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Spatial Registration
Published in Praveen Kumar, Jay Alameda, Peter Bajcsy, Mike Folk, Momcilo Markus, Hydroinformatics: Data Integrative Approaches in Computation, Analysis, and Modeling, 2005
Given a set of corresponding features, one has to select a registration transformation model that will compensate for geometric distortions. The choice of a registration transformation model is usually based on assumptions about data acquisition. In general, the transformation models can be categorized as linear or nonlinear. Linear transformations are functions between two vector spaces that respect the arithmetical operations addition and scalar multiplication defined on vector spaces. Examples of linear transformation models would include rigid body, affine, perspective, or projective mappings. A rigid body transformation takes into account translational and rotational deformations and it preserves all distances. An affine transformation maps parallel lines on to parallel lines and can compensate for any translation, rotation, scale, and shear distortions. A perspective transformation maps an image from the 3D world coordinate system to the 2D graphics display with projection lines converging at the eye point. For example, objects appear smaller if they are further away from the eye point. A projective transformation is the composition of a pair of perspective projections and corresponds to the perceived positions of observed objects when the viewpoint of the observer changes. This type of transformation preserves incidence (the relationships describing how objects meet each other) and cross-ratio (the ratio of ratios of point distances) but does not preserve sizes and angles. If the data acquisition process distorts images by nonlinear warping then nonlinear transformation models are appropriate. An example of a nonlinear transformation function would be a rational polynomial (cubic) model that is used for mapping 3D terrain datasets to 2D images.
Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A common example of a projective transformation is given by a perspective transormation (Figure 4.7). Strictly speaking this gives a transformation from one plane to another, but, if we identify the two planes by (for example) fixing a Cartesian system in each, we get a projective transformation from the plane to itself.
Multiple-View Geometry
Published in Jian Chen, Bingxi Jia, Kaixiang Zhang, Multi-View Geometry Based Visual Perception and Control of Robotic Systems, 2018
Jian Chen, Bingxi Jia, Kaixiang Zhang
The matrix has nine elements with only their ratio significant, so the transformation is specified by eight parameters. A projective transformation between two planes can be computed from four point correspondences, with no three collinear on either plane.
Non-destructive inspection system for MAG welding processes by combining multimodal data
Published in Quantitative InfraRed Thermography Journal, 2021
Katharina Simmen, Benjamin Buch, Andreas Breitbarth, Gunther Notni
The image acquisition by three different cameras with three different resolutions and three different alignments to the DUT leads to a perspective-based distortion. In order to make the images comparable, every single image must be rectified in relation to a reference image. Distorted images can be rectified in perspective (such as translation, rotation and scaling) by means of projective transformation. This transformation is defined as a relationship between points of two images: distorted image and undistorted image .