Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multi-resolution stochastic processes in the form of infinite series: Xt=n=0infinity lnDnt en, where ln Deltan(t) is the integral of the nth Haar wavelet from 0 to t and epsilon n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions for Xt to converge uniformly with probability one are provided. Xt ∈ Hu , the collection of all almost sure uniform limits, retains the second moment properties and the roughness of sample paths of Brownian motion, yet lacks some of the features of Brownian motion, e.g. does not have independent and/or stationary increments or is not Gaussian. Three major tools are developed to analyze elements of Hu . First, a notion of dyadic quadratic variation is used to prove that Brownian motion is the only martingale in Hu . The other two, the nth level self similarity of the bridges and the tree structure of dyadic increments of different levels, are essential in establishing sample paths results such as Holder continuity and fractional dimensions of graphs. Simulation of the approximants Xnt of seven different types of random variables epsilonn reveals different behaviors of the sample paths.