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Method to Compute Pointing Displacement, Smear, and Jitter Covariances for Optical Payloads

Peter Seiler, Mark E. Pittelkau, Felix Biertümpfel

TL;DR

Addresses how to quantify image motion effects in optical payloads during exposure through displacement, smear, and jitter covariances. The approach models pointing as a stationary LTI process driven by white noise and uses an augmented Lyapunov differential equation, solved efficiently by a matrix exponential, to obtain covariances. The work generalizes Bayard’s displacement covariance to include smear and jitter, offers a smaller-dimension variant for displacement, and demonstrates the method on first-order, MIMO, and satellite examples. The resulting covariances parameterize the image motion MTF and support verification of pointing requirements for space missions and other high-precision imaging systems.

Abstract

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.

Method to Compute Pointing Displacement, Smear, and Jitter Covariances for Optical Payloads

TL;DR

Addresses how to quantify image motion effects in optical payloads during exposure through displacement, smear, and jitter covariances. The approach models pointing as a stationary LTI process driven by white noise and uses an augmented Lyapunov differential equation, solved efficiently by a matrix exponential, to obtain covariances. The work generalizes Bayard’s displacement covariance to include smear and jitter, offers a smaller-dimension variant for displacement, and demonstrates the method on first-order, MIMO, and satellite examples. The resulting covariances parameterize the image motion MTF and support verification of pointing requirements for space missions and other high-precision imaging systems.

Abstract

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.
Paper Structure (15 sections, 46 equations, 5 figures, 2 tables)

This paper contains 15 sections, 46 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Model
  4. Displacement, Smear, and Jitter
  5. Double Integrator Form
  6. Pointing Metrics (Covariances)
  7. Computation of the Covariances
  8. Augmented Lyapunov Differential Equation
  9. Block State Space Form
  10. Matrix Exponential Solution
  11. Numerical Examples
  12. First-Order Pointing Process
  13. Multi-Input Multi-Output Example
  14. Satellite Example
  15. Conclusion

Figures (5)

  • Figure 1: Image motion (black) decomposed into displacement , smear , and jitter.
  • Figure 2: Normalized pointing covariances for a first-order process as a function of non-dimensional exposure time $\left|aT\right|$.
  • Figure 3: Covariance analysis interconnection
  • Figure 4: Pointing covariances for increasing exposure times about x-axis (left) and z-axis (right).
  • Figure 5: Comparison of input sensitivity functions.