The monogenic signal consists of an image and its first-order Riesz transform. It describes signal structure as a sinusoid with a particular amplitude, phase and orientation; however, the orientation estimate is poor around certain phase values. We describe a novel method of estimating this sinusoidal signal model using higher-order Riesz transforms, such that amplitude, phase and orientation estimates are improved under noise conditions. Furthermore, the method leads to novel intrinsically-1D (line and edge) and intrinsically-2D (corner and junction) detectors.