STEP-STARE FPA IMAGER SYSTEM EFFECTIVE RESOLUTION
Image errors result due to spacecraft-induced quiescent and dynamic disturbances that cause the imaging instrument's Line of Sight (LOS) motion. These errors are characterized by the low frequency component called "drift" and the high frequency component called "Jitter". Drift is a consequence of motion between frame dwells. The drift errors can be accommodated by over scanning the frame sample area in an overlap region that ensures that the frame contains the intended image region. Jitter consists of motions that have predominant effects during the pixel exposure time. Jitter causes a blurring effect in the image and results in degradation of image resolution. MITRE's proposed enhancements in the technology and concept design of the imaging instrument are aimed at achieving a high resolution. In order to achieve the desired resolution, a technique for precise estimation for the jitter component of the LOS errors is needed. Thus our main focus is to determine the jitter component of the LOS error and to compute its effect on the image resolution; that is, to determine the imager's effective resolution.
B-2 DRIFT AND JITTER COMPONENTS
B-2.1 Evaluation of Drift and Jitter Components of LOS Motion
The total LOS motion has two components, namely, the drift and the jitter. The drift causes displacement of the scene whereas the jitter causes distortion of the scene. Let the total error in the LOS motion, at any time during the picture taking interval, T, be denoted by e(t). Then the mean value of LOS error in the scene represents the actual drift of the LOS. As the drift occurs due to low frequency disturbances, the drift component may be considered constant during the picture taking interval T. The error component, d(t), due to jitter effect is a time-dependent component as it occurs during the picture taking interval. The total jitter, D, is the RMS value of d(t) over the time T and its mean square value is of interest to us in computing its effect on the image quality.
The expressions for drift and jitter could be computed simply if e(t) were a known function, which is usually not the case. In general, e(t) is a stationary random process, and the quantities of interest are the expected values of mean square drift and of mean square jitter.
In order to compute these quantities, we proceed as follows:
* We Compute the auto correlation function of e(t):
* The Fourier Transform of and its inverse function are given by
where is the spectral density, also called the power spectral density, of the process e(t), and is a function of , where is the frequency in radians/sec.
* Setting in the above relation yields
which, from the fundamental definition of , gives
where f is the frequency in Hz.
Thus the area under the curve, vs. f, # gives the expected value of the mean square LOS error. The units of are those of mean square quantity, in this case e^(2), per unit of frequency, Hz. In physical sense, the plot of vs. is the spacecraft's Point Spread Distribution of the LOS to nadir and is the counterpart of the spacecraft's Point Spread Function (PSF), which is a plot of the spatial distribution of the LOS error.A diagrammatic representation of the spacecraft PSF and Point Spread Distribution are shown in figures B-1 (A Typical Spacecraft Line-of-Sight Error PSF) and B-2 (Typical Frequency Spectrum of LOS Point Spread Distribution).
The concept of the point spread distribution of the LOS error is of interest to us because it allows us to perform the analysis in the frequency domain. It may be noted that the Point Spread distribution ranges for values of f from - to + . The mean square spectral density of a stationary random process, e(t), is a real, even and nonnegative function of frequency, f. Let Power Spectral Density (PSD) equivalent of for values of f between 0 and + be denoted by . The equivalent expression for mean square of the LOS error, E^(2) , can be written as:
Mean Square Error:
We introduce here a weighting function for Jitter and denote it by Wd(C), where is a function of the picture exposure time T. The expected value of the mean square drift and jitter can now be computed by multiplying the area under the PSD curve by the appropriate weighting function. Thus,
Mean Square Drift:
and the mean square jitter can be expressed as:
Mean Square Jitter:
is the weighting function for jitter and [1- ] is the weighting function for Drift. Derivation of the expressions for the weighting functions is shown in Appendix C.
B-2.2 Crossover Frequency Between Drift and Jitter
We notice that the weighting functions for Jitter and Drift are a function of the parameter C which represents the total angular shift of the LOS motion during the picture taking time interval, T. A plot of the weighting functions Wd(C) and 1-Wd(C) vs. C is shown in Figure B-3 (Weighting Function for Jitter and Drift).
At the point of intersection of the two curves the value of the weighting function for jitter and drift are equal (= 0.5). The value of C corresponding to this point is 2.78. It may be noted that for values of C higher than 2.78, the values for the Jitter weighting function is considerably high whereas the Drift weighting function values are negligible. For values of C lower than 2.78, the reverse is the case. The value of C corresponding to the intersection point justifiably can be considered as the crossover point for Drift and Jitter. The frequency corresponding to this is called the crossover frequency. Thus we define
The crossover frequency, which is dependent on the picture taking interval, T, of the Imaging instrument, plays a significant role in separating the frequencies which contribute to Jitter from those that contribute primarily to drift errors of the LOS disturbance spectrum.
