3D reconstruction from multiple images is the creation of three-dimensional models from a set of images. It is the reverse process of obtaining 2D images from 3D scenes.

This article's factual accuracy may be compromised due to out-of-date information. (October 2019)

The essence of an image is a projection from a 3D scene onto a 2D plane, during which process the depth is lost. The 3D point corresponding to a specific image point is constrained to be on the line of sight. From a single image, it is impossible to determine which point on this line corresponds to the image point. If two images are available, then the position of a 3D point can be found as the intersection of the two projection rays. This process is referred to as triangulation. The key for this process is the relations between multiple views which convey the information that corresponding sets of points must contain some structure and that this structure is related to the poses and the calibration of the camera.

In recent decades, there is an important demand for 3D content for computer graphics, virtual reality and communication, triggering a change in emphasis for the requirements. Many existing systems for constructing 3D models are built around specialized hardware (e.g. stereo rigs) resulting in a high cost, which cannot satisfy the requirement of its new applications. This gap stimulates the use of digital imaging facilities (like a camera). An early method was proposed by Tomasi and Kanade.^[2] They used an affine factorization approach to extract 3D from images sequences. However, the assumption of orthographic projection is a significant limitation of this system.

The task of converting multiple 2D images into 3D model consists of a series of processing steps:

Camera calibration consists of intrinsic and extrinsic parameters, without which at some level no arrangement of algorithms can work. The dotted line between Calibration and Depth determination represents that the camera calibration is usually required for determining depth.

Depth determination serves as the most challenging part in the whole process, as it calculates the 3D component missing from any given image – depth. The correspondence problem, finding matches between two images so the position of the matched elements can then be triangulated in 3D space is the key issue here.

Once you have the multiple depth maps you have to combine them to create a final mesh by calculating depth and projecting out of the camera – registration. Camera calibration will be used to identify where the many meshes created by depth maps can be combined to develop a larger one, providing more than one view for observation.

By the stage of Material Application you have a complete 3D mesh, which may be the final goal, but usually you will want to apply the color from the original photographs to the mesh. This can range from projecting the images onto the mesh randomly, through approaches of combining the textures for super resolution and finally to segmenting the mesh by material, such as specular and diffuse properties.

Given a group of 3D points viewed by N cameras with matrices ${\displaystyle \{P^{i}\}_{i=1\ldots N}}$, define ${\displaystyle m_{j}^{i}\simeq P^{i}w_{j}}$ to be the homogeneous coordinates of the projection of the ${\displaystyle j^{th}}$ point onto the ${\displaystyle i^{th}}$ camera. The reconstruction problem can be changed to: given the group of pixel coordinates ${\displaystyle \{m_{j}^{i}\}}$, find the corresponding set of camera matrices ${\displaystyle \{P^{i}\}}$ and the scene structure ${\displaystyle \{w_{j}\}}$ such that

${\displaystyle m_{j}^{i}\simeq P^{i}w_{j}}$ (1)

Generally, without further restrictions, we will obtain a projective reconstruction.^[4][5] If ${\displaystyle \{P^{i}\}}$ and ${\displaystyle \{w_{j}\}}$ satisfy (1), ${\displaystyle \{P^{i}T\}}$ and ${\displaystyle \{T^{-1}w_{j}\}}$ will satisfy (1) with any 4 × 4 nonsingular matrix T.

A projective reconstruction can be calculated by correspondence of points only without any a priori information.

In auto-calibration or self-calibration, camera motion and parameters are recovered first, using rigidity. Then structure can be readily calculated. Two methods implementing this idea are presented as follows:

Recently, new methods based on the concept of stratification have been proposed.^[10] Starting from a projective structure, which can be calculated from correspondences only, upgrade this projective reconstruction to a Euclidean reconstruction, by making use of all the available constraints. With this idea the problem can be stratified into different sections: according to the amount of constraints available, it can be analyzed at a different level, projective, affine or Euclidean.

Inevitably, measured data (i.e., image or world point positions) is noisy and the noise comes from many sources. To reduce the effect of noise, we usually use more equations than necessary and solve with least squares.

For example, in a typical null-space problem formulation Ax = 0 (like the DLT algorithm), the square of the residual ||Ax|| is being minimized with the least squares method.

In general, if ||Ax|| can be considered as a distance between the geometrical entities (points, lines, planes, etc.), then what is being minimized is a geometric error, otherwise (when the error lacks a good geometrical interpretation) it is called an algebraic error.

Therefore, compared with algebraic error, we prefer to minimize a geometric error for the reasons listed:

1. The quantity being minimized has a meaning.
2. The solution is more stable.
3. The solution is constant under Euclidean transforms.

All the linear algorithms (DLT and others) we have seen so far minimize an algebraic error. Actually, there is no justification in minimizing an algebraic error apart from the ease of implementation, as it results in a linear problem. The minimization of a geometric error is often a non-linear problem, that admit only iterative solutions and requires a starting point.

Usually, linear solution based on algebraic residuals serves as a starting point for a non-linear minimization of a geometric cost function, which provides the solution a final “polish”.^[11]

The 2-D imaging has problems of anatomy overlapping with each other and do not disclose the abnormalities. The 3-D imaging can be used for both diagnostic and therapeutic purposes.

3-D models are used for planning the operation, morphometric studies and has more reliability in orthopedics.^[12]

• 3D pose estimation – Process of determining spatial characteristics of objects
• 3D reconstruction – Process of capturing the shape and appearance of real objects
• 3D photography
• 2D to 3D conversion – Process of transforming 2D film to 3D form
• 3D data acquisition and object reconstruction – Scanning of an object or environment to collect data on its shapePages displaying short descriptions of redirect targets
• Epipolar geometry – Geometry of stereo vision
• Camera resectioning – Process of estimating the parameters of a pinhole camera model
• Computer stereo vision – Extraction of 3D data from digital images
• Structure from motion – Method of 3D reconstruction from moving objects
• Comparison of photogrammetry software
• Visual hull
• Human image synthesis – Computer generation of human images