INTRODUCTION
3D models are used in many fields such as animation, computer games, virtual reality, film
industry, computer vision, etc. The generating of 3D models can be done in many ways, from a
series of images, manually and also 3D models can be generated from video [1],[2].
The generating three-dimensional structure from a series of 2D images or video of scenes is
interesting area for research and this problem is known as Structure from Motion (SfM). This
process recovers the 3D structure of a scene and the orientation and position of the camera when
each image was captured. SfM is trying to recover, from segments of images the 3D structure of a
scene and the position and pose (orientation) of camera at the moment of capturing each image.
Applications for SfM can be broadly split into two categories, those that require geometric
accuracy and those that require photorealism:
• Geometric accuracy: These types of application are generally less concerned with the
visual appearance of the model but require the scene structure and camera motion to be
reconstructed with a high degree of accuracy. Robot navigation, for instance, requires
high– accuracy models, but the visual appearance of the model is unimportant. The reverse
engineering of existing objects for use in CAD requires the structure of the object to be
recovered with a high degree of accuracy. Film special effects that place computer–
generated objects into the film and other ‘augmented reality’ applications require the
camera motion to be very accurately reconstructed but the appearance of the structure is
irrelevant as it is never seen in the finished product.

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• Photorealism: In contrast, there are a growing number of situations where the geometric
accuracy of the underlying reconstruction is less important as long as, for the purposes of the
application, the model visually resembles the real scene. This is the case for applications such
as virtual reality, simulators, computer games and special effects that require a virtual set
based on a real scene.
The purpose of structure from motion is getting cloud of 3D points in the scene, which in the
process of feature matching and meshing, give the final 3D model. So, given the position of a
feature in one image (sequence) we need to find the corresponding position of the same feature in
successive image. This is correspondence problem is based on principles of multiple view
geometry.
Image formation process is not generally invertible: from its projected position in a camera image
plane, a scene point can only be recovered up to a one-parameter ambiguity corresponding to its
distance from the camera. The additional information that we need can be received in two ways.
One possibility is to exploit prior knowledge about the scene to reduce the number of degrees of
freedom. For example, parallelism and coplanarity constraints can be used to reconstruct simple
geometric shapes such as line segments and planar polygons from their projected positions in
individual views.
Another possibility is to use corresponding image points in multiple views. Given its image in
two or more views, a 3D point can be reconstructed by triangulation [3]. An important
prerequisite is the determination of camera calibration and pose, which may be expressed by a
projection matrix. The geometrical theory of structure from motion allows projection matrices
and 3D points to be computed simultaneously using only corresponding points in each view.
Structure from motion techniques is used in a wide range of applications including
photogrammetric survey, the automatic reconstruction of virtual reality models from video
sequences, and for the determination of camera motion (e.g. so that computer-generated objects
can be inserted into video footage of real-world scenes).
2. THE CORRESPONDENCE PROBLEM
A central tenet of structure from motion is that, given the position of a feature in one image, it is
possible to find the corresponding position of the same feature in successive images. This
problem, known as the correspondence problem.
Fig.1 Points of interest detected by SIFT (coloured green)
An underlying assumption of structure from motion is that, given the position of a scene feature
in one image, it is possible to find the corresponding position of the same feature in successive

