In　this　paper,　we　study　the　problem　of　low-rank　tensor　completion　with　the　purpose　of　recovering　a　low-rank　tensor　from　a　tensor　with　partial　observed　items.　To　date,　there　are　several　different　definitions　of　tensor　ranks.　We　focus　the　study　on　the　low　tubal　rank　tensor　completion　task.　Previous　works　solve　the　low　tubal　rank　tensor　completion/recovery　problems　by　convex　tensor　nuclear　norm　minimization.　However,　this　kind　of　tensor　nuclear　norm　is　orientation　dependent,　which　is　originally　due　to　the　definition　of　tensor-tensor　product.　Based　on　the　convex　tensor　nuclear　norm　minimization,　the　tensor　recovery　performance　varies　when　the　orientation　of　the　input　data　is　different.　However,　in　practice,　it　is　generally　hard　to　choose　the　best　way　of　the　data　input.　To　address　this　issue,　we　propose　a　new　convex　model　which　is　based　on　the　sum　of　tensor　nuclear　norm　minimization.　It　includes　the　existing　tensor　nuclear　norm　minimization　model　as　a　special　case　which　is　corresponding　to　an　orientation　of　the　input　data.　The　proposed　model　is　convex　and　thus　can　be　solved　efficiently.　Numerical　experiments　on　images　and　video　sequences　demonstrate　the　effectiveness　of　our　proposed　method.　©　2013　IEEE.