Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices   Quaternion matrices are employed successfully in many color image processing
applications. In particular, a pure quaternion matrix can be used to represent
red, green and blue channels of color images. A low-rank approximation for a
pure quaternion matrix can be obtained by using the quaternion singular value
decomposition. However, this approximation is not optimal in the sense that the
resulting low-rank approximation matrix may not be pure quaternion, i.e., the
low-rank matrix contains real component which is not useful for the
representation of a color image. The main contribution of this paper is to find
an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion
matrix (a color image). Our idea is to use a projection on a low-rank
quaternion matrix manifold and a projection on a quaternion matrix with zero
real component, and develop an alternating projections algorithm to find such
optimal low-rank pure quaternion matrix approximation. The convergence of the
projection algorithm can be established by showing that the low-rank quaternion
matrix manifold and the zero real component quaternion matrix manifold has a
non-trivial intersection point. Numerical examples on synthetic pure quaternion
matrices and color images are presented to illustrate the projection algorithm
can find optimal low-rank pure quaternion approximation for pure quaternion
matrices or color images.

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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: stat
29
The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin