This thesis develops a quaternion wavelet transform (QWT) as a new multiscale analysis tool for geometric image features. The QWT is a near shift-invariant tight frame representation whose coefficients sport a magnitude and three phases: two phases encode local image shifts while the third contains textural information. The QWT can be computed using a dual-tree filter bank with linear computational complexity. We develop the QWT by applying an alternative theory of 2-D Hilbert transforms and analytic signals to the 1-D complex wavelet transform. To demonstrate the properties of the QWT's coherent magnitude/phase representation, we develop a efficient and accurate algorithm for estimating the local geometrical structure of images and a multiscale algorithm for estimating the disparity between a pair of images. The latter algorithm has potentials for various applications in image registration and flow estimation. It uses an interesting multiscale phase unwrapping procedure and features linear complexity and sub-pixel accuracy.