  The main purpose of our paper is a new approach to design of algorithms of
Kaczmarz type in the framework of operators in Hilbert space. Our applications
include a diverse list of optimization problems, new Karhunen-Lo\`eve
transforms, and Principal Component Analysis (PCA) for digital images. A key
feature of our algorithms is our use of recursive systems of projection
operators. Specifically, we apply our recursive projection algorithms for new
computations of PCA probabilities and of variance data. For this we also make
use of specific reproducing kernel Hilbert spaces, factorization for kernels,
and finite-dimensional approximations. Our projection algorithms are designed
with view to maximum likelihood solutions, minimization of "cost" problems,
identification of principal components, and data-dimension reduction.
