Perception Lecture Notes: Spatial Frequency Channels

Professor Michael Landy

What you should know about this lecture

Sine wave gratings

When we first discussed linear systems theory, it was in the context of audition where we describe an auditory signal as a single value, pressure, as a function of a single variable, time. Linear systems theory has also been applied extensively to vision, but there the stimulus is substantially more complicated. At a minimum, you talk about an image (i.e., a picture, a retinal image, a neural image) which is a function of the two spatial dimensions x and y. But, if you are interested in temporal sensitivity as well (e.g., to understand visual motion), then you have a signal that is a function of three variables: x, y, and t (i.e., a movie, or sequence of images). In audition, the basic stimulus used in linear systems theory is the sine wave. The analogous stimulus for vision is the sine wave grating. Such gratings can vary in spatial frequency (measured in cycles/degree, for a retinal image), orientation, phase and contrast.
Low and High Spatial Frequency Gratings

Low and high spatial frequency sine wave gratings

Contrast for sine wave gratings is usually defined as Michelson contrast for which the formula is (Imax-Imin)/(Imax+Imin)  or  (Imax-Imin)/(2 Imean). This is a number that ranges from zero (the bright and dark bars have the same intensity as the mid-gray, in other words the grating is invisible) to one (the bright bars are twice the intensity of the mean and the dark bars are black). Note that this definition is the same as the definition we've used previously, Weber contrast or  ΔI / I, where ΔI  means the increment of the bright bar above the mean (Imax-Imean) and I stand for the mean intensity (I=Imean).
Contrast=1
Contrast = .5
Contrast = 0

Gratings with contrasts of 1 (top), .5 (middle) and 0 (bottom)

The characterization of a system in linear systems theory is the Modulation Transfer Function (MTF), i.e., the degree to which different frequencies are amplified or attenuated by the system. The behavioral analogy to the MTF is the contrast sensitivity function (CSF) which describes how sensitive an observer is to sine wave gratings as a function of their spatial frequency. This is measured using a contrast detection experiment wherein one determines the minimum contrast required to detect sine wave gratings of various spatial frequencies. As usual, sensitivity is defined as 1/(threshold contrast) (so if threshold is low, sensitivity is high).
Sweep grating

Sweep grating

The figure above shows a pattern that increases in spatial frequency from left to right (the bars get narrower), and decreases in contrast from bottom to top (the bars get fainter). By tracing out the boundary between visible and invisible you can make out the curved shape of your CSF. The typical results of such a measurement follow:

contrast sensitivity function

Contrast sensitivity function

The typical CSF is bandpass in nature. That is, you are most sensitive for an intermediate range of spatial frequencies (around 4-6 cycles/degree), and less sensitive to spatial frequencies both lower and higher than this, much like the audiogram. The highest spatial frequency you can see (the high frequency cutoff) determines your spatial acuity, i.e., the finest spatial patterns you can see. This acuity limit typically worsens with age.

Multiple spatial frequency channels

The CSF is typically not thought of as the MTF of a single kind of neuron, but rather an envelope of sensitivity over several underlying mechanisms, each corresponding to neurons with differing preferred spatial frequencies (i.e., with different sizes of receptive field; larger = lower spatial frequency preference). A graph illustrating this follows:
spatial frequency channels
In the figure above, four spatial frequency channels are illustrated, and the notion is that the CSF represents the sensitivity pooled over those underlying channels, i.e., sensitivity is primarily determined by whatever channel (or set of neurons) is most sensitive to the stimulus. Now, what does this parsing of the stimulus into different frequency bands do for the observer?
filtered images

Filtering by spatial frequency channels - the neural image

As you can see, the low frequency filters provide information about large objects, shadows, and other smooth, gradual changes in intensity across the image. The higher spatial frequency filters emphasize progressively finer details.

What evidence is there for the existence of multiple spatial frequency channels? Well, first of all, there is the physiological evidence. In V1 and beyond, for each location in the visual field there are neurons varying in preferred spatial frequency, orientation, direction of motion, and so on. But, there is behavioral evidence as well. First, consider the effects of adapting to a particular spatial frequency. One begins by measuring the CSF as illustrated in a figure above. Then, you have the observer stare at a particular sine wave grating (e.g., 8 cycles/degree) for an extended period of time. The visual system adapts to that pattern, and any neurons or mechanisms that were sensitive to that pattern become desensitized temporarily. If one re-measures the CSF while in that adapted state, the results are as follows:
post-adaptation contrast sensitivity function
The dotted curve represents the post-adaptation CSF and, as you can see, sensitivity is reduced, but only for gratings with spatial frequencies near that of the adapting grating. The idea is that the spatial frequency channels sensitive to the adapting grating now have reduced sensitivity due to all that stimulation, but those with spatial frequency preferences distant from the adapter remain unaffected. Spatial frequency adaptation not only affects threshold, but also affects the appearance of supra-threshold gratings. After adaptation to 10 cycles/degree  (i.e., even narrower bars), an 8 cycle/degree grating will appear to have an even lower spatial frequency (even wider bars). Likewise, after adapting to 6 cycles/degree (i.e., wider bars) an 8 cycle/degree grating will appear to have an even higher spatial frequency (even narrower bars). The explanation is illustrated here:
size aftereffect explanation
Explanation of the size aftereffect

The idea is that adaptation to the higher spatial frequency desensitizes the higher spatial frequency channels (those that "see" the higher frequency grating). Then, when you display the 8 cycle/degree grating, the center of the response profile shifts to lower frequencies (bottom-right graph above). When you adapt to a lower spatial frequency grating, you see the opposite shift in the response (upper-right graph). Similar shifts happen with orientation and direction of motion, indicating that there are channels tuned for various orientaitions and directions of motion.

There are two other common methods of demonstrating the existence of multiple spatial frequency channels psychophysically. The first is called summation. There, the idea is that if you ask an observer to detect a combination of two grating (literally added together on the screen), the sensitivity is much higher if the two gratings are close in spatial frequency (so that they are detected by the same channel) than when they are far different in spatial frequency (so that they are detected by separate channels). A third psychophysical paradigm is called masking. In this type of experiment, the observer is asked to detect one test grating (with frequency f1) in the presence of another masking grating (with frequency f2). The masking grating is always present, and the question is to what degree does it mask the test grating, making it harder to detect. It turns out that the results lead to the same model. When the masking grating is similar in spatial frequency to the test grating, masking is strong, and when it is dissimilar, masking is weak. In other words, one grating masks another only to the extent that both are detected by the same channel.