# Color Image Compression via the Higher-Order Singular Value Decomposition (HOSVD)

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This Demonstration shows a method for data compression via the higher-order singular value decomposition (HOSVD) with color images interpreted as third-order tensors. The data compression is achieved by decreasing the tensor ranks in the pixel dimensions, while the third dimension (RGB channels) is not changed.

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In the source code of this Demonstration, a basic general-purpose HOSVD toolbox is included as a collection of functions. It works with arbitrary tensor dimensions.

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Contributed by: Michael Weyrauch and Alexander Scholz (June 2013)
Open content licensed under CC BY-NC-SA

## Details

Higher-order singular value decomposition (HOSVD) is a generalization of the well-known singular value decomposition (SVD) for matrices. HOSVD defines a decomposition for tensors of higher dimension (greater than two). This decomposition enables reducing the rank of the tensors, which can be used as a method for data compression. In this Demonstration, color images are interpreted as third-order tensors. The third dimension corresponds to RGB channels and is not rank reduced. You can vary the ranks of the first and second dimensions, the pixels.

For a matrix, the column and row ranks are equal, and a unique matrix rank can be defined. For higher-dimensional tensors this does not hold, but a rank for each of its dimensions can be defined. This concept is called the -rank. For a detailed description of this rank concept, see [1].

For matrices, a low-rank approximation is optimal. An -rank approximation of a tensor is not necessarily optimal. The higher-order orthogonal iteration (HOOI) is an optimization algorithm that improves the low-rank approximation. For HOOI, see [2].

Explanation of the Controls

"compression rate" slider: adjusts the ranks of the first and second dimensions 1–128; 128 means full-rank HOSVD (no compression), 1 means rank-(1,1,3)-HOSVD (maximum compression in this Demonstration)

"optimization" checkbox: controls whether the HOOI optimization algorithm (with 10 iterations) is executed or not

"test image" buttons: choose one of three different test images from Mathematica's ExampleData

References

[1] L. De Lathauwer, B. De Moor, and J. Vandewalle, "A Multilinear Singular Value Decomposition," SIAM Journal on Matrix Analysis and Applications, 21(4), 2000 pp. 1253–1278. doi:10.1137/S0895479896305696.

[2] L. De Lathauwer and J. Vandewalle, "Dimensionality Reduction in Higher-Order Signal Processing and Rank-(R1,R2,...,RN) Reduction in Multilinear Algebra," Linear Algebra and Its Applications, 391, 2004 pp. 31–55. doi:10.1016/j.laa.2004.01.016.

## Permanent Citation

Michael Weyrauch and Alexander Scholz

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