Image Quality Criteria –Which Metric Should I use? Optical Sciences 521 Jed J. Hancock Introduction

Image Quality Criteria –Which Metric Should I Use?
Optical Sciences 521

Jed J. Hancock

Introduction
Image quality criteria define the expected imaging performance of an optical system. It may also be described as the set of requirements that characterizes how well the imaging system works. It is imperative to have image quality criteria when designing an optical imaging system, this is how one knows when to be done designing and begin building the system. The image quality criteria are usually defined by a set of image quality metrics such as wave front error (WFE), point spread function (PSF), and modulation transfer function (MTF).
During the design of the optical system the metrics chosen to define the image quality are usually modeled. During this modeling process an error analysis is performed in order to quantify the system performance as a function of planned and unplanned errors. Once the system is built, those metrics also need to be tested to verify that the imaging system will be able to meet its objectives and that the engineering practices used to design the system are valid and can be used again with confidence. Selection of metrics that can be accurately modeled and tested (on schedule and within cost) is important to the success of the imaging program.
The optical engineer must understand how the different physical parts and functions of an imaging system design will influence the image quality metrics. Ultimately, the image quality metrics used to quantify the performance are affected by almost all aspects of engineering disciplines required to build an optical system (electrical, mechanical, thermal, contamination, etc...) For example, material selection of the metering rods in a traditional Cassegrain telescope determines the primary-secondary mirror separation as a function of temperature. This may significantly impact the telescope WFE. Communicating engineering requirements that flow down from the system image quality metrics to other engineers is also critical to the success of the program.
Understanding the image quality of an optical system can be a complicated process. Imaging systems are built for almost any number of applications. Image quality criteria should be selected that is appropriate for the application. Also, one metric is usually not enough of a descriptor to guarantee that the system will be designed appropriately. It is not uncommon for image quality criteria to be explained using multiple metrics. Selecting the appropriate image quality metrics for a particular application is an important first task.

Image Quality Metrics
There are many kinds of image quality metrics. They can be categorized into two distinct groups, geometric and diffraction. Some metrics are both geometric and diffraction based, these metrics can be especially useful for setting the image quality criteria of certain imaging applications.
Geometric
Geometric image quality metrics can be computed or solved using simple ray tracing through an optical system by following the practices of geometric optics. The geometric optics methods are the most valid methods for applications where the expected wavefront error is greater than 1 wave. Geometric methods are usually very easy to compute with modern computers and especially with the use of ray trace programs.
One of the most common geometric image quality metrics is spot size. Geometric spot size is a useful metric for systems investigating the image of points in different wavelengths and from different field locations as illustrated in Figure 1. The geometric spot size is estimated by tracing rays though the optical system and plotting their coordinates at the image plane. The spot size is usually quantified by the geometric and/or RMS radius.

Figure 1. A plot of spot diagrams for 3 different field points.

Spot diagrams can be used to compute the ensquared or encircled energy. Ensquared energy is used as a metric because it enables a geometric estimate of how many of the geometric rays of an imaged spot will land inside a square that represents a pixel. Once again, in the geometric regime, this is usually a good metric for systems with WFE’s of greater than 1 wave. Figure 2 illustrates a simulated spot with a square overlayed.

Figure 2. The ensquared energy calculated with geometric methods.

Perhaps one of the most widely used geometric methods for measuring image quality is the WFE. The WFE is the optical path difference measured from the reference wavefront in the exit pupil of the imaging system. As can be seen in Figure 3, the reference wave is perfectly spherical and the aberrated wave is not. The optical path difference is shown on the figure by ΔW(x,y) where x and y are the exit pupil coordinates. This is a geometric measurement that can be used for imaging systems that expect less than 1 wave of error. Some advantages of using the WFE metric is that the types of system aberrations can be learned by studying the wave fan plots, lens manufacturing tolerances such as surface figure can be converted to WFE using rules of thumb, and finally the WFE can be related to other metrics, even diffraction metrics such as the Strehl ratio. It is often required to convert between image quality metrics using rules of thumb rather than detailed models.

Figure 3. An illustration of the wave front error optical path.

