We　propose　the　eigenvalue　problem　of　an　anisotropic　diffusion　operator　for　image　segmentation.　The　diffusion　matrix　is　defined　based　on　the　input　image.　The　eigenfunctions　and　the　projection　of　the　input　image　in　some　eigenspace　capture　key　features　of　the　input　image.　An　important　property　of　the　model　is　that　for　many　input　images,　the　first　few　eigenfunctions　are　close　to　being　piecewise　constant,　which　makes　them　useful　as　the　basis　for　a　variety　of　applications,　such　as　image　segmentation　and　edge　detection.　The　eigenvalue　problem　is　shown　to　be　related　to　the　algebraic　eigenvalue　problems　resulting　from　several　commonly　used　discrete　spectral　clustering　models.　The　relation　provides　a　better　understanding　and　helps　developing　more　efficient　numerical　implementation　and　rigorous　numerical　analysis　for　discrete　spectral　segmentation　methods.　The　new　continuous　model　is　also　different　from　energy-minimization　methods　such　as　active　contour　models　in　that　no　initial　guess　is　required　for　in　the　current　model.　A　numerical　implementation　based　on　a　finite-element　method　with　an　anisotropic　mesh　adaptation　strategy　is　presented.　It　is　shown　that　the　numerical　scheme　gives　much　more　accurate　results　on　eigenfunctions　than　uniform　meshes.　Several　interesting　features　of　the　model　are　examined　in　numerical　examples,　and　possible　applications　are　discussed.　©　2016　IEEE.