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images. In any image of a real–world scene, there will be a considerable number of pixels lying in
regions of homogeneous texture. This means that each pixel is surrounded by pixels of very
similar intensity (and colour), making it virtually impossible to differentiate between the correct,
corresponding, pixel and other nearby pixels. Problems are also caused by occlusion, where
features lying on part of a scene may be obstructed in subsequent views by other parts of the
scene and therefore not actually visible. These factors restrict the number of pixels that it is
possible to match to a small percentage of the total pixels in an image sequence, resulting in a
sparse set of point correspondences. This set of correspondences is also likely to contain a
significant number of mismatches.First, we will need to detect a number of key points in each
image (8 minimum). This cold be corners or SIFT points. The scale-invariant feature transform
(SIFT) is a computer vision algorithm to detect, describe, and match local features in images.
This stage is called feature extraction. Then, we will need to match each key point to its
equivalent in each point of view. This is called feature matching and can be performed using
template matching, optical flow.
Fig.2 Corresponding position of the same feature in successive images
Another basic assumption made in SfM is that the image sequence represents a rigid scene. In
other words, either the entire scene is static and the camera moves through the scene or the entire
scene moves past the camera as one object. This is equivalent to multiple cameras each taking one
image of a different view of the scene. Along with other assumptions, such as a pinhole camera
model, the principles of multiple view geometry can be used to recover the geometry of the
cameras and structure of the scene from the known correspondences, up to a projective
reconstruction.
The geometrical theory of structure from motion assumes that one is able to solve the
correspondence problem, which is to identify points in two or more views that are the projections
of the same point in space. One solution is to identify corresponding points interactively in each
view. An important advantage is that surfaces can be defined simultaneously with
correspondences. A disadvantage is that the interactive approach is time consuming; also, the
accuracy of the resulting reconstruction will depend critically on how carefully the user positions
the image points.
Structure from Motion algorithms apply the principles of multiple view geometry to features
matched across a sequence of images to recover the structure of the scene and the motion of the
cameras. These features can include points [4], lines [5] and higher–level primitives such as
planes [6]. In general, by far the most common approach is that of point–matching. Points
features are easily extracted from images using a corner detector [7], they are widespread in most
scene types and they don’t suffer from partial occlusion problems in the same way as lines and
curves.

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3. STRUCTURE AND CAMERA CALIBRATION
Camera calibration is the first step towards computational computer vision. Although some
information concerning the measuring of scenes can be obtained by using uncalibrated cameras,
calibration is essential when metric information is required. The use of precisely calibrated
cameras makes the measurement of distances in a real world from their projections on the image
plane possible. Some applications of this capability include: dense reconstruction, visual
inspection, object localization and camera localization. Camera calibration is divided into two
phases. First, camera modelling deals with the mathematical approximation of the physical and
optical behaviour of the sensor by using a set of parameters. The second phase of camera
calibration deals with the use of direct or iterative methods to estimate the values of these
parameters. The most used techniques for camera calibration are: Non-linear optimization
techniques, Linear techniques which compute the transformation matrix and Two-step techniques.
The basic formulae governing the geometric constraints for the perspective projection of scene
features in two views originated in the fields of projective geometry and photogrammetry. A
method of computing the epipolar geometry relating two images from 7-point correspondences is
produced from Hesse [8] and Sturm [9].
Fig.3 Epipolar geometry of two views
An important prerequisite is the determination of camera calibration and position, which may be
expressed by a projection matrix. A matrix which encapsulates the epipolar geometry of two
calibrated cameras is call Essential matrix. Initial research in SfM was limited almost exclusively
to binocular stereo using cameras that had been previously calibrated with carefully manufactured
calibration objects of known geometry. The essential matrix could also be generalised to the case
of uncalibrated cameras led to the creation of the Fundamental matrix [10]. This work had
important implications as it meant that it was possible to simultaneously recover the projective
structure of a scene and camera motion solely from image correspondences, without any
knowledge of the parameters of the camera.
Following the discovery of the fundamental matrix, that encapsulates the geometry of two views,
the Trifocal tensor [11] for three views and the Quadrifocal tensor [12] for four views were
derived. Although the quadrifocal tensor represents the limit for the number of views for which a
closed–form solution is available, the need to produce reconstructions from long image sequences
has resulted in the creation of methods that merge sets of camera matrices, derived from
fundamental matrices or trifocal tensors, into one projective frame.

