A Hybrid Hyperquadric Model for 2-D and 3-D Data Fitting

We present in this paper a new curve and surface implicit model. This implicit model is based on hyperquadrics and allows a local and global control of the shape and a wide variety of allowable shapes. We define a hybrid hyperquadric model by introducing implicitly some local properties on a global shape model. The advantage of our model is that it describes global and local properties through a unique implicit equation, yielding a representation of the shape by means of its parameters, independently of the chosen numerical resolution. The data fitting is obtained through the minimization of energy, modeling the attraction to data independently of the implicit description of the shape. After studying the geometry of hyperquadrics and how the shape deforms when we modify slightly its implicit equation, we are able to define an algorithm for automatic refining of the fit by adding an adequate term to the implicit representation. This geometric approach makes possible an efficient description of the data points and an automatic tuning of the fit according to the desired accuracy.

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