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In order to fully exploit the advantages of massive multiple-input multiple-output (mMIMO), it is critical for the transmitter to accurately acquire the channel state information (CSI). Deep learning (DL)-based methods have been proposed for CSI compression and feedback to the transmitter. Although most existing DL-based methods consider the CSI matrix as an image, structural features of the CSI image are rarely exploited in neural network design. As such, we propose a model of self-information that dynamically measures the amount of information contained in each patch of a CSI image from the perspective of structural features. Then, by applying the self-information model, we propose a model-and-data-driven network for CSI compression and feedback, namely IdasNet. The IdasNet includes the design of a module of self-information deletion and selection (IDAS), an encoder of informative feature compression (IFC), and a decoder of informative feature recovery (IFR). In particular, the model-driven module of IDAS pre-compresses the CSI image by removing informative redundancy in terms of the self-information. The encoder of IFC then conducts feature compression to the pre-compressed CSI image and generates a feature codeword which contains two components, i.e., codeword values and position indices of the codeword values. Subsequently, the IFR decoder decouples the codeword values as well as position indices to recover the CSI image. Experimental results verify that the proposed IdasNet noticeably outperforms existing DL-based networks under various compression ratios while it has the number of network parameters reduced by orders-of-magnitude compared with various existing methods.