The axis-angle representation is a convenient format to represent rotations as it only has 3 elements and therefore keeps the number of elements used to represent a rotation to a minimum. Note that while the rotation matrix has 9 elements, due to the orthogonality and unit norm constraints, only 3 are independent.
Here, are the 3D coordinates of the point, are the focal length of the camera and the coordinates of the center of projection respectively (we assume square pixels and zero skew), are the parameters of the rotation matrix in the axis-angle representation and are the elements of the translation vector between the object and the camera coordinate system. For points each of which is visible in images, the number of independent parameters is: Let’s first consider the number of independent parameters that we need to optimize over.
MATLAB SYMBOLIC TOOLBOX JACOBIAN HOW TO
The primary purpose of this post is to show how to compute the jacobians and practical information about some of their properties. During my research on Bundle Adjustment, I found plenty of references that describe the structure of the problem and the techniques used to solve it, but none that show how to compute these jacobians that are needed by any practical implementation. The LM algorithm needs the jacobians i.e., the partial derivatives of the image coordinates wrt the intrinsic and extrinsic parameters of the camera, and the coordinates of the 3D points. īundle Adjustment methods typically employ the Levenberg Marquardt (LM) algorithm to find the minimum of the optimization function. Henceforth, we’ll assume that our camera is a rectilinear camera with square pixels. Together, these parameters are combined as follows to project a 3D world point on the image: The vector consists of the rotation and translation parameters to transform the world point in the camera coordinate system and the intrinsic parameters of the camera that is the focal length (in the x and y dimensions), coordinates of the center of projection and the skew factor (if any). Here, is the predicted projection of point on the image and denotes the Euclidean distance between the image points represented by the vectors and. Then, Bundle Adjustment involves minimizing the following objective function: In the first post, we’ll derive expressions for the elements of the Jacobian matrix used in BA, in the second post, we’ll plot some of these Jacobians and consider their properties and in the third and fourth post, we’ll apply BA to simulated data sets and analyze the results.Īssume each camera is parameterized by a vector and each 3D point by a vector.
MATLAB SYMBOLIC TOOLBOX JACOBIAN SERIES
In this series of posts, we’ll examine BA in detail. Bundle Adjustment (BA) is a well established technique in Computer Vision typically used as the last step in many feature based 3D reconstruction algorithms to tweak the intrinsic and extrinsic camera parameters as well as the position of the reconstruction points to minimize the reconstruction error between the projected and observed image points 1
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