Providing sample shape statistics with FCA and ISA approaches

Providing sample shape statistics for time warped signals or more generally for heterogeneous shape signals can be useful in many signal processing applications. In fact, the classical mean and the L distance are inadequate in describing and quantifying the time or shape variability among such signals. Specific methods derived from the curve registration (CR) theory were developed to deal with this warping or shape problem. More recently, two approaches seem similar in many points. Functional convex averaging (FCA) and integral shape averaging (ISA) proposed an alternative to CR methods in averaging and comparing time warped signals. Both methods estimate a sample mean without requiring that observed signals are well structured and consider the warping as a stochastic process. The objective of the study is to compare ISA and FCA methods with respect to shape equality condition. For this purpose, shape equality is defined and the used methods are recalled with an emphasis on the similarities. After, a corrected ISA (CISA) approach providing a mean and a distance is proposed and compared to the FCA approach. This comparison with simulations shows that the CISA approach is more suitable for quantifying shape variability than FCA one since it respects shape equality condition