The　ship　wake　detection　of　SAR　images　is　useful　not　only　in　estimating　the　speed　and　the　direction　of　moving　ships,　but　also　in　finding　small　ships　which　are　hard　to　be　detected.　The　traditional　ship　wake　detection　methods　of　SAR　images　can　achieve　satisfactory　results　in　simple　backgrounds,　but　hardly　work　in　complex　backgrounds.　In　this　paper,　we　propose　a　novel　method　based　on　the　morphological　component　analysis　and　the　dictionary　learning　to　detect　ship　wakes　in　complex　backgrounds.　In　our　method,　the　SAR　image　is　decomposed　into　a　cartoon　component　containing　ship　wakes　and　a　sea-background　texture　component　by　adaptively　learning　the　ship　wake　dictionary　and　the　sea-background　texture　dictionary;　and　then　the　shearlet　transform　is　used　to　enhance　ship　wakes　in　the　cartoon　component.　Experimental　results　show　our　method　outperforms　the　traditional　methods　for　SAR　images　in　complex　backgrounds.