VISUAL PSYCHOPHYSICS

1. Mystery photograph.

2. Sine wave. The sine wave is any waveform, but to use a practical and specific example, we start with sounds. Pure tones are sinusoidal variations in air pressure (Y axis) with time (X axis). A sine wave is defined by cycle time (or, more commonly, its inverse, which is frequency), amplitude, and phase.

3. Combination of pure tones (A, B, and C) to make more complex sounds (A+B+C). This is an example of Fourier synthesis-the creation of complex waveforms from sine waves chosen for frequency, amplitude, and phase. Fourier analysis is the opposite-the determination of which sine waves are present in a complex waveform.

4. Further examples of Fourier techniques. Left: The appropriate combination of pure sine waves can be added to form a square wave. Middle: The appropriate combination of pure sine waves can also be added to form a triangle wave. Actually, the sine waves used for triangle and square waves are identical except for their phase relationships. For the square wave, each alternate sine wave is added so that peaks line up with troughs (dashed lines at left), while in the triangle wave, peaks line up with peaks and troughs with troughs (dashed lines in middle). Right: Formation of arbitrarily complex waveform from sine waves. For sound, this is an example of the fact that any arbitrary sound-a click, Pavoratti sneezing, a baby's cry-can be created by adding appropriately selected pure tones. This is the basis of music synthesizers.

5. Fourier transform. A convenient way to represent a temporal wave form (e.g., a complex sound) is to convert it to its component sine waves (i.e., pure tones), and this involves transforming the X axis from time to frequency. Shown are the pure tones making up the complex sound, and these are simply plotted at the lower right in terms of the amplitude for each frequency present. Also included in the Fourier transform (but not shown graphically) is the relative phase of each tone.

6. Middle C on the piano shown both in time (top) and frequency (bottom). The bottom is the Fourier transform of the top, and the top is the reverse Fourier transform of the bottom.

7. Example of modulation transfer functions (MTFs). This example actually compares the performance of two audio amplifiers. Each curve plots the relative amplification (amplitude of the output divided by the amplitude of the input) of each pure tone component of any sound. The better amplifier (blue) has a flatter MTF, meaning that it provides the same amplification for all tones in its range. The poorer amplifier (red) provides differential amplification, and this will distort the amplified sounds.

8. Fourier transforms and MTFs in action, simple version. Left column: Fourier transform of a simple sound to be amplified. Middle column: MTFs of two amplifiers (solid red and blue lines). Superimposed are the Fourier transforms of the input sounds. To determine the Fourier transform of the output, simply multiply each input frequency by the height of the MTF at that frequency to determine the output frequency. Right column: MTF of the output.

9. Fourier transforms and MTFs in action, more general version. the general rule is tha Fourier transform of the input multiplied by the MTF of the amplifier for each frequency yields the Fourier transform of the output. A: Fourier transform of sound to be amplified. B,C: MTFs of two amplifiers. The better one (B) has a consistent level of amplification for all frequencies in its range. The poorer amplifier (C) will distort the sound by boosting middle frequencies relative to higher and lower ones. D: Fourier transform of output of better amplifier. E: Fourier transform of output of poorer amplifier.

10. Fourier transforms and MTFs in action, yet again. Go clockwise, starting from the lower left with the input sound. Above this are the three pure tones comprising the input, and above these is the input's Fourier transform. When multiplied by the MTF of the amplifier, we get the Fourier transform of the output (top right). Under this are the three output tones represented and at the bottom is the reconstructed output sound. In other words, if you know the MTF of an amplifier, it is possible to completely describe how any input (e.g., lower left) will be reproduced (e.g., lower right).

11. Examples of amplifiers that are low pass, band pass, and high pass. The low pass amplifies only lower frequencies, the band pass amplifies only middle frequencies, and the high pass amplifies only higher frequencies.

12. The luminance profile for a sine wave grating. The profile is a sine wave with the same parameters noted in Figure 2, but the terminology is slightly different. Frequency is now a spatial function and it is a reflection of the number of cycles of the sine wave per degree of visual angle. Amplitude is now expressed as contrast.

13. Sine and square wave gratings with their luminance profiles below.

14. Higher and lower frequency sine wave gratings.

15. Higher and lower contrast sine wave gratings.

16. Addition of two sine wave gratings (top and middle) to make a more complex visual pattern (bottom). This is an example of Fourier synthesis and is precisely analogous to adding to tones to make a more complex sound.

