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Three-dimensional steerable discrete cosine transform with application to 3D image compression

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Abstract

This work introduces the three-dimensional steerable discrete cosine transform (3D-SDCT), which is obtained from the relationship between the discrete cosine transform (DCT) and the graph Fourier transform of a signal on a path graph. One employs the fact that the basis vectors of the 3D-DCT constitute a possible eigenbasis for the Laplacian of the product of such graphs. The proposed transform employs a rotated version of the 3D-DCT basis. We then evaluate the applicability of the 3D-SDCT in the field of 3D medical image compression. We consider the case where we have only one pair of rotation angles per block, rotating all the 3D-DCT basis vectors by the same pair. The obtained results show that the 3D-SDCT can be efficiently used in the referred application scenario and it outperforms the classical 3D-DCT.

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Notes

  1. Such axes labels are used here with the purpose of distinguishing the coordinates in a three-dimensional (eigen)space only; we are not referring to axes x, y and z usually employed to characterize \(\mathbb {R}^3\).

  2. Unless otherwise stated, from this point forward, the eigenvectors of \(\mathbf {L}(\mathcal {L}_N)\) are expressed as \(1\times N^3\) vectors, formed by the entries of the corresponding \(N\times N\times N\) structure taken in the lexicographical order.

  3. The indexes are the same ones we have used to identify the eigenvalues and eigenvectors of \(\mathbf {L}(\mathcal {L}_N)\).

  4. A practical rule for obtaining the other two coefficients of a triple, from a coefficient \(F_{k,l,m}\) lying in the red part of the cubical structures we have considered, is to apply two sequential cyclic shifts to the left to indexes k, l, m; in this manner, one obtains coefficients \(F_{l,m,k}\) and \(F_{m,k,l}\).

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Acknowledgements

This work was supported by CNPq under Grants 309598/2017-6 and 409543/2018-7.

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Authors and Affiliations

  1. Department of Electronics and Systems, Federal University of Pernambuco, Recife, Brazil

    Verusca S. Lima & Juliano B. Lima

  2. Polytechnic School of Pernambuco, University of Pernambuco, Recife, Brazil

    Verusca S. Lima & Francisco Madeiro

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  1. Verusca S. Lima
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Lima, V.S., Madeiro, F. & Lima, J.B. Three-dimensional steerable discrete cosine transform with application to 3D image compression. Multidim Syst Sign Process 32, 491–519 (2021). https://doi.org/10.1007/s11045-020-00746-9

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