Images as bags of pixels

We propose modeling images and related visual objects as bags of pixels or sets of vectors. For instance, gray scale images are modeled as a collection or bag of (X, Y, I) pixel vectors. This representation implies a permutational invariance over the bag of pixels, which is naturally handled by endowing each image with a permutation matrix. Each matrix permits the image to span a manifold of multiple configurations, capturing the vector set´s invariance to orderings or permutation transformations. Permutation configurations are optimized while jointly modeling many images via maximum likelihood. The solution is a uniquely solvable convex program, which computes correspondence simultaneously for all images (as opposed to traditional pairwise correspondence solutions). Maximum likelihood performs a nonlinear dimensionality reduction, choosing permutations that compact the permuted image vectors into a volumetrically minimal subspace. This is highly suitable for principal components analysis which, when applied to the permutationally invariant bag of pixels representation, outperforms PCA on appearance-based vectorization by orders of magnitude. Furthermore, the bag of pixels subspace benefits from automatic correspondence estimation, giving rise to meaningful linear variations such as morphings, translations, and jointly spatio-textural image transformations. Results are shown for several datasets.