## Abstract

For an optical imaging system, the critical output is the image itself, and therefore, the quality of that image is of utmost importance. To estimate or predict the image quality (IQ), a simulation/model is typically created to yield an output image, given an imaging system and an object/scene. The IQ is typically graded based on the imaging system along with the scene. Developing an imaging simulation and creating input scenes to produce IQ results can be time-consuming, leading to a desire for a simple method to estimate the predicted IQ. This work develops a national image interpretability rating scale (NIIRS) IQ value based on simplifying assumptions for remote-sensing purposes. While the results are on the optimistic side, this back-of-the-envelope IQ estimation allows the process of developing a new imaging system to move forward more in parallel rather than in series, i.e., developing an imaging full simulation/model in parallel with designing/procuring hardware.

© 2019 Optical Society of America

## Corrections

Jason Mudge, "Back-of-the-envelope image quality estimation using the national image interpretability rating scale: erratum," Appl. Opt.**58**, 8839-8839 (2019)

https://www.osapublishing.org/ao/abstract.cfm?uri=ao-58-32-8839

## 1. INTRODUCTION

Imaging systems have been in existence for centuries and for a variety of purposes. The desire for various types of imaging systems gives rise to the need for an image quality (IQ) estimation or prediction of the particular imaging system. This naturally leads to adjusting the IQ by changing aspects of the imaging system for either improved IQ or to gain in cost reduction by reducing IQ. In a sense, the image produced should be correct, not too good or not too poor, for the particular application. Unfortunately, there is not a perfect method to judge, predict, or measure, for that matter, the IQ. However, there are some generally accepted metrics [1]. The IQ metric applied for this work is the national image interpretability rating scale (NIIRS), which appears to have gained the most attention [1,2] in remote sensing. For this type of sensing, a simplified system is provided and shown in Fig. 1.

There are several versions of the NIIRS equation, known as the general image quality equation (GIQE), in existence for the visible spectrum. In this work, GIQE version 3.0 is utilized due to its simpler and possibly more accurate [3] form and is

In place of a complicated image generating simulation, we propose a simplified technique to estimate the components in Eq. (1) and generate a design area for the desired/required IQ. With this, the hardware design can progress forward, and the initial vendor identification and procurement process of required hardware can begin, which typically has a long lead time. At the same time, a detailed simulation can be created to fine-tune the potential design at a later date [4]. In the following section, approximations are made to yield such an estimated IQ for an initial stab at an imaging system design from an IQ perspective.

## 2. IQ ESTIMATION

Estimation of the GIQE components is the first step in generating a back-of-the-envelope imaging system design. For this analysis, it is assumed that there is not a sharpening filter turned on, i.e., $H=1$ and $G=1$, but this could be added to the analysis should the desire be there. As stated, the application in the work is for remote sensing, leading to the approximation that the object is far away, i.e., $z\gg f$. For GSD, the value is

where $f$ is the imaging system’s effective focal length, $z$ is the distance to the object from the imaging system (see Fig. 1), and ${p}_{p}$ is the pixel pitch. Next, the signal-to-noise ratio (SNR) is calculated based on Refs. [2,5], and the result isFinally, the last component required in Eq. (1) is RER, and this is a bit trickier to estimate. Using the definition of the Fourier transform given in Goodman [6], it is assumed the optical transfer function (OTF) is a triangle function, which implies a square aperture in place of the more traditional circular aperture for mathematical ease and is diffraction-limited for this same ease. We are concerned with an approximate IQ estimate, so we leave out all other distortions and aberrations again (that is the job of a detailed simulation). The OTF is

