Recognition and Positioning of Rigid Objects using Algebraic and Moment Invariants

of “Recognition and Positioning of Rigid Objects using Algebraic and Moment Invariants,” by Gabriel Taubin, Ph.D., Brown University, May 1991 In this thesis we describe twomodel-based approaches to 3D rigid object recognition and positioning from range data, particularly in cluttered environments. Following a model-based approach, we recognize and locate objects in the data data set by comparing and geometrically matching small regions of the data with corresponding regions of known models, stored in a database. Due to the problem of occlusion, a known object is represented in the database as a hierarchical collection of regions, each of them approximated by a parameterized model. We use two types of parameterized models. With the first type we consider objects whose boundaries can be well approximated by piecewise algebraic curves or surfaces, or both, in which case the preliminary recognition and matching is based on comparing the coefficients of the corresponding polynomials; the final recognition and matching is based on determining how well the data fits a stored polynomial model. With the second type, we consider more general objects, objects which do not fall into the previous group, and usemoments as the region descriptors. In order to develop practical systems for object recognition and position estimation, a number of problems must be solved first. Problems solved for this purpose, in this thesis, are the following. The first of these problems is how to fit models to regions of the data sets. Although computing moments from a data set is relatively straightforward, fitting algebraic curves and surfaces to regions of a data set is difficult. We show several different efficient and numerically stable algorithms for fitting implicit 3D curves and surfaces to data sets, in particular, for fitting algebraic 3D curves and surfaces to data sets. The algorithms also apply to 2D curves, and to curves and surfaces in fourth and higher dimensions. We also show how these fitting methods can be used in segmentation algorithms. The second problem is that when the coordinate system changes, the coefficients of the polynomial which define a given curve or surface change, and the same happens with the moments of a region. However, both coefficients and moments change in a well known fashion. The recognition and matching is based on computing and comparing invariants of, either the coefficients of the polynomials, or the moments. Invariants are functions of the coefficients or the moments, which are independent of the coordinate system. We introduce computationally efficient algorithms for computing invariants. The third problem solved is the problem of recovering the coordinate transformation which best aligns two matching curves or surfaces, or two vectors of moments. We solve these problems by defining an intrinsic coordinate system for both algebraic curves, and moment vectors. The parameters of this intrinsic coordinate system are functions of the coefficients of the polynomials, or the moments, and are very inexpensive to compute. Finally, the fourth problem dealt with is preliminary ideas on how to organize all of these tools to build object recognition and positioning systems.