  Recently proposed L2-norm linear discriminant analysis criterion via the
Bhattacharyya error bound estimation (L2BLDA) is an effective improvement of
linear discriminant analysis (LDA) for feature extraction. However, L2BLDA is
only proposed to cope with vector input samples. When facing with
two-dimensional (2D) inputs, such as images, it will lose some useful
information, since it does not consider intrinsic structure of images. In this
paper, we extend L2BLDA to a two-dimensional Bhattacharyya bound linear
discriminant analysis (2DBLDA). 2DBLDA maximizes the matrix-based between-class
distance which is measured by the weighted pairwise distances of class means
and meanwhile minimizes the matrix-based within-class distance. The weighting
constant between the between-class and within-class terms is determined by the
involved data that makes the proposed 2DBLDA adaptive. In addition, the
criterion of 2DBLDA is equivalent to optimizing an upper bound of the
Bhattacharyya error. The construction of 2DBLDA makes it avoid the small sample
size problem while also possess robustness, and can be solved through a simple
standard eigenvalue decomposition problem. The experimental results on image
recognition and face image reconstruction demonstrate the effectiveness of the
proposed methods.
