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Toward a comparative analysis of three-dimensional shape measures

Measuring a system's capability to acquire accurate three- dimensional shape is important for validating the system for a particular application. Various system factors are reviewed that contribute to inaccurate shape. As shown in this paper, different shape measures do not do a complete evaluation but provide different information depending on the type of error. A partial-directed hausdorf (PDH) and complex inner product (CIP) measure that were previously introduced to measure two-dimensional shapes are now extended to measure three-dimensional shapes. PDH measures how close the 3D surface is to the ideal 3D surface within a predefined acceptable error margin, while the CIP measures how well the 3D surface correlates to the ideal 3D surface. Two variants of the CIP measure are used in this paper including a pure phase only filter and a normalized matched filter. The CIP measure is compared to the Procrustes metric for comparing shapes. Using a test case shape, the measures are compared and shown to provide varying information. Alone, any one measure cannot provide complete shape information. Combining measures provides a more robust three-dimensional shape measurement system. The shape measures are demonstrated first on three-dimensional data with controlled variation and then on laser ranging data.