Display Modulation by Fourier Transform

A preferred method for determining the grating modulation of a rear-projection display using a grating image and Fourier analysis is presented. This method is insensitive to spatial image noise and is in better correspondence with the response of the human visual system than is the standard technique. This method is not limited to any particular display type and can be applied to any technology.

by Thomas G. Fiske and Louis D. Silverstein

THE Modulation Transfer Function (MTF) or grating modulation versus the spatial frequency of a display system is commonly used as a metric of the ability of the system to faithfully reproduce information as a function of spatial frequency.1,2 The data are often gathered by using a scanning slit device to record the spatial variation of luminance across a grating image. A grating image (also called a grille pattern) is an image with a repeating pattern (horizontal or vertical) of alternating black and white lines. Usually, a series of images are measured with a range of widths from 1 to 5 or more pixels. A properly calibrated charged-coupled-device (CCD) based imaging device can also be used to acquire the data.

The maximum and minimum values from such a luminance pattern are estimated, and the Michelson contrast or modulation is calculated using Eq. (1):


This direct method for estimating grating modulation works well for displays that present a relatively well-behaved and regular spatial pattern. The VESA FPDM Standard (Version 2.0) uses this concept to define and quantify "effective resolution" in Section 303-7.3 If the display presents an image that is relatively noisy, accurate results are difficult to achieve with this method. In addition, while sine-wave gratings have typically been used for cathode-ray-tube displays (CRTs) and analog imaging systems, square-wave gratings are often specified for direct-view matrix displays3 and matrix image sources for projection systems. If the intended square-wave spatial pattern departs from a 50% duty cycle or if it has a shape that is significantly different than a square-wave (an outcome that is highly likely for projection displays based on matrix image sources), this method will give results that either overestimate or underestimate the perceptually relevant modulation.

We propose a Fourier analysis method as a preferred alternative to the standard method. Square-wave grating images are used. In this method, the 2-D modulation spectrum is computed from a 2-D image. The calculation is based on 2-D Fast Fourier Transform (FFT) analysis implemented in Matlab.4 One could also use a 1-D analysis performed on a high-resolution slit-scan-type measurement.5 The amplitude of the fundamental sinusoidal component of the Fourier transform (suitably normalized) is reported as the modulation.

The Fourier analysis method has many advantages. A display with a noisy image makes the Lmin and Lmax values of the pattern difficult to determine. The Fourier analysis method is immune from such a difficulty as long as the noise is uncorrelated with and is in a different spatial frequency regime than the grating test pattern.

It is well established that the human visual system (HVS) analyzes spatial patterns by decomposing the patterns into their sinusoidal frequency components6i.e., its response is correlated with the spatial-frequency content of a presented image as determined by the HVS contrast-sensitivity function. For displays where a square-wave grating test image significantly departs from a square-wave shape or a 50% duty cycle (as might occur in a microdisplay-based projection system), the Fourier content of the grating image will obviously differ from that of a nominal square-wave grating. The HVS will respond to the differences and an observer will report a corresponding difference in perception. The Fourier method will accurately capture the changes in spatial-frequency content and reflect any perceptual differences if they exceed the frequency-dependent thresholds of the HVS. The standard method, on the other hand, will yield a modulation result that is insensitive to any perceptually significant changes in shape or duty cycle as long as Lmin and Lmax do not change.

Thomas G. Fiske is Principal Systems Engineer at Rockwell Collins Display Systems, 2701 Orchard Parkway, MS-40, San Jose, CA 95134; 408/532-4986, fax -4105, e-mail: tgfiske@rockwellcollins.com. Louis D. Silverstein is founder and Chief Scientist at VCD Sciences, Inc., Scottsdale, AZ.

Measurement of Modulation by Direct Method

Figure 1 shows views of a one-line-on/one- line-off grating image and a five-line-on/ five-line-off grating image. Figure 2 shows the resulting luminance vs. distance plots; the data are averaged across the non-modulated direction. The data are from a prototype three-chip LCoS-based rear-projection display for avionics applications with addressable resolution of about 128 dots per inch (dpi).7 The screen is a 5-mil bulk diffuser. The data were obtained by using a ProMetric Color 1421-1 measurement system from Radiant Imaging.8 The field of view is 0.573 in. in the

horizontal and 0.382 in. in the vertical directional, and for this particular display there are 22 sensor pixels per display pixel. The maximum and minimum luminance values are shown in Fig. 2, and the resulting modulation values are 0.55 for the one-line grating and 0.73 for the five-line grating as calculated by using Eq. (1). The measurement set-up is shown in Fig. 3.

