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For many vision tasks that involve object shapes, such as alignment of object landmarks, Principal Component Analysis (PCA) is a standard technique to model the distribution of their locations. Prior to PCA, a training data set of landmarks is preprocessed by Procrustes Analysis, which removes the differences of scale, rotation and translation among them, and consequently maps all the training data onto a hypersphere. A data distribution on a hypersphere, however, cannot be appropriately modeled by the major linear axes found by PCA, which assumes volumetric ellipsoidal distribution of data. Proper considerations of the nature of the hypersphere data distribution should be taken. Principal Nested Spheres (PNS) is a data decomposition technique specifically designed for data on a hypersphere. We demonstrate the effectiveness of the PNS by incorporating it with existing object alignment methods.

PCA and PNS applied to data scattered on a 3-dimensional sphere (toy example)
The linear axes obtained by PCA cannot appropriately capture the data distribution. PNS fits a (hyper)circle which explains the data variance on a (hyper)sphere.

The first component of PNS and PCA (upper:PNS, lower:PCA)

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