Models of geometric shapes are required for many fields that involve the description and understanding of spatial phenomena. In computer graphics applications, models that produce realistic impressions of the objects are desired. In psychology, general models with perceptually relevant parameters are needed so that psychologically significant shape characteristics can be quantified. To automate the extraction of shape information from real images, we must have geometric models that are broad enough to encompass naturally produced shapes, but simple enough to identify their parameters. Unified approaches to shape modeling that meet all these criteria are very desirable.

[Superqudrics]

Problem: Assumption of intrinsical symmetry!

{Extended Superquadrics}

Formulation: Extending the exponents of superquadrics to functions (Bezier curves) of latitude angle and longitude angle in the spherical coordinate system in order to avoid the symmetry problem in superquadrics and the many-to-one problem in hyperquadrics.

Shape Control and Local Deformations: Since the exponents of extended superquadrics are no longer constants, but Bezier splines, only a small portion of the 2D extended superquadric shape is affected by varying one control point in the Bezier spline exponets. Hence, local deformations can be obtained by adjusting the corresponding control points in the exponent functions.

Local deformation by ajusting one control point in f1.
The whole sequence | Animated GIF

Local deformation by ajusting one control point in f2.
The whole sequence | Animated GIF

More animations (AVI movie).

Realistic Modeling:

Four realistic images generated with extended superquadric modeling: sofa pillow, bread slice, test tube, and mushroom (32K).

[Extended Superquadrics Fitting]

Algorithm:

• Inside-outside function is used to define an error function.
• Levenberg-Marquardt non-linear optimization method is utilized to accomplish the minimization.
• A hierarchical iteration algorithm is used ( increase the number of control points hierarchically).

Steps of fitting a convex synthetic curve which is generated by I3DM, an Interactive 3D Modeler in Silicon Graphics' Workstation (7K).

Steps of fitting a partially concave synthetic curve which is generated by I3DM, an Interactive 3D Modeler in Silicon Graphics' Workstation (7K).

3D Real Data:

Fitting a spoon from range data using extended superquadrics (14K).

Fitting a duck from range data using extended superquadrics (21K).

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