Video based human motion analysis has been actively studied over the past decades. We propose novel approaches that are able to analyze human motion under such challenges and apply them to surveillance and security applications. Part I analyses the cyclic property of human motion and presents algorithms to classify humans in videos by their gait patterns. Two approaches are proposed. The first employs the omputationally efficient periodogram, to characterize periodicity. In order to integrate shape and motion, we convert the cyclic pattern into a binary sequence using the angle between two legs when the toe-to-toe distance is maximized during walking. Part II further extends the previous approaches to analyze the symmetry in articulation within a stride. A feature that has been shown in our work to be a particularly strong indicator of the presence of pedestrians is the X-junction generated by bipedal swing of body limbs. The proposed algorithm extracts the patterns in spatio-temporal surfaces. In Part III, we present a compact characterization of human gait and activities. Our approach is based on decomposing an image sequence into x-t slices, which generate twisted patterns defined as the Double Helical Signature (DHS). It is shown that the patterns sufficiently characterize human gait and a class of activities. The features of DHS are: (1) it naturally codes appearance and kinematic parameters of human motion; (2) it reveals an inherent geometric symmetry (Frieze Group); and (3) it is effective and efficient for recovering gait and activity parameters. Finally, we use the DHS to classify activities such as carrying a backpack, briefcase etc. The advantage of using DHS is that we only need a small portion of 3D data to recognize various symmetries.