As always, the effects of veiling glare on the measurement system should be accounted for in a measurement such as this.3 In this particular configuration of detector, lens, and measurement distance, the veiling glare reduces the modulation transfer factor for a

high-spatial-frequency grating by about 10%. This was determined by measuring the spatial luminance distribution of a uniformly backlit ronchi ruling [e.g., 50 line pairs per inch (lpi), part no. G56-592 from Edmund Optics.9] that is similar in spatial frequency to the display grating of interest. We can make stray-light corrections on the full grating image by making some simplifying assumptions. An image

of the full grating was acquired and compared to an image where all but one line was masked. We use the bright and dark parts of the one-line image to establish luminance correction factors for the full image. If we assume that the correction factors are uniform

Figure1(left)_tif Figure1(right)_tif

Fig. 1: One-line (left) and five-line (right) grating images for a rear-projection display.

p25_fig2a_tif p25_fig2b_tif

Fig. 2: Luminance vs. distance for the one-line (left) and five-line (right) grating images. The data is averaged across the non-modulated direction. The heavy red horizontal lines indicate the maximum and minimum luminance values.

across the image and depend only on the luminance value at any particular point, we can apply them in a fairly straightforward manner. However, one must come up with correction factors for every spatial-frequency regime and every lens and f-stop used for data acquisition. Because the effects of veiling glare do not bear on the main issues addressed in this paper, it will not be discussed further.

Modulation by Fourier Analysis

When using the Fourier analysis technique, one must take care to only include an integral number of cycles in the image data to be analyzed. If this is not done, the extra partial cycle will lead to spurious results in calculating the amplitude of the fundamental sinusoidal component of the grating image. If one wants to calculate a modulation transfer factor as opposed to modulation of the displayed grating, the resulting amplitude must be normalized by a factor of 4/π, which is the amplitude of the fundamental sinusoidal component of the square-wave input. When the Fourier technique is applied to the data in Fig. 2, the normalized modulation is 0.46 and 0.72 for the one- and five-line grating, respectively.

Figure 4 shows the normalized modulation data for gratings of up to five lines calculated with both techniques. The horizontal axis is in cycles per degree (cpd) calculated for a viewer at 25 in. Notice that the normalized modulation values coincide quite well at low spatial frequencies (3-, 4-, and 5-line gratings). The values begin to depart at higher spatial frequencies (1- and 2-line gratings) with the direct method overstating the normalized modulation by about 20% compared to the Fourier analysis method for a one-line grating. This difference is significant andthe Fourier result is preferred as being more perceptually correct.

The difference in modulation values between the methods at higher spatial frequencies is due to a couple of factors: the detailed waveform shape and the duty cycle of the waveform. In general, for waveforms with the same maximum and minimum values, the one with the more sinusoidal shape will have a smaller normalized modulation as computed by the Fourier method compared to that computed by the direct method. The modulation calculated via the direct method does not change when the shape of the waveform changes. The modulation calculated with the Fourier method does track the change in shape of the waveform.

We adapt the methodology of Section 303-7 from the VESA FPDM Standard3 and use the suggested threshold value of 0.5 for the normalized modulation to obtain an effective resolution for adequate text and graphics rendering. The first thing to note is that for the direct method, the modulation never falls below 0.5 and the effective resolution is the same as the addressable resolution; namely,


Fig. 3: Measurement configuration showing the ProMetric 1421 and a rear-projection display.


Fig. 4: Modulation for the grating images of Figs. 1 and 2. The blue dashed line is direct method modulation; the black solid line is Fourier method modulation. Hx refers to the number of lines in the grating image.