where $\sigma $ is a one-dimensional special frequency (working in one spatial dimension, either ${\sigma}_{x}$- or ${\sigma}_{y}$-, simplifies the problem and is not limiting), and $\lambda $ is the optical wavelength (typically this is the average wavelength). It should also be noted that a square aperture and circular aperture have the same incoherent cutoff frequency of $\frac{1}{\lambda f/\#}$ when the length of the sides of the square equals the diameter of the circular aperture. However, the circular aperture drops down a shade below the square aperture OTF in between the zero frequency and the cutoff [6]. To generate the RER, the object being imaged is an edge or step function also known as the Heaviside function. The Fourier transform of the Heaviside function is where $F\{\xb7\}$ is the Fourier transform operator and $H(x)=\frac{1}{2}(1+\mathrm{sgn}(x))$. The $\mathrm{sgn}(x)$ is the sign function and is defined in Ref. [6] but should call out that $\mathrm{sgn}(0)=0$, which is equivalent to $H(0)=\frac{1}{2}$. Again, the Heaviside function represents an edge object in one dimension (effectively a slanted edge object in two dimensions) that, along with the OTF, determines the edge response. Multiplying the Fourier transformed object with the OTF in the frequency domain gives the Fourier transform of the image asBefore moving to applying the above formula with an example, there is some optimization that should be examined or at least thought through. For most of the optical parameters, it is clear how the adjustments affect the IQ. For example, increasing $D$ improves IQ. This increases the SNR and RER, yielding improved IQ with respect to NIIRS (although practically, this is to a point in that depth of focus can become an issue, and creating a diffraction-limited system with large diameter optics has its difficulties). On the other hand, the system focal length plays a role in the GSD, RER, and SNR. Increasing the focal length reduces the GSD and increases the RER, both of which are good for IQ, but it has the ill effect of reducing the SNR. These values can be played against each other to reach an optimum. To do this, the first derivative is taken with respect to the system focal length and set equal to zero

This equation is then solved for the optimum focal length. After some calculus and algebra, the optimum focal length is not analytically attainable because it is a transcendental equation in focal length (as is the case for a pixel pitch optimization). Nonetheless, in the following section Eq. (10) is plotted as a function of focal length (or $f/\#$), which shows a maximum numerically.## 3. APPLICATION

As is customary, a simple example to show the usefulness of Eq. (10) is provided, and it is well worth the effort in this case. Using values for a low-Earth orbit (LEO) nadir looking satellite for a visible wavelength gives the parameters as $z=800\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, $\lambda =600\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, ${p}_{p}=4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, $D=0.25$ m, ${r}_{2}=0.15$, ${r}_{1}=0.07$, $\mathit{QE}=0.6$, $t=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{msec}$, and $L=1.7\times {10}^{19}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\mathrm{ph}/\mathrm{s}}{{\mathrm{m}}^{2}\xb7\mathrm{sr}}$(a slightly overcast day on Earth). As for the focal length chosen, a set of reasonable focal lengths are spanned ($0.25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{m}\le f\le 8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{m}$). For these focal lengths explored, the largest lower limit of $x$ for a valid Eq. (7) is 1.88 μm, and the value $\frac{{p}_{p}}{2}$ is evaluated at 2 μm. This meets the constraint discussed above in the paragraph after Eq. (7), i.e*.*, $x$ is large enough such that Eq. (7) is valid.

Each of the components that goes into computing the NIIRS score is plotted in Figs. 2–4, and as is traditionally done, the values are plotted as a function of f-number ($f/\#$). The first plot is shown in Fig. 2 for the GSD. The RER is fundamentally bounded between zero and unity, and the plot in Fig. 3 abides by this fact.

The SNR is reduced as the f-number is increased (shown in Fig. 4) and is the last component for the GIQE. It should be recognized that the equation used from Ref. [5, see p. 84] has a small angle approximation (high f-numbers) imbedded within it. This implies the improvement of SNR as the f-number is reduced and is not to the extent shown in Fig. 4 or given in Eq. (3) at very low f-numbers.