The modulation transfer function MTF is both a geometric and diffraction metric for image quality. The MTF is the modulus of the Fourier Transform of the point spread function (PSF) and the complex auto-correlation of the complex pupil function. Therefore the diffraction effects from the imaging system are shown in the MTF. The MTF is plotted as modulation (contrast) versus spatial frequency. This metric is most useful for application intended to image extended scenes. It gives you the ability to understand how other aspects of the imaging system like the detector will affect image quality.
Figure 4 shows the utility of the MTF by plotting the optics MTF and the detector MTF on the same graph. The optics imaging ability is cutoff by the system F# and wavelength. This particular curve shows that the pixel is small enough that the detector cutoff occurs at a larger spatial frequency than the optics. The blue line shows a diffraction limited optics MTF and the red line shows how aberrations will affect the contrast performance at certain spatial frequencies. To obtain a system MTF the pixel curve is multiplied by the optics curve point by point. To obtain even more accurate modeling of the imaging application the MTF of the atmosphere, jitter environment, and other imperfections can be included in the MTF plot and multiplied to obtain a system estimate including all the prevalent errors.

Figure 4. The modulation transfer function.

The PSF of the optical system describes the effects of diffraction and all aberrations. A picture of a typical PSF is shown in Figure 5. The width of the central core for a diffraction limited system is a function of the system F# and wavelength. It should be noted that ensquared energy or encircled energy can also be a computed using the PSF when systems have a WFE less than about ¼ of a wave. Therefore, the ensquared energy can be both a diffraction and geometric image quality metric.

Figure 5. A 3D and 2D view of the point spread function.

The Strehl ratio is a common way of measuring image quality using the PSF. The Strehl ratio is computed by taking the ratio of the height of the central core of the actual system PSF with respect to the central core of an aberration free PSF. Figure 6 illustrates this ratio.

Figure 6. The Strehl ratio.

The RMS WFE of the system can be related to the Strehl ratio by the following relationship,

.

The W_rms is RMS WFE of the system in units of λ. This is a convenient way to relate the WFE to the PSF or ensquared energy metric.

In summary, Table 1 lists the image quality metrics that have been discussed. MTF and ensquared energy are metrics that are both geometric and diffraction based. The RMS WFE, while best used when the errors are expected to be less than 1 wave is a geometric image quality metric. A spot diagram is a geometric method used to describe the systems ability to image a point when the expected system errors are greater than 1 wave. The PSF is a diffraction metric used to describe the systems ability to image a point when the expected errors are less than 1 wave and more ideally closer to ¼ of a wave.
Table 1. A comparison of the geometric and diffraction image quality metrics.

Geometric	Diffraction
MTF	MTF
% Ensquared energy	% Ensquared energy
RMS WFE	---
Spot Diagram	PSF

Conclusion –Select Wisely
In conclusion, a simple example is given to express the need to carefully select the image quality metric or metrics used to establish the image quality criteria of an imaging system. Figure 7 shows the MTF for several aberration free optical systems. However, each optical system has a varying size of central obscuration. It is noticed in the plot that the MTF drastically decreases as the obscuration size grows. The MTF analysis clearly shows this phenomenon. Because of the obscuration, the central core of the PSF gets smaller, but more of the energy in the PSF is diffracted outside of the central core causing a blurred or less sharp image. In other words there is a reduction of contrast in the mid spatial frequency range of the images produced by this system.
The image quality of a telescope system intended for imaging extended scenes is affected by the size of the central obscuration. If WFE was the only metric chosen to establish the image quality criteria of the system, the obscuration affect could have been overlooked resulting in poorer than expected imagery. For a case such as this, the WFE metric might be used to set the optical element, mechanical, thermal, and other pertinent tolerancing requirements that influence the image quality. However, diffraction metrics such as the MTF or PSF must also be used to understand the affects of the central obscuration to the systems image quality.

Figure 7. MTF degradation as a function of obscuration size.

References
Lightsey, Paul A., “Image for large segmented space telescopes”, Proceedings of SPIE, Vol 4850, 2003.
Grievenkamp, John, “Field Guide to Geometric Optics”, SPIE Vol. FG01.
ZEMAX Lens Design Manual
Burge, Jim, “Opti 521 course notes”, University of Arizona.