Of great importance to a wide variety of computer vision and image analysis problems is the ability to represent two-(2D) and three-dimensional (3D) data or objects. Implicit polynomial curves and surfaces are two of the most useful representations available. Their representational power is evidenced by their ability to smooth noisy data and to interpolate through sparse or missing data. Furthermore, their associated Euclidean and a ne invariants are powerful discriminators, making implicit polynomials a computationally attractive technology for recognizing objects in arbitrary positions with respect to cameras or range sensors. In this paper, we introduce a completely new approach to tting implicit polynomials to data. The algorithm represents a signi cant advancement of implicit polynomial technology for three important reasons. First, it is orders of magnitude faster than existing methods. Second, it has signi cantly better repeatability and numerical stability than current methods. Third, it can easily t polynomials of high, such as 14 th to 18 th , degree. In addition, this approach provides a completely new way of thinking about and handling implicit polynomials.
KEYWORD: free-form modeling, irtspection, pose esiimation, implicit polqn.omials, algebraic in.varianis, shape represemiation, mv2tiscale analysis, Bajesian o bjeci recogmiiion. AB STRACTIn this paper we summarize our results and methods in modeling and representing free-form objects by implicit polynomials, and outline the application of this approach to the industrial inspection problem. Implicit polynomials have been widely used in the computer vision and graphics communities for many years with their flexibility, robustness and invariant features. The effectiveness of the use of this representation hinges upon two factors: the stability of the implicit polynomial fitting procedure and the segmentation of the object shape (or boundary) into manageable pieces. In much of the previous work, algebraic invariants of implicit polynomials have been used for estimation or recognition. In this paper, we propose a general multi-scale free-form object modeling and recognition framework where shape information at different scale is captured by different degree and size polynomials (patchlets). We also show that pose estimation and in turn object recognition can be done without using the algebraic invariants. To accomplish this a closed-form pose solution is obtained by comparing the corresponding patchiets (invariant recognition without using invariants). An overview of a free-form object recognition and inspection system built upon these results is given in the paper.
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