Assignment Task
Abstract
This paper proposes a new image sequence stabilization (ISS) scheme based on filtering of absolute frame positions. The proposed ISS scheme removes undesired motion effects in real-time, while preserving desired gross camera displacements. The well-known finite impulse response (FIR) filter is adopted for filtering. The proposed ISS scheme provides a filtered position and velocity with fine inherent properties. It is demonstrated that the filtered position is not affected by the constant velocity. It is also shown that the filtered velocity is separated from the position. Via numerical simulations, the performance of the proposed scheme is shown to be superior to the existing Kalman filtering scheme.
Introduction
Recently, video communication and processing have been playing a significant role in mobile platforms such as mobile phones, handheld PCs, digital consumer camcorders, and so on. Thus, cameras have become an inherent part in acquiring video images. However, image sequences acquired by a camera mounted on a mobile platform are usually affected by undesired motions causing unwanted positional fluctuations of the image.
To remove undesired motion effects and to produce compensated image sequences that expose requisite gross camera displacements only, various image sequence stabilization (ISS) schemes have been developed and widely used, such as the motion vector integration (MVI) in and the discrete-time Fourier transform (DFT) filtering in. However, in the MVI scheme, the filtered position trajectory is delayed owing to filter characteristics imposing larger frame shift than actually required for stabilization, in which stabilization performance is degraded. The DFT filtering scheme is not suited for real-time application since off-line processing is required. Therefore, recently, the ISS scheme using the Kalman filtering has been made by posing the optimal filtering problem due to the compact representation and the most efficient manner .
However, the Kalman filter has an infinite impulse response (IIR) structure that utilizes all past information accomplished by equaling weighting andalso has a recursive formulation. Thus, the Kalman filter tends to accumulate the filtering error with the progression of time. In addition, the Kalman filter has been known to be sensitive and demonstrate divergence phenomenon for temporary modeling uncertainties and round-off errors
Therefore, in the current paper, an alternative ISS scheme is proposed. The proposed ISS scheme gives the filtered absolute frame position in real-time, removing undesired motion effects, while preserving desired gross camera displacements. For the filtering, the proposed ISS scheme adopts the well known finite impulse response (FIR) filter that utilizes only finite information on the most recent window [7-9]. The proposed ISS scheme provides the filtered velocity as well as the filtered position. These filtered positions and velocities have good inherent properties such as unbiasedness, efficiency, time-invariance, deadbeat, and robustness due to the FIR structure. It is revealed that the filtered position is not affected by the constant velocity. It is also shown that the filtered velocity is separated from the position. These remarkable properties cannot be obtained from the Kalman filtering based scheme in [3, 4]. Via numerical simulations, the performance of the proposed ISS scheme using the FIR filtering is shown to be superior to the existing Kalman filtering scheme.
The paper is organized as follows. In Section 2, an alternative ISS scheme is proposed. In Section 3, inherent properties of the proposed scheme can be seen. In Section 4, numerical simulations are performed. Finally, concluding remarks are made in Section 5.
Theorem 1: When the velocity is constant on the observation window[i – M, i] the filtered position in xp (i) is not affected by the velocity.
Proof: When the velocity is constant as v x on the observation window[i – M, i] , the finite observations ZMi in (5) can be represented in
ZM(i) I { XV (i) = XV for [ i – M,i]}
=
LM X P ( i -M) + NM XV(i) + GM WP(i) + V(i)]
Then, the Filtred Position XP(i) is derived from as
XP(i)
= Hv Zm (i)
= H
v [ LM X P ( i -M) + NM XV(i) + GM WP(i) + V(i)]
= HP LM XP(i – M) + HP NM XV + HP[ GMWP(i) + V(i)]
= Hv [NM XV(i) + GM WP(i) + V(i)],
= XP (i – M ) + M XV + Hp [ GM WP (i) + V(i)].
From (1), the actual position XP(i) can be represented on
[ i – M,i]
as follows:
XP (i) { XV (i) = Xv for [ I – M, i]
= XP(i – M) + M XV + GM WP (i)
Where GM = [ I I …. I 0] . Thus using and the error of the filtered position XP (i) is
XP(i) – XP (I)
= HP [GM WP (i) + V(i) ] – GM WP (i)
Which does not include the velocity term. This completes the proof.
The velocity itself can be treated as a variable which should be filtered. In this case, the filtered velocity is shown to be separated from the position term.
Theorem 2: The filtered velocity xv (i) in (2) is separated from the position term
Proof: The filtered velocity xv (i) is derived from (2) and (7) as
xv (i)
= Hv Zm (i)
= H
v [ LM X P ( i -M) + NM XV(i) + GM WP(i) + V(i)]
= Hv [NM XV(i) + GM WP(i) + V(i)],
which does not include the position term. This completes the proof.
The above remarkable properties of the proposed ISS scheme using the FIR filtering cannot be obtained from the existing Kalman filtering scheme in. In addition, as mentioned previously, the proposed scheme has the deadbeat property, which indicates the fast tracking ability of the proposed scheme. Furthermore, due to the FIR structure and the batch formulation, the proposed scheme might be robust to temporary modeling uncertainties and to round-off errors, while the Kalman filtering scheme might be sensitive to these situation
The noise suppression of the proposed ISS scheme might be closely related to the window length M . However, although the proposed ISS scheme can have greater noise suppression as the window length M increases, too large M may yield the long convergence time of the filtered position and velocity, which degrades the filtering performance of the proposed scheme. This illustrates the proposed ISS scheme’s compromise between the noise suppression and the tracking speed of the filtered position and velocity. Since M is an integer, fine adjustment of the properties with M is difficult. Moreover, it is not easy to determine the window length in systematic ways. In applications, one method to determine the window length is to take the appropriate value that can provide enough noise suppression.