B-2.3 Typical LOS Disturbance Power Spectral density (PSD) Curves
Equations for straight line envelopes of typical LOS disturbance PSD curves may be written as a function of frequencies, f , fr, and fs as:
where f represents all frequencies in the spectrum below the rollover frequency , and are all the frequencies in the spectrum above . is the constant value of the Power Spectrum between f = 0 to f = . Setting the dimensionless frequency ratios as:
the equation for a typical spectrum curves can be written in terms of the dimensionless parameters b and X as:
Thus we can draw envelopes of disturbance, PSD, for the range of all spectrum frequencies normalized to the rollover frequency, fr, and for various values of b from b=0 to b=1. Plots of various typical PSD curves for b=0, b=0.2, b=0.5 and b=1, are shown in figure B-4 (Normalized LOS PSD Curves).
B-2.4 MODIFIED DRIFT AND JITTER WEIGHTING FUNCTIONS FOR TYPICAL PSD CURVES
Expressions for weighting functions, Wd(C) and 1-Wd(C), for Jitter and Drift respectively, were derived earlier in equations B-2.8 through B-2.10. By substituting the analytical expression for the typical PSD spectrum, B^(2) (f), equation B-2.15, in the integral for the mean square LOS Error, equation B-2.7, we can express the mean square error for each of the typical PSD curves in terms of the parameters b and C. A very insightful simplification results if we introduce a new normalized frequency parameter a = fr/fcross, where fcross is the crossover frequency between drift and jitter components of the error spectrum, as determined in section B-2.2. We define C1 at the crossover frequency pertinent for each typical spacecraft, i.e., C1=2*pi*fcross*T. We can now express the weighting functions for drift and Jitter as a function of the parameters a, b, and C1. The resulting expression for the modified weighting function, G1(a,b), for drift is given by:
Accordingly, the modified Jitter weighting function is given by 1-G1(a,b). A plot of the drift and jitter weighting functions for b=0 ,b=0.2, b=0.5 and b=1 for parameter, a ranging from 0.1 to 10.0 is shown in figures B-5 (Modified Drift Weighting Function: G1(a,b)) and B-6 (Modified Jitter Weighting Function [1-G1(a,b)]).
It can be seen from these plots that the value of jitter weighting function increases exponentially as the rollover frequency, Froll, of a specific PSD curve for LOS disturbance is closer to or greater than the crossover frequency, Fcross , whereas the drift weighting function shows exponential drop in value. The LOS disturbance PSD curve of a specific spacecraft has a flat plateau for frequencies up to the rollover frequency. For frequencies above the rollover frequency, the PSD curve slopes down as per the equation for the power spectrum. The spacecraft attitude control system effectively controls the low frequency contents of the disturbance spectrum and keeps the power spectrum at a constant B0^(2) power level. Above the rollover frequencies, the control system is no more effective and the power spectrum contributes primarily to the jitter content in the LOS error.
B-2.5 Evaluation of Drift and Jitter Components for Typical Spacecraft
The area under the PSD curves gives the total mean square value of the LOS error. That is, the total scene shift is given by
which, for values of spectrum frequencies, fs, greater than fr, the value of the integral for the typical PSD curves can be expressed as
Mean Square Error: (B-2.18)
The drift and jitter components can be computed by multiplying the mean square error value by the respective weighting functions, i.e.,
Mean Square Drift: (B-2.19)
Mean Square Jitter: (B-2.20)
B-3 IMAGE RESOLUTION
B-3.1 Algorithm to Compute Jitter Component of LOS Error
A step by step procedure to compute the Jitter component of the LOS error is summarized below:
Step 1: Given the exposure time, T, we use Eqn. (2.11) to compute Fcross: Fcross = 1/(2.3 T)
Step 2: From the PSD curve of the LOS spectra, obtain the following Characteristic data:
i) Roll-over Frequency, Fr
ii) B0^(2) in units of Arcsec^(2)/Hz.
iii) Slope of the section of the PSD curve from Fr to Fs
iv) Highest frequency of the PSD, Fs
Step 3: From the above, compute the dimensionless quantities a and b:
a = Fr / Fcross
b = Fr / Fs
Step 4: Compute the LOS mean square error, E^(2), using Eqn. (B-2.18)
E^(2) = Area under the PSD Curve
Step 5: Compute modified Weighting Function for Jitter and Drift using Plots of G1(a,b) and [1- G1(a,b)] shown in Figures B-5 and B-6, section B-2.4.
