In this document, we reproduce a result as seen in Tenenbaum, Silva and Langford (2000). By using dimensionality reduction technique (Isomap in particular), we find the 2 intrinic dimensions which contain the information of the orientation of the face images. The data we have are 698 face images each of which is of resolution 64 x 64 pixels.
rm(list = ls())
library(maniTools)
library(R.matlab)
util_path <- system.file("face_orientation", "util_face_orientation.R", package = "maniTools")
source(util_path)
# load data
fpath <- system.file("face_orientation", "face_data.mat", package = "maniTools")
matlab_file <- readMat(fpath)
image_mat <- t(matlab_file$images)
# perform dimensionality reduction
proj_data <- RDRToolbox::Isomap(image_mat, 2, 6)$dim2 #set d = 3 to learn lighting directions.
plot(proj_data, pch = 19, cex = 0.5)
# plot results
sample_image_ind <- sample(seq(nrow(image_mat)), 20)
purrr::walk(sample_image_ind, ~plot_faces(., proj_data, image_mat))
In the above, we see the projected dimension 1 relates to the left-right orientation of the faces while the projected dimension 2 relates to the top-bottom orientation of the faces. Isomap successfully extracts the information about the true variation of the face images.
Paper: Tenenbaum, J. B., De Silva, V., & Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. science, 290(5500), 2319-2323.