This paper presents a method to assess the pointing and image motion performance of optical payloads in the presence of image displacement (shift), smear, and jitter. The method assumes the motion is a stationary random process over an image exposure interval. Displacement, smear, and jitter covariances are computed from the solution to a Lyapunov differential equation. These covariances parameterize statistical image motion modulation transfer functions (MTFs), and they can be used to verify pointing and image motion MTF requirements. The method in the present paper extends a previous method to include smear, as well as displacement, and hence jitter. The approach in the present paper also leads, as a special case, to a more efficient method to compute the displacement covariance than the previous method. Numerical examples illustrate the proposed method.
T he imaging performance of an optical payload is potentially limited by image motion during an exposure. Image motion reduces the modulation transfer function (MTF) -the spatial frequency response -of the optical system, thus adversely affecting image quality [1]. The image motion is due to bus and optical payload pointing control system errors driven by various noise and disturbance sources. See, for example, [2] for a survey of noise and disturbance sources on spacecraft. Here, the "bus" refers to any kind of platform that carries an optical payload, including spacecraft, aircraft (planes, drones, balloons), ships, ground vehicles, robots, and even hand-held cameras. In this work, the optical system is a isoplanatic (shift-invariant) noncoherent (incoherent) imaging system.
Components of image motion over an exposure interval include displacement (shift), smear (linear), jitter (Gaussian blur), smile (quadratic), frown (cubic), exponential decay, tonal and multi-tonal motion, and fractional cycles. Tonal and multi-tonal motion may be included in jitter under certain conditions [3]. In previous work [4][5][6][7], “jitter” includes image smear motion. Smear and jitter are treated separately in [1] due to their different effects on image quality. In this work we consider displacement, smear, and jitter as defined in [1].
In this work, the model of the coupled bus and payload pointing control system is a continuous-time, linear time-invariant system. The output of the control system is a two-dimensional image motion on a focal plane. Since we assume the optical system is linear shift-invariant, image motion is related to the angular motion of an optical line of sight. Some (improper) requirements may define the output to be three-axis angular motion of a specified reference frame or two-axis angular motion of a vector direction (pointing). Any of these definitions can be accommodated. The image motion is obtained through a linear optical model (LOM), which is a matrix that maps structural and rigid-body modes to image motion. The image motion is used to compute the image motion MTF [4]. The total disturbance input is typically white noise or broadband noise plus spectral lines due to harmonic and subharmonic disturbances. Broadband noise can be modeled as the output of a shaping filter driven by zero mean white noise. We assume that the system has reached a statistical steady-state so that the output is stationary.
The covariance of the displacement, smear, and jitter and the mean smear vector (called pointing metrics in [1]) parameterize statistical image motion MTFs. The covariance matrices are computed from simulated pointing motion data, and it is possible to compute them from telemetry. The displacement, smear, and jitter covariances and the image motion MTFs can be used to verify pointing and image motion MTF requirements.
In [1], the covariances are computed from the power spectral response of a dynamic system to disturbances comprising spectral lines, which are due to various vibration sources. Since the power spectrum of the system response comprises spectral lines, the integral of the frequency-weighted power spectrum becomes a summation of terms, and there is no approximation in the integration. However, this approach is approximate and inefficient when the power spectral density is continuous, since numerical integration such as rectangular or trapezoidal integration would be used, and is inaccurate when the system has lightly-damped structural modes.
The contribution of this Note is a method to efficiently and accurately compute the displacement, smear, and jitter covariance for a combined bus and payload pointing control system driven by stationary white noise. Bayard [7] derived algorithms to compute displacement and jitter covariance, where the jitter motion includes smear motion. This work revises and extends the results in [7] to compute covariance matrices for displacement, smear, and jitter. The results of the present Note also lead, as a special case, to a more efficient method to compute the displacement covariance than the method in [7]. The algorithm derived in this work is provided in the publicly available Image Motion OTF and Pointing Performance Analysis Toolbox (IMOTF-PPA) [8], which runs in Matlab and Octave. The the algorithm is implemented in the function covss.m and demonstrated in the example ex_covss.m.
The present Note is structured as follows. In Section II, the system model is defined and expressions for displacement, smear, and jitter and their covariances are defined. In Section III, we define a Lyapunov differential equation (LDE) whose solution is the covariance for accuracy, displacement, and smear, from which the jitter covariance is computed.
The finite-horizon solution is obtained by rewriting the LDE in block state-space form, which is then solved by using a matrix exponential. This step provides a more efficient and robust solution using well-established
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