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The question can be formulated as a statistical hypothesis asserting that the distribution of the shapes of closed curves representing outlines of cell nuclei in a spatial domain is independent of the distribution of their locations. The key challenge in developing a procedure to test the hypothesis from a sample of spatially indexed curves (e.g. from an image) lies in how symmetries in the data are accounted for: shape of a curve is a property that is invariant to similarity transformations and reparameterization, and the shape space is thus an infinite-dimensional quotient space. Starting with a convenient geometry for the shape space developed over the last few years, I will discuss dependence measures and their estimates for spatial point processes with shape-valued marks, and demonstrate their use in testing for spatial independence of marks in a breast cancer application.
