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Correlators have been used for detecting shapes but not as often for measuring shape similarity. The complex inner product (CIP) has been used in various formulations as a shape similarity measure. The CIP is essentially a one-dimensional correlation approach to measuring similarity. One-dimensional variants of the correlation techniques including the matched filter (MF), phase-only filter (POF), and amplitude-modulated phase only filter (AMPOF) are shown to measure shape similarity in a trend that approaches human perception, however, clear performance differences are noted. The results show that the best correlator for measuring shape similarity is not the best correlator for detecting a shape. It is suggested that detection and shape similarity are fundamentally different functions that are in opposition to some degree. Ideal detection and ideal similarity measurement functions are explored. The degree to which various formulations of correlators approach the ideal functions of detection and similarity measurement are shown as well as results from human psychophysical experiments.

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. The system factors are classified into various classes based on types of measurement errors produced. 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 3-D surface is to the ideal 3-D surface within a predefined acceptable error margin while the CIP measures how well the 3-D surface correlates to the ideal 3-D 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.

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