A new multiplication-free transform derived from DHT is introduced: the RHT. Investigations on the properties of the RHT led us to the concept of weak-inversion. Using new constructs, we show that RHT is not involutional like the DHT, but exhibits quasi-involutional property, a new definition derived from the periodicity of matrices. Thus instead of using the actual inverse transform, the RHT is viewed as an involutional transform, allowing the use of direct (multiplication-free) to evaluate the inverse. A fast algorithm to compute RHT is presented. This algorithm show embedded properties. We also extended RHT to the two-dimensional case. This permitted us to perform a preliminary analysis on the effects of RHT on images. Despite of some SNR loss, RHT can be very interesting for applications involving image monitoring associated to decision making, such as military applications or medical imaging.