Display Modulation by Fourier Transform
A preferred method for determining the grating modulation of a rearprojection 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 chargedcoupleddevice (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):
(1)
This direct method for estimating grating modulation works well for displays that present a relatively wellbehaved and regular spatial pattern. The VESA FPDM Standard (Version 2.0) uses this concept to define and quantify "effective resolution" in Section 3037.^{3} If the display presents an image that is relatively noisy, accurate results are difficult to achieve with this method. In addition, while sinewave gratings have typically been used for cathoderaytube displays (CRTs) and analog imaging systems, squarewave gratings are often specified for directview matrix displays^{3} and matrix image sources for projection systems. If the intended squarewave spatial pattern departs from a 50% duty cycle or if it has a shape that is significantly different than a squarewave (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. Squarewave grating images are used. In this method, the 2D modulation spectrum is computed from a 2D image. The calculation is based on 2D Fast Fourier Transform (FFT) analysis implemented in Matlab.^{4} One could also use a 1D analysis performed on a highresolution slitscantype 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 L_{min} and L_{max} 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 components^{6}; i.e., its response is correlated with the spatialfrequency content of a presented image as determined by the HVS contrastsensitivity function. For displays where a squarewave grating test image significantly departs from a squarewave shape or a 50% duty cycle (as might occur in a microdisplaybased projection system), the Fourier content of the grating image will obviously differ from that of a nominal squarewave 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 spatialfrequency content and reflect any perceptual differences if they exceed the frequencydependent 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 L_{min} and L_{max} do not change.
Thomas G. Fiske is Principal Systems Engineer at Rockwell Collins Display Systems, 2701 Orchard Parkway, MS40, San Jose, CA 95134; 408/5324986, fax 4105, email: 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 onelineon/one lineoff grating image and a fivelineon/ fivelineoff grating image. Figure 2 shows the resulting luminance vs. distance plots; the data are averaged across the nonmodulated direction. The data are from a prototype threechip LCoSbased rearprojection display for avionics applications with addressable resolution of about 128 dots per inch (dpi).^{7} The screen is a 5mil bulk diffuser. The data were obtained by using a ProMetric Color 14211 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 oneline grating and 0.73 for the fiveline grating as calculated by using Eq. (1). The measurement setup 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
highspatialfrequency 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. G56592 from Edmund Optics.^{9}] that is similar in spatial frequency to the display grating of interest. We can make straylight 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 oneline image to establish luminance correction factors for the full image. If we assume that the correction factors are uniform
Fig. 1: Oneline (left) and fiveline (right) grating images for a rearprojection display.
Fig. 2: Luminance vs. distance for the oneline (left) and fiveline (right) grating images. The data is averaged across the nonmodulated 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 spatialfrequency regime and every lens and fstop 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 squarewave 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 fiveline 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 5line gratings). The values begin to depart at higher spatial frequencies (1 and 2line gratings) with the direct method overstating the normalized modulation by about 20% compared to the Fourier analysis method for a oneline 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 3037 from the VESA FPDM Standard^{3} 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,