29
Fig.4 Geometry from N-views
Reconstruction with only knowledge of feature correspondences is only possible up to a
projective reconstruction and there are many ways to obtain projection matrices from a
geometry constraint, i.e. a fundamental matrix or a focal tensor. Hence projective reconstruction
is mainly the recovery of fundamental matrices or focal tensors. Methods, implementation hints,
and evaluations are well discussed by Hartley and Zisserman in [13]. If the input, i.e. feature
correspondences, includes outliers, robust methods such as RANSAC, LMS can be employed to
reject them.
4. AUTO-CALIBRATION
Camera auto-calibration is the process of determining internal camera parameters directly from
multiple uncalibrated images of unstructured scenes. The intrinsic parameters and hence upgrade
the reconstruction from projective to metric. Most auto–calibration methods are based on the
concept of the absolute conic; a conic that is invariant to Euclidean transformations. One of the
most important concepts for self-calibration is the Absolute Conic (AC) and its projection in the
images. Since it is invariant under Euclidean transformations, its relative position to a moving
camera is constant. For constant intrinsic camera parameters its image will therefore also be
constant. This means that its relative position to a moving camera is constant and therefore its
image in any view depends only on the intrinsic parameters of the camera. Recovery of the image
of the absolute conic in a given view allows the intrinsic parameters of the camera for that view to
be determined. A particularly convenient representation, the absolute dual quadric, projects to the
dual image of the absolute conic in any view, and calculating this enables all cameras and 3D
structure to be upgraded to a metric reconstruction. Accurate calibration ensures precise
measurements and reliable analysis by correcting distortions and estimating intrinsic and extrinsic
camera parameters.
5. 3DMODEL FROM SERIES OF IMAGES
We made experiment to create 3D model from several images. For resolve the correspondence
problem we used fundament matrix and using several algorithms for SfM we create 3D model of
the object which was capture with hand-held camera (Figure 5). This step is actually the main step
in 3D modelling from several images, because in this step we must choose which algorithm to be
used for find corresponding points of more images with moving cameras at different points in
time. We used RANSAC, MSAC and MLESAC. Random sample consensus (RANSAC) is
algorithm which continuously generate hypothetical solutions estimated from randomly selected,
minimal data sets and testing each solution. That means the solution can be computed from the
smallest sample and the likelihood of a sample containing distance is minimized. A M–estimator
known as MSAC (M–estimator SAmple Consensus) solves this problem with giving outliers a

30
fixed penalty related to the threshold and scoring inliers according to their error. The Maximum
Likelihood Sample Consensus (MLESAC) algorithm is a version of RANSAC that is
probabilistic. Using mixture model, it maximises a likelihood. The comparasion of these
algorithms are given in Table I. The standard deviation of the prediction errors is Root Mean
Square Error (RMSE).
Table I. Comparison of the speed and RMSE parameter of the algorithms for elimination of unnecessary
points (outliers)
Algorithm Speed RMSE
MSAC 35s 0,817
RANSAC 75s 0,729
MLESAC 30s 0,699
Fig.5 Cloud of 3D points in several images
After getting cloud of 3D points, we connect and match them in order to create 3D model of the
object from several images. The result is shown in Figure 6.
Fig.6 Final 3D model

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6. CONCLUSIONS
Structure and motion recovery recovers the structure of the scene and the motion information of
the camera. The motion information is the position, orientation, and intrinsic parameters of the
camera at the captured views. The structure information is captured by the 3D coordinates of
features. Because we want to get 3D model from several 2D images, for this step we must
research 3D reconstruction from multiple views i.e. multiple view geometry. The first step is
getting cloud of 3D points (using algorithms for the elimination of outliers and their comparison),
and with their connecting and matching, create 3D model of object that is capture in those
images. Structure from Motion (SfM) stands as a captivating realm within computer vision,
bridging the gap between two-dimensional images and the intricate three-dimensional world.
The future of Structure from Motion (SfM) is poised for transformative advancements through the
integration of deep neural networks. Recent developments, such as Gaussian splatting techniques
or the recent paper from Oxford university and Meta AI, showcase a paradigm shift in SfM
methodologies.
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2231 – 6604 Volume 7, Issue 3, pp: 680-686 ©IJESET.
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AUTHOR
Prof. Svetlana Mijakovska is a Doctor of Technical Sciences in Graphic Engineering
at Faculty of Technical Sciences in Bitola, Macedonia. She is interested in computer
graphics, visualization, 3d Virtual reality.

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