As　one　of　the　next-generation　multimedia　technology,　high　dynamic　range　(HDR)　imaging　technology　has　been　widely　applied.　Due　to　its　wider　color　range,　HDR　image　brings　greater　compression　and　storage　burden　compared　with　traditional　LDR　image.　To　solve　this　problem,　in　this　paper,　a　two-layer　HDR　image　compression　framework　based　on　convolutional　neural　networks　is　proposed.　The　framework　is　composed　of　a　base　layer　which　provides　backward　compatibility　with　the　standard　JPEG,　and　an　extension　layer　based　on　a　convolutional　variational　autoencoder　neural　networks　and　a　post-processing　module.　The　autoencoder　mainly　includes　a　nonlinear　transform　encoder,　a　binarized　quantizer　and　a　nonlinear　transform　decoder.　Compared　with　traditional　codecs,　the　proposed　CNN　autoencoder　is　more　flexible　and　can　retain　more　image　semantic　information,　which　will　improve　the　quality　of　decoded　HDR　image.　Moreover,　to　reduce　the　compression　artifacts　and　noise　of　reconstructed　HDR　image,　a　post-processing　method　based　on　group　convolutional　neural　networks　is　designed.　Experimental　results　show　that　our　method　outperforms　JPEG　XT　profile　A,　B,　C　and　other　methods　in　terms　of　HDR-VDP-2　evaluation　metric.　Meanwhile,　our　scheme　also　provides　backward　compatibility　with　the　standard　JPEG.　©　2020　IEEE.