Background　subtraction　has　been　a　fundamental　and　widely　studied　task　in　video　analysis,　with　a　wide　range　of　applications　in　video　surveillance,　teleconferencing,　and　3D　modeling.　Recently,　motivated　by　compressive　imaging,　background　subtraction　from　compressive　measurements　(BSCM)　is　becoming　an　active　research　task　in　video　surveillance.　In　this　paper,　we　propose　a　novel　tensor-based　robust　principal　component　analysis　(TenRPCA)　approach　for　BSCM　by　decomposing　video　frames　into　backgrounds　with　spatial-temporal　correlations　and　foregrounds　with　spatio-temporal　continuity　in　a　tensor　framework.　In　this　approach,　we　use　3D　total　variation　to　enhance　the　spatio-temporal　continuity　of　foregrounds,　and　Tucker　decomposition　to　model　the　spatio-temporal　correlations　of　video　background.　Based　on　this　idea,　we　design　a　basic　tensor　RPCA　model　over　the　video　frames,　dubbed　as　the　holistic　TenRPCA　model.　To　characterize　the　correlations　among　the　groups　of　similar　3D　patches　of　video　background,　we　further　design　a　patch-group-based　tensor　RPCA　model　by　joint　tensor　Tucker　decompositions　of　3D　patch　groups　for　modeling　the　video　background.　Efficient　algorithms　using　the　alternating　direction　method　of　multipliers　are　developed　to　solve　the　proposed　models.　Extensive　experiments　on　simulated　and　real-world　videos　demonstrate　the　superiority　of　the　proposed　approaches　over　the　existing　state-of-the-art　approaches.