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Recently, quaternions have been applied to multi-channel image processing such as color image registration, edge detection, and image matching. The general approach consists of using up to four image components as separate components in a quaternion-valued image. Techniques from quaternion algebra and analysis, such as quaternion Fourier transforms and correlation, are exploited to implement the desired image processing operations. The motivation underlying use of quaternions involves the concept of holistic data processing. That is, information in separate channels is processed jointly, thereby leveraging correlations across channel. Furthermore, quaternion transforms, such as the Fourier and wavelet transforms, have been defined and can be applied to process quaternion-valued imagery. For example, analogues from classical signal processing, such as the convolution theorem, have been derived and can be used to reduce computational burden. Such quaternion-based image processing techniques can provide performance superior to classical approaches which process image components independently. However, under certain conditions, performance of quaternion-based methods fails to meet performance of independent channel processing methods. In this paper, we perform a theoretical analysis of the performance of quaternion image matching. We present conditions under which quaternionbased processing outperforms, or falls short of, independent channel processing methods. We produce numerical image matching results validating the theoretical performance analysis using synthetic data. We define and solve a linear system of equations to generate the synthetic data. Receiver Operating Characteristics curves are provided to demonstrate the improved performance of the quaternion image matching approach under the favorable conditions predicted by the theory.
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