17. Fourier transform of visual luminance profile. This is precisely analogous to Figure 5, except the wave form is spatial instead of temporal and the Fourier transform is to spatial frequency instead of temporal frequency.

18. How to measure an optical MTF for a lens. Compare the ratio of contrast in the image to that in the object for every spatial frequency (this ratio must be 0-1), and plot this for every spatial frequency. This provides the optical MTF, which is precisely analogous to the amplifier MTFs described above.

19. Example of simple optical MTF based on image/object contrast ratios at three spatial frequencies.

20. As with amplifiers, the Fourier transform of an object multiplied by the optical MTF of a lens produces the Fourier transform of the image. thus the optical MTF of a lens, or a human eye, fairly completely describes its ability to form images.

21. Examples of optical systems that are low pass, band pass, and high pass. this is analogous to Figure 11. However, it should be noted that actual optical lenses and natural eyes are never band pass or high pass, but they can be low pass. It is possible electronically to create the band pass or high pass effects while imaging objects, and examples are given below.

22. Effects of optical blur and diffusion. Blur is simple defocus and is the result if you try to view the world without your contact lenses. Diffusion is the result of looking through a foggy or dirty window. Shown are the normal MTF for a lens (thick, black line), the MTF after blur (green lines; less blur is the solid line and more blur is the dashed line), and the MTF after diffusion (red lines; less diffusion is the solid line and more diffusion is the dashed line). Blur results in a loss of contrast selectively for higher spatial frequencies. (Note that it creates a low pass MTF.) Diffusion results in an equal loss of contrast for all spatial frequencies.

23. Example of perceptual MTF. This figure shows a "grating" with spatial frequency increasing logarithmically on the X axis and with contrast decreasing logarithmically along the Y axis. The entire rectangle is filled with grating, but you can see only a gray patternless field when the contrast gets too low. Interestingly, the shape of the upper envelope is like a band pass MTF, meaning that we see middle spatial frequencies better than lower or higher ones. this perceptual MTF is also known as a contrast sensitivity function (CSF).

24. effects of blur and diffusion on the CSF (see Figure 22). Top: Normal CSF (solid green). This is not limited by the normal optical MTF (solid blue line). Middle: Blur. The CSF is now reduced at high frequencies because of the optics. Bottom: Diffusion. The CSF is now reduced at all frequencies except the very lowest because of the optics.

25. Mach band. The luminance profile (lower) shows a simple ramp between a darker region to a brighter one. The appearance of the image (upper) is a transition between dark and bright that is bounded by a darker bar (D) to the left and a brighter bar (B) to the right.

26. Explanation of Mach band based on Fourier analysis. From top to bottom, we have: the human CSF (percetptual MTF); the Fourier transform of the perceived image; and the luminance profile of the perceived image.

27. Ambiguous painting. In focus, the high frequencies dominate, and you see a figure looking out a window. Blurring eliminates the high frequencies, and the remaining low frequencies paint a different picture.

28. Low and high pass MTFs with an abrupt transition between frequencies that will be imaged and those that will not. Note that the transition is at the halfway point so that each image has the same range of frequencies. This MTF pair will be used in the following Figures.

29. Photograph of Dr. K. This is in focus and all here spatial frequencies are present.

30. Same photograph as Figure 29, but after passing through the high pass MTF shown in Figure 28. Notice how hard it is to recognize her when only her high frequencies are present. This, incidentally, is the same as Figure 1.

31. Same photograph as Figure 29, but after passing through the low pass MTF shown in Figure 28. Notice how easy it is to recognize her when only her low frequencies are present. Some of her sharp details are missing. Even though the amount of spatial frequency removed is the same as in Figure 30, this is much easier to perceive than Figure 30. Thus lower spatial frequencies are far more important to our visual perception than are higher ones.

32. Summary of Figures 29-31.

33. Another example of Figure 32.

34. Another example of Figure 32, showing that this applies to all objects and not just faces.

35. Limits of acuity measures vs CSFs to assess vision. The acuity measures (20/nn) are a single point on a CSF where it crosses the X axis. The whole CSF is a much better predictor of vision. For instance, the red CSF denotes poorer overall vision than does the blue one despite higher acuity (20/40 versus 20/160).

36. Cat versus human CSF.

37. CSFs of "malingerers" and "workers", showing that the "malingerers" really could not see well.