First introduce the example of geometric registration and calibration:
Feature-based registration is the
problem of estimating motion from a set of two or more matching 2D or 3D points .
1. Use least squares 20 registration
Given a set of matching feature points \ ({X_i, x_ {i} ^ {'}}} and plane parameter transformation of the following form:
In order to best estimate the motion parameters, the commonly used method is least squares, which is to minimize the sum of squared residuals.
The corresponding matrix and Jacobian matrix for different transformations are summarized as follows:
2. Application: Panorama
One of the simplest applications of image registration is a special form of image stitching, called "panorama". In the panorama, before the simple average is used for fusion, the image will be translated, and there may be rotation and scaling.
For example, the following figure:
Rotation and scaling transformation formula:
Three, iterative algorithm
Linear least squares is the simplest parameter estimation method. However, most problems in computer vision
do not have a simple linear relationship between the measured value and the unknown value. The problem obtained in this case is called "non-linear least squares" or "non-linnear regression".
4. Robust least squares and RANSAC
Conventional least squares is a suitable choice for measurements where the noise conforms to a normal (Gaussian) distribution. However, when there are outliers in the corresponding points, a more robust least squares is needed. In this case, Green uses *** M-estimate ** (it applies a robust penalty function \ (p (r) \) to the residuals instead of their squares.)
M-estimation can often reduce the impact of outliers, but in some cases, starting from too many
outliers will prevent IRLS (or other gradient descent algorithms) from converging to the global optimal. A better way is to find a set of interior points that do not correspond, that is, points that are consistent with the active motion estimation. There are two main methods (both randomly select a subset of k corresponding points, then calculate the estimated \ (p \) , and then calculate the residual of the corresponding set of points);
- Random Sample Consensus
- Minimum median variance
Five, 3D registration
Unlike 2D image feature registration, many computer vision applications require registration of 3D points. In the case where 3D transformation is linear to motion parameters, eg. Translation, similarity, and affine transformation, conventional least squares can be used.
For the case of European sports, called the "absolute direction" problem, the following two methods are required:
- Orthogonal Procrustes algorithm : By calculating the singular value decomposition (SVD) decomposition of the 3 × 3 correlation matrix, the rotation matrix is obtained.
- Absolute direction algorithm : estimate the unit quaternion corresponding to the rotation matrix \ (R \) and find the eigenvector with the largest positive eigenvalue.
Experiments have compared the other two methods to prove that the accuracy difference between the two methods is negligible.