Bundle Adjustment: Optimizing Visual Data for Precise Reconstruction

By Fouad Sabry

What is Bundle Adjustment

In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images, given a set of images depicting a number of 3D points from different viewpoints.Its name refers to the geometrical bundles of light rays originating from each 3D feature and converging on each camera's optical center, which are adjusted optimally according to an optimality criterion involving the corresponding image projections of all points.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Bundle adjustment

Chapter 2: Levenberg-Marquardt algorithm

Chapter 3: Gauss-Newton algorithm

Chapter 4: Newton's method in optimization

Chapter 5: Iteratively reweighted least squares

Chapter 6: 3D reconstruction from multiple images

Chapter 7: Homography (computer vision)

Chapter 8: Chessboard detection

Chapter 9: Perspective-n-Point

Chapter 10: Powell's dog leg method

(II) Answering the public top questions about bundle adjustment.

(III) Real world examples for the usage of bundle adjustment in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Bundle Adjustment.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 6, 2024

Book preview

Chapter 1: Bundle adjustment

Given a collection of images depicting a number of 3D points from different viewpoints, bundle adjustment in photogrammetry and computer stereo vision is the simultaneous refinement of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images. Named after the optimality criterion involving the corresponding picture projections of all points, it involves the geometric bundles of light rays that originate from each 3D feature and converge on each camera's optical center.

The final phase of most feature-based 3D reconstruction methods is bundle adjustment.

In essence, it's an optimization problem for the 3D structure and the parameters that determine how it's viewed, perspective, intrinsic calibration, and radial distortion (from the camera itself), to obtain a reconstruction which is optimal under certain assumptions regarding the noise pertaining to the observed: 2

The goal of bundle adjustment is to reduce the discrepancy between the expected and observed positions of picture points, It can be written as the square root of a very large number of nonlinear, functions with real values.

Thus, Nonlinear least-squares methods are used to perform the minimization.

Of these, Due to its simplicity and the effectiveness of the damping strategy it employs, Levenberg-Marquardt has become one of the most popular methods. This allows it to swiftly converge from a large sampling of initial assumptions.

Minimizing a function requires iteratively linearizing it around the current estimate, The normal equations are linear systems whose solution is at the heart of the Levenberg-Marquardt algorithm.

Minimization issues in the context of bundle adjustment framework, The lack of correlation between the parameters for various 3D points and cameras results in a sparse block structure for the normal equations.

Using a sparse form of the Levenberg-Marquardt technique that takes advantage of the zeros pattern in normal equations might greatly improve computational efficiency thanks to this, avoiding storing and operating on zero-elements.: 3

During bundle adjustment, the camera and the structure's initial parameter estimations are collaboratively refined to determine which parameters best forecast the observed points' locations over the available images.

More formally, assume that n 3D points are seen in m views and let {\mathbf {x}}_{{ij}} be the projection of the i th point on image j .

Let \displaystyle v_{{ij}} denote the binary variables that equal 1 if point i is visible in image j and 0 otherwise.

Assume also that each camera j is parameterized by a vector {\mathbf {a}}_{j} and each 3D point i by a vector {\mathbf {b}}_{i} .

Reprojection errors can be reduced across the board by using bundle adjustment, which takes into account all 3D point and camera settings, specifically

\min _{{{\mathbf {a}}_{j},\,{\mathbf {b}}_{i}}}\displaystyle \sum _{{i=1}}^{{n}}\;\displaystyle \sum _{{j=1}}^{{m}}\;v_{{ij}}\,d({\mathbf {Q}}({\mathbf {a}}_{j},\,{\mathbf {b}}_{i}),\;{\mathbf {x}}_{{ij}})^{2},

where {\mathbf {Q}}({\mathbf {a}}_{j},\,{\mathbf {b}}_{i}) is the predicted projection of point i on image j and d({\mathbf {x}},\,{\mathbf {y}}) denotes the Euclidean distance between the image points represented by vectors \mathbf {x} and \mathbf {y} .

Since the minimum is calculated across a large number of points and images,, As its name implies, bundle adjustment doesn't mind if some of your picture projections are