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This book develops monogenic wavelet transform (MWT) with extension to multispectral signals as a new multiscale analysis tool for geometric image features. Monogenic wavelets offer geometric representation of grayscale images through an AM/FM model allowing invariance of coefficients to translations and rotations. The underlying concept of local phase includes a fine contour analysis into a coherent unified framework. Starting from a link with structure tensors, the book proposes a non-trivial extension of the monogenic framework to vector-valued signals to carry out a non-marginal color monogenic wavelet transform. They also give a practical study of this new wavelet transform in the contexts of sparse representations and invariant analysis, which helps to understand the physical interpretation of coefficients and validates the interest of our theoretical construction. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitude, the local orientation and local phases. Both the monogenic wavelet transform and the structure multivector are combined with an appropriate scale-space approach, resulting in multi-hyperspectral images.