In this diploma thesis, the use of the singular value decomposition of matrices with quaternion coefficients in colour image processing is studied. The quaternion representation enables the colour components of an image to be treated jointly, which is a natural extension of the usual matrix representation of images. After introducing the basics of quaternions and matrices with quaternion coefficients, the complex adjoint matrix is introduced, allowing the computation of the singular value decomposition to be transferred to complex matrices. Two algorithms for computing the singular value decomposition of quaternion matrices are considered. The first is based on singular value decomposition of the complex adjoint matrix, while the second one employs Givens rotations and Householder reflections. Both algorithms are implemented in MATLAB and applied to examples of image processing. The final part is devoted to a comparison of the algorithms and an interpretation of the obtained results.
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