The Jitter weighting function, G1(a,b), and the Drift Weighting function, [1- G1(a,b)], can be extrapolated for the appropriate b value of the LOS power spectral density curve.
Step 6: From step 4 and 5, we compute the Jitter component of mean square error, Eqns. (B-2.19) and (B-2.20), section B-2.5.
Mean Square Drift:
Mean Square Jitter:
The RMS value for Jitter = D
B-3.2 Effect of Jitter on Image Resolution
We assume that the disturbance Power Spectrum, B^(2) (f), is known for the particular spacecraft and that the crossover frequency, Fcross, between drift and jitter of the spectrum have been computed following the procedure outlined in the previous section.
Step 1: INR System's Point Spread Function , PSF(INR):
* Assuming a Gaussian distribution of the LOS error e(t), we denote the Probability Distribution Function, PDF, of e(t) as GD(e).
* If x(t) is the actual spatial position of the Imager's LOS at time, t, then the real position without LOS error would be (x(t)-e(t)).
* The effective Point Spread Function, PSF(INR), of the Image Navigation and Registration (INR) system is obtained from the convolution of the imager's PSF, PSF(INST), with GD(e), i.e.,
Step 2: INR System Resolution:
* The most commonly used measure of image resolution is the Full-Width-Half-Maximum (FWHM) of the PSF. Assuming a Gaussian PDF of the Jitter component of LOS error, the FWHM value is 2.35 times the standard deviation 2.35 D
* Then the RMS value of the INR system's effective Resolution is given by the RSS of the FWHM values for the imaging instrument, itself, and the spacecraft:
The computational techniques discussed in the foregoing were applied to assumed data for HIRIS, a NASA project, and MITRE's concepts as shown in examples 1 and 2 below:
B-4.1 Example - 1. The High Resolution Imaging Spectrometer (HIRIS)
Following data was assumed for the HIRIS instrument and the spacecraft disturbance spectrum:
T = 0.1 sec
Fr = 15 Hz.
B0^(2) = 1.0 Arcsec^(2) / Hz.
Fs = 1000 Hz.
Following step 1 through 6 of section B-3.1, we compute the RMS value of the Jitter, D:
Fcross = 1/(2.3 T) = 4.34 Hz.
a = Fr/Fcross = 3.46
b = Fr/Fs = approximately zero.
Mean Square Error: = 23.56 Arcsec^(2)
Using the figures in section 2.4, we get the values of the weighting functions for drift and jitter as follows:
For log(a) = log(3.46) = 0.54, and using the curve, b=0, the values obtained from the plots are:
Drift Weighting Function: G1(a,b) = 0.18
Jitter Weighting Function: [1-G1(a,b)] = 0.82
Mean Square Drift: = 4.24 Arcsec^(2)
Mean Square Jitter: = 19.32 Arcsec^(2)
RMS value of Jitter: D = 4.395 Arcsec = 2.13 Microrads.
HIRIS INR System Resolution:
We make the assumption that the HIRIS instrument's FWHM = 10 microrads.
Then using Eqn. (B-3.2) of section B-3.2, we get = 11.18 microrads
B-4.2 Example - 2. Comparison with MITRE's Concept
Using the same disturbance characteristics as used in example-1, we compute the INR system resolution.
T = 0.0005 sec
Fr = 15 Hz.
B0^(2) = 1.0 Arcsec^(2) / Hz.
Fs = 1000 Hz.
Following steps 1 through 6 of section B-3.1, we compute the RMS value of the Jitter, D:
Fcross = 1/(2.3 T) = 869.56 Hz.
a = Fr/Fcross = 0.01725
b = Fr/Fs 0
Mean Square Error: = 23.56 Arcsec^(2)
Using Figures B-5 and B-6 of section B-2.4, we get the values of the weighting functions for drift and jitter as follows:
For log(a) = log(0.01725) = -1.763, and using the curve b=0, the values obtained from the plots are:
Drift Weighting Function: G1(a,b) 0.9
Jitter Weighting Function: [1-G1(a,b)] 0.1
Mean Square Drift: = 21.2 Arcsec^(2)
Mean Square Jitter: = 2.356 Arcsec^(2)
RMS value of Jitter: D = 1.535 Arcsec = 0.744 Microrads.
MITRE's INR System Resolution:
We make the assumption that the instrument's FWHM = 10 microrads, (same as for HIRIS instrument in Example 1). Then using equation B-3.2 of section B-3.2, we get
= 10.15 microrads
Thus in using a staring sensor the instrument's FWHM is not modified significantly by the effect of spacecraft jitter.
It may be noted that the data used in the above two examples is only for illustrating the technique. The procedure can be used easily to validate the INR system performance once the real instrument's data and the real platform disturbance spectrum are available.