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Volumetric image compression has become an urgent task to effectively transmit and store images produced in biological research and clinical practice. At present, the most commonly used volumetric image compression methods are based on wavelet transform, such as JP3D. However, JP3D employs an ideal, separable, global, and fixed wavelet basis to convert input images from pixel domain to frequency domain, which seriously limits its performance. In this paper, we first design a 3-D trained wavelet-like transform to enable signal-dependent and non-separable transform. Then, an affine wavelet basis is introduced to capture the various local correlations in different regions of volumetric images. Furthermore, we embed the proposed wavelet-like transform to an end-to-end compression framework called aiWave to enable an adaptive compression scheme for various datasets. Last but not least, we introduce the weight sharing strategies of the affine wavelet-like transform according to the volumetric data characteristics in the axial direction to reduce the amount of parameters. The experimental results show that: 1) when cooperating our trained 3-D affine wavelet-like transform with a simple factorized entropy module, aiWave performs better than JP3D and is comparable in terms of encoding and decoding complexities; 2) when adding a context module to further remove signal redundancy, aiWave can achieve a much better performance than HEVC.