Finally, Fig. 5 shows the NIIRS values for a string of $f/\#$’s. Notice that none of the components that make up NIIRS has a maximum or minimum, but the NIIRS IQ score does. For the optimum with respect to IQ purposes, the NIIRS score is 4.1 for an $f/15$ system. From Ref. [2], the ability to identify, detect, etc., is detailed out for various NIIRS scores. Given the 4.1 NIIRS value for this example, the performance can be evaluated based on Ref. [2], and an example of a NIIRS value greater than or equal to 4 and less that 5 is “Detect an open missile silo door.”

However, by looking at Eq. (10), we find another interesting value to plot the components of NIIRS as well as NIIRS itself against instead of the traditional f-number. The value is dimensionless and is $f/{p}_{p}$, which is equal to $\mathrm{GSD}/z$ (similar triangles), which is basically the system angular sampling and is a sole quantity (f-number is not because the function is not strictly a function of the ratio of focal length to aperture diameter). This is shown to be the case in Eq. (12) below,

## 4. DISCUSSION

Without question, this is an optimistic IQ predictor/estimator. It is for a square aperture, not the more typical circular aperture [6], and the result does not include wavefront aberrations, atmospheric turbulence, focus error, motion blur, and pixel blurring [4], to name a few distortions that comprise the total OTF (or RER). With regard to the SNR, there has been no consideration to transmission losses due to the atmosphere and optical throughput, which would find their way into a more detailed simulation.

Another popular method to help determine imaging performance is the Rayleigh criterion. This criterion for a square aperture is $\lambda f/\#$ (or $1.22\lambda f/\#$ for a circular aperture) when diffraction-limited. This value represents the minimum distance the imaging system can separate two points sources (provides a minimum angular resolution). This presumes sufficient sampling (minimum of two and preferably three samples across the two incoherently summed point spread function (PSF) peaks) and enough SNR. In some respects, the criterion can be thought of as a single value that represents the entire OTF. We have not discussed sampling or “$Q$” [4] for that matter, but they are important when it comes to IQ. In fact, increasing IQ can be traded for an amount of acceptable aliasing [4]. Continuing with $Q$, Eq. (8) can be rewritten in terms of $Q=\frac{\lambda f/\#}{{p}_{p}}$, giving,

Extending the example above, the NIIRS value can be plotted as a function of $Q$ and is done in Fig. 7. There is a nice discussion on the “Story of $Q$” given in Ref. [4], but in this work, we wanted to show how NIIRS varies with $Q$ with this back-of-the-envelope solution.The methodology used in the visible can be readily applied in the infrared. The analogous reference for the infrared is again written by Leachtenauer *et al.*, provided in Ref. [8], where the infrared GIQE is defined. There are some differences, since the “$a$” and “$b$” values along with other constants are not the same as they are in the visible 3.0 equation [8]. The GIQE in the infrared is more like the visible version 4.0 than the 3.0 and is

Lastly, only IQ is considered in this work, and the focal length optimum is also with respect to just IQ. But in a complete design the field-of-view (FoV) IQ also must be considered. Another important aspect not considered is size, weight, and power consumption, known as SWaP.

## 5. SUMMARY

A first-cut IQ design tool for an imaging system is provided as given in Eq. (10) to generate a general design area. The tool provides an optimistic NIIRS IQ score without the need for a complicated numerical simulation/model. This work provides an upper bound on the IQ, allowing a vendor’s search for hardware procurement and detailed simulation to progress more in parallel than in a series. Using this tool, an optimum focal length (or $f/\#$) or $f/{p}_{p}$ ratio can be determined numerically for IQ.

## Acknowledgment

I would like to thank many of the individuals who were encouraging over the many years of working in the area of optical sciences and other technical areas. These conversations were not necessarily related to this topic in particular but simply a seed. Without these intelligent individuals in my life such as Richard L. Kendrick of Raytheon Space and Airborne Systems, this work would not have been possible. Thanks to all in my life particularly Georgina Baca. Parts of this research were supported by Golden gate Light Optimization, LLC. In memory of Theodore Tarbell.

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