Fresnel　incoherent　correlation　holography　(　FINCH)　was　proposed　to　break　the　barrier　of　spatial　incoherent　digital　holographic　imaging　and　show　the　potential　of　super-resolution　imaging　preferences.　We　developed　FINCH　as　a　compressive　sensing　modality　and　reconstruction　procedure　as　an　inverse　problem　in　order　to　realize　3D　tomographic　imaging.　Improved　axial　resolution　is　obtained　via　compressive　reconstruction.　Reconstruction　guarantees　and　accuracy　of　the　proposed　method　are　discussed.　Compared　with　the　real-valued　signal　operation,　the　signal-to-noise　ratio　of　the　results　is　increased　when　reconstructing　from　the　complex-valued　hologram　obtained　from　the　FINCH　system.　(C)　2017　Optical　Society　of　America