The multiscale edges of a signal are the sharp variation points measured at different scales. This thesis studies a model of multiscale edge representation based on the local maxima wavelet transform. The wavelet transform is a mathematical formulation of a multiscale decomposition. It decomposes a signal into multiple components indexed by a scale parameter. A particular class of wavelets are used such that each of these components is the first derivative of a smooth version of the signal, with the scale parameter indicating the degree of smoothing. The local maxima of this wavelet transform is therefore a multiscale edge representation. This thesis shows that the local maxima not only identify the edges but also characterize the edges. An algorithm to reconstruct a signal from its local maximum representation is developed. The experimental results show that the algorithm reconstructs the original signal, and this reconstruction is stable. This implies that the local maximum representation is a reorganization of the signal information. Therefore, various pattern analysis algorithms can be developed uniquely based on the properties of edges. Image processing can also be done through the multiscale edge representation. An application to image coding is described.