A Fast Discriminant Feature Extraction Framework Combining Implicit Spatial Smoothness with Explicit One for Two-Dimensional Image

Images have two-dimensional inherent spatial structures, and the pixels spatially close to each other have similar gray values, which means images are locally spatially smooth. To extract features, traditional methods usually convert an original image into a vector, resulting in the destruction of spatial structure. Thus 2D image-based feature extraction methods emerge, typically, such as 2DLDA and 2DPCA, which reduce time complexity significantly. However，2D-based methods manipulate on the whole raw (or column) of an image, leading to spatially under-smoothing. To overcome such shortcomings, spatial regularization is proposed by explicitly imposing a Laplacian penalty to constrain the projection coefficients to be spatially smooth and has achieved better performance than 2D-based methods, but sharing the genetic high computing cost with 1D methods. Implicit spatial regularization (ISR) constrains spatial smoothness within each local image region by dividing and reshaping image and then executing 2D-based feature extraction methods, resulting in a performance improvement of the typical bi-side 2DLDA over SSSL (a typical ESR method). However, ISR obtains the spatial smooth implicitly but has lack of explicit spatial constraints such that the feature space obtained by ISR is still not smooth enough. The optimization criteria of bi-side 2DLDA are not jointly convex simultaneously, resulting in high computing cost and globally optimal solution cannot be guaranteed. Inspired by statements above, we introduce a novel linear discriminant model called fast discriminant feature extraction framework combining implicit spatial smoothness with explicit one for two-dimensional image recognition (2D-CISSE). The key step of 2D-CISSE is to preprocess spatial smooth for images, then ISR is executed. 2D-CISSE not only retains spatial smooth explicitly, but also reinforces the explicit spatial constraints. Not only can it achieve globally optimal solution, but it also have generality, i.e. any out-of-shelf image smoothing methods and 2D-based feature extraction methods can be embedded into our framework. Finally, experimental results on four face datasets (Yale, ORL, CMU PIE and AR) and handwritten digit datasets (MNIST and USPS) demonstrate the effectiveness and superiority of our 2D-CISSE.