Spurious resolution

Lens resolution is often expressed in terms of line pairs per millimeter [1]. The criterion used to determine whether the lines of a test chart are resolved can involve an objective measurement of contrast in the recorded image. Another criterion is subjective and involves an observer who visually distinguishes darker and brighter lines in the image. Either way, the researcher should be aware of the possibility of spurious resolution. Spurious resolution can occur with a microscopic subjects which are blurred due to lens aberrations or inaccurate focusing, but also with defocused macroscopic subjects.

Figure 1 shows a little figure in front of a test chart, photographed with a 100-mm macro lens used on a 35-mm camera. The scene is captured at four different lens apertures from f/22, via f/11 and f/8, up to f/5.6. The focus is on the eyes of the model, and the test chart in the background is progressively blurred as the aperture increases. The background in the f/22 image is only mildly blurred and reveals a pattern of horizontal bars on the left side with alternating brightness, complemented with a smaller pattern of just four vertical black bars at the bottom right corner. The upper right-hand side consists of plain text. At f/11 there is more background blur, reducing the contrast between the dark and the bright bars. The text is still legible. With a little imagination the writing is still readable at f/8, but the line pattern has degenerated into a nearly uniform gray blur. At f/5.6 one expects the background blur to increase even further. Indeed, the text is now blurred beyond recognition. Surprisingly, however, there is an apparent restoration of the line pattern. The test chart creates the impression of being resolved. A false impression, which is known as spurious resolution.

Occurrence of spurious resolution
Figure 1. A willing model in front of a test chart, photographed at four lens apertures. Spurious resolution occurs at f/5.6.

Upon closer examination of the f/5.6 photograph, it is not just the reemergence of the line pattern that is curious. The pattern is also subject to phase reversal. Dark bars have become the brighter ones, and vice versa. This is clearly seen at the rear tip of the bindle stick. Although mysterious at first sight, the occurrence of spurious resolution is readily explained by examining the influence of a uniform blur disk on the light intensity distribution. Figure 2 visualizes what happens when a pattern of four bars is defocused via a mathematical operation known as convolution. The width of a single bar is W, and the defocusing is examined for blur disks with a diameter of D = 0 (perfect focus), 1.4W, 2.0W, and 2W. In all cases, each point of the original target is spread out over a width D. The horizontal axis uses a dimensionless scale of distance divided by W, and the horizontal axis uses a dimensionless scale of intensity divided by the maximum value for the D = 0 scenario.

For clarity, the figure only shows a one-dimensional cross section and uses different colors to discriminate between the intensities of the individual bars. The bars broaden with an increasing blur disk diameter D. The graphs at the right show the total light intensity, which is simply the sum of the four separate curves at the left. The reduction in contrast at D = 1.4W, the uniform blur intensity at D = 2W, the reappearance of a line pattern at D = 2.8W and its phase reversal, all observations of the photographs in Fig. 1 are reproduced. That includes the reduction of four bars to three spuriously resolved bars at D = 2.8W, which is in agreement with the bottom right-hand side pattern in Fig. 1.

Explanation of spurious resolution
Figure 2. The effect of defocusing on a periodic structure of four bars. Left: Individual bars are discriminated by the use of colors. Right: Total (summed) intensity. The quantity D is the diameter of the blur disk (point spread function).

Notice that the background chart in Fig. 1 has four black bars against a white background, whereas Fig. 2 illustrates the phenomenon for four bright bars against a dark background. The explanation is the same. Also notice that the values of D are inversely proportional to the f-number. The uniform gray blur of f/8 requires a blur disk diameter of twice the bar width, and the values of 1.4W and 2.8W simply follow from the aperture values f/11 and f/5.6.

Spurious resolution is not restricted to line patterns, but can occur with any periodic structure. The example in this article uses a macroscopic object subject to background blur, but spurious resolution can equally occur with a microscopic line pattern which is blurred due to lens aberrations. The reduction of N lines on a test chart to N–1 lines in an image is a strong, but not infallible [2] indicator of spurious resolution and a good reason to use charts with a few lines only. Few people will attempt to count the lines in the left-hand side of the photographs in Fig. 1, but a reduction from four to three lines is easily noticed. Phase reversal is also a sign of spurious resolution, but like line counting it is not infallible. If a periodic pattern is progressively blurred, spurious resolution manifests itself repeatedly, alternatingly with and without phase reversal, and with an ever decreasing contrast.

Researchers examining the resolution of photographic lenses, human eyes [3], or whatever imaging system, should be aware of the possibility of spurious resolution. Unrecognized spurious resolution leads to a considerable overestimate of lens resolution or visual acuity. In subjective evaluations the traditional line pattern may be dropped altogether in favor of an aperiodic test chart, such as plain text at various font sizes.

References

[1] Norman Koren, http://www.imatest.com/docs/sharpness.html.
[2] R. N. Hotchkiss, F. E. Washer, and F. W. Rosberry, “Spurious resolution of photographic lenses,” J. Opt. Soc. Am. 41, 600–603 (1951).
[3] M. Bach, S. Waltenspiel, and A. Schilw├Ąchter, “Detection of defocused gratings — Spurious resolution, a pitfall in the determination of visual acuity based on preferential looking or VEP,” in J. J. Kulikowski, C. M. Dickinson, and I. J. Murray (eds), Seeing contour and colour, Pergamon press, 562–565 (1989).