Approach: 2. Construction of Cellular Morphometric Types and Cellular Morphometric Context

For a set of WSIs and corresponding nuclear segmentation results, let M be the total number of segmented nuclei; N be the number of morphometric descriptors extracted from each segmented nucleus, e.g. nuclear size, and nuclear intensity; and X be the set of morphometric descriptors for all segmented nuclei, where . The construction of cellular morphometric types and cellular morphometric context are described as follows,

  1. Construct cellular morphometric types D, where D=[d1,...,dk]T are the K cellular morphometric types to be learned by the following optimization:

    where  Z=[z1,...,zM]T indicates the assignment of the cellular morphometric type,  is a cardinality constraint enforcing only one nonzero element of  is a non-negative constraint on the elements of zm, and |zm| is the L1-norm of zm. During training, this equation is optimized with respect to both Z and D; In the coding phase, for a new set of X, the learned D is applied, and this equation is optimized with respect to Z only.
  2. Construct cellular morphometric context vis SPM. This is done by repeatedly subdividing an image and computing the histograms of different cellular morphometric types over the resulting subregions. As a result, the spatial histogram, H, is formed by concatenating the appropriately weighted histograms of all cellular morphometric types at all resolutions.

In our experiment, K is fixed to be 64. Meanwhile, given the fact that each patient may contain multiple WSIs, SPM is applied at a single scale for the convenient construction of cellular morphometric context as well as the integrative analysis at patient level, where both cellular morphometric types and the subtypes of cellular morphometric context are associated with clinical outcomes, and molecular information.