128 dpi or about 28 cpd at a viewing distance of 25 in. This leads one to conclude that this display meets the modulation threshold for rendering text and graphics. From the Fourier method data, however, we find that the modulation does fall below 0.5 at the highest addressable resolution and that the maximum effective resolution is only 110 dpi or 24 cpd at a viewing distance of 25 in. (using an algorithm adapted from the VESA document, where we linearly interpolate to find the effective resolution at the chosen modulation threshold). In this case, we see that the Fourier method, which is more perceptually meaningful, gives a substantially different result than the direct method. The second thing to notice is that if one uses the effective resolution calculation as strictly defined in the VESA document, the Fourier method data gives an effective maximum resolution of 103 dpi – which underestimates the previous value by about 6%. This can be attributed to calculating effective resolution by using modulation data plotted against grating pixel width as seen in Fig. 1 of Section 303-7 of the VESA document. We recommend that the procedure be modified so as to take into account a nominal viewing distance and that the modulation be plotted vs. cycles per visual degree as is done here.

Figure 5 shows a one-line grating image for a beaded projection screen. The optically relevant portion of the screen is a layer of small glass beads embedded in a black material. The structure of the screen is such that it provides for excellent ambient light rejection. The bead size and spacing are typically many times smaller than a display pixel, so the beads provide a random, high-frequency sampling of the image. The luminance data are averaged across the non-modulated direction to give luminance vs. position data in Fig. 6. We see that the beaded structure of the screen results in a waveform that is quite noisy. Even though the noise from the bead-to-bead spacing is beyond visible spatial frequencies at a typical viewing distance of 25 in., it does cause a problem in attempting to assign maximum and minimum luminance values to the grating image in order to calculate a modulation via the direct method. One can attempt to perform a moving average over small distances in order to reduce the noise without affecting the modulation. The problem comes in trying to determine how much averaging is enough. Table 1 shows the results of the direct modulation calculation as a function of moving-average window width as well as the result of the Fourier method. We see that the direct modulation method gives a wide range of results, depending on how much averaging is done. The Fourier method gives a result that is near the low end of the range of results from the direct method. Because the frequency of the spatial noise is much greater than the frequency of the grating, the contribution to the modulation due to each is easily distinguished in the Fourier method. In a case where some element of the optical system imparts enough noise to the image to make it difficult to determine the modulation by the direct method – whether or not the noise is visually perceptible and degrades the modulation – the Fourier method will give unambiguous results. This feature, as well as the uncertainty in perceptually significant modulation due to the shape of the waveform for this high-frequency grating pattern, makes the use of the direct method problematic. This case highlights the advantages of the Fourier method for obtaining reliable and unambiguous results for display modulation.


The Fourier method for calculating display modulation provides a robust, non-ambiguous and perceptually meaningful characterization of the spatial image quality of a display. The Fourier method is preferred over the conven-


Fig. 5: Highly magnified view of a one-line grating image on a beaded projection screen.


Fig. 6: Luminance vs. distance for the one-line grating image on a beaded projection screen. The smooth blue line is the 3/4-pixel-wide moving average.

Table 1: Direct modulation vs. smoothing level for the data in Fig. 6 and the modulation from the Fourier method










1/4 pixel




1/2 pixel




3/4 pixel




1 pixel








tional direct method of calculating grating modulation since it correctly handles spatial noise in the waveform and can be directly related to the spatial frequency sensitivity of the human visual system.


1T. G. Fiske and L. D. Silverstein, "Display Modulation by Fourier Transform: A Pre-ferred Method, J. Soc. Info. Display 14, No. 1, 101-105 (2005).

2K. Ichikawa, Y. Kodera, and H. Fujita, "MTF measurement method for medical displays by using a bar-pattern image," J. Soc. Info. Display 14, No. 10, 831-837. (2006).

3Flat Panel Display Measurements Standard, Version 2.0, (2001), VESA Sections 303-2, 303-7, and A101.

4Matlab®, Version 6.5, The Mathworks, Inc., http://www.mathworks.com/.

5A. M. Nuijs and J. J. L. Horikx, "Diffraction and scattering at anti-glare structures for display devices," Applied Optics 33, No. 18, 4058-4068. (1994).

6F. W. Campbell and J. G. Robson, "Application of Fourier Analysis to the Visibility of Gratings, J. Physiology 197, 551-556 (1968).

7M. H. Kalmanash, "Status of development of LCOS Projection Displays for F-22A, F/A-18E/F, and JSF Cockpits," Proc. SPIE 4362, 161-169 (2001).

8Radiant Imaging, Inc., http://www.radimg.com/.

9Edmund Optics, www.edmundoptics.com/. •