Robust　principal　component　analysis　(PCA)　is　one　of　the　most　important　dimension　reduction　techniques　to　handle　high-dimensional　data　with　outliers.　However,　the　existing　robust　PCA　presupposes　that　the　mean　of　the　data　is　zero　and　incorrectly　utilizes　the　Euclidean　distance　based　optimal　mean　for　robust　PCA　with　1-norm.　Some　studies　consider　this　issue　and　integrate　the　estimation　of　the　optimal　mean　into　the　dimension　reduction　objective,　which　leads　to　expensive　computation.　In　this　paper,　we　equivalently　reformulate　the　maximization　of　variances　for　robust　PCA,　such　that　the　optimal　projection　directions　are　learned　by　maximizing　the　sum　of　the　projected　difference　between　each　pair　of　instances,　rather　than　the　difference　between　each　instance　and　the　mean　of　the　data.　Based　on　this　reformulation,　we　propose　a　novel　robust　PCA　to　automatically　avoid　the　calculation　of　the　optimal　mean　based　on　1-norm　distance.　This　strategy　also　makes　the　assumption　of　centered　data　unnecessary.　Additionally,　we　intuitively　extend　the　proposed　robust　PCA　to　its　2D　version　for　image　recognition.　Efficient　non-greedy　algorithms　are　exploited　to　solve　the　proposed　robust　PCA　and　2D　robust　PCA　with　fast　convergence　and　low　computational　complexity.　Some　experimental　results　on　benchmark　data　sets　demonstrate　the　effectiveness　and　superiority　of　the　proposed　approaches　on　image　reconstruction　and　recognition.