Mechanisms that signal hue variation are less sensitive to high temporal frequencies than mechanisms that signal luminance or brightness variation (e.g., King-Smith,
1975; King-Smith & Carden,
1976; Noorlander, Heuts, & Koenderink,
1981). This difference is apparent in temporal contrast-sensitivity functions (TCSFs) that show how sensitivity to sinusoidal flicker varies with frequency. Chromatic TCSFs obtained with equiluminant stimuli are generally low-pass in form and fall more steeply with increasing frequency than achromatic or luminance TCSFs, which are band-pass in form and extend to higher frequencies than chromatic TCSFs (e.g., de Lange,
1958; Kelly & van Norren,
1977; King-Smith & Carden,
1976; Metha & Mullen,
1996; Regan & Tyler,
1971; Smith, Bowen, & Pokorny,
1984; Sternheim, Stromeyer, & Khoo,
1979; Stockman, MacLeod, & DePriest,
1991; Tolhurst,
1977). TCSFs are often taken to indicate the attenuation characteristic of temporal filtering in the visual system and are modeled as a cascade of filters. The relative loss of high-frequency chromatic sensitivity is typically attributed to extra filtering stages or to stages with longer time constants, both of which increase the sluggishness of the chromatic system (e.g., Lee, Sun, & Zucchini,
2007; Yeh, Lee, & Kremers,
1995).
Petrova, Henning, and Stockman (
2013) and Stockman, Petrova, and Henning (
2014) refined the chromatic-filter model by using the distortion in contrast-modulated flicker to measure separately the attenuation characteristics of early and late filters in the chromatic pathway. They found that the early filter acts as a band-pass filter peaking at 10–15 Hz, while the late filter acts as a two-stage low-pass filter that begins to attenuate chromatic flicker above about 3 Hz. Much of the additional loss of high-frequency sensitivity in the chromatic system (when compared to the achromatic or luminance system) appears to be due to the late, low-pass filter. As part of their work, Stockman et al. (
2014) identified two nonlinear stages in the chromatic pathway: an early expansive nonlinearity, linked to the half-wave rectification that occurs when signals are segregated into ON and OFF pathways, and a central compressive nonlinearity that follows the late low-pass filter.
In a companion article, we recently reported and characterized a new color phenomenon (Stockman et al.,
2017) in which asymmetries in sawtooth-shaped, flickering L-cone- or M-cone-isolating waveforms produce shifts in the mean hue of the waveforms. Between about 4 and 13 Hz, slowly-on/rapidly-off or slowly-off/rapidly-on sawtooth waveforms that are modulated around a mean yellow-appearing chromaticity exhibit shifts in mean hue in the direction of the slowly changing part of the sawtooth: toward red for slowly-on L-cone and slowly-off M-cone waveforms, and toward green for slowly-off L-cone and slowly-on M-cone waveforms. Measurements of the dependence of the mean hue shifts on modulation depth and on the slope of the sawtooth revealed that the discriminability of the hue changes depended mainly on the second harmonic of the sawtooth. Consequently, in this article only the first and second harmonics of the sawtooth waveforms are used.
The first two harmonics of sawtooth waveforms are given by
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicodeTimes]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\begin{equation}\tag{1}y\left( t \right) = \sin \left( {2\pi ft} \right) + {1 \over 2}\sin \left( {2\pi 2ft + \theta } \right),\end{equation}
where
f is frequency (Hz),
t is time (s),
Display Formula\(\theta = 0^\circ \) for a slowly-off waveform, and
Display Formula\(\theta = 180^\circ \) (
π radians) for a slowly-on waveform. The slowly-off and slowly-on waveforms are shown in
Figure 1 as the blue and brown lines, respectively. For both waveforms, the second harmonic has half the amplitude of the first harmonic, so that the amplitude spectra of the two waveforms are identical. The difference between the waveforms lies in the phase of their second harmonics, which differ by 180°. In our experiments, we vary
Display Formula\(\theta \).
The main panel of
Figure 1 shows two cycles of five waveforms each with a period of 1 second: The dashed black line is the fundamental component alone (the sinusoidal first harmonic), and its phase is fixed at 0
Display Formula\(^\circ \). The other four waveforms (colored lines) are each composed of the first and second harmonics with amplitudes of 1 and 0.5, respectively, but each composite waveform has a different second-harmonic phase. The icons and labels on the right illustrate these composite waveforms and give the names we use to describe them: slowly-off (blue line), peaks-align (red line), slowly-on (brown line), and troughs-align (green line). Each waveform is generated by a different second-harmonic phase.
From
Equation 1 it can be seen that positive values of phase (
Display Formula\(\theta \)) slide the second harmonic to the left (i.e., advance it), while negative values slide it to the right (i.e., delay it). We shall call negative values of phase
phase delays and will often specify the phase delay of the second harmonic relative to the phase of the first harmonic. (This simply means finding a time when the first harmonic is in sine phase—crossing zero in the positive-going direction as in
Figure 1—calling that time 0, and noting the resulting phase
Display Formula\(\theta \) of the second harmonic, because the waveform can then be described as in
Equation 1.) Several features of these waveforms are worth noting: The slowly-off (blue) and slowly-on (brown) waveforms have the greatest asymmetry between their rising and falling sides, while the peaks-align (red) and troughs-align (green) waveforms have the greatest asymmetry between their excursions above and below the mean.
In this article, we made hue matches as a function of the phase of the second harmonic using waveforms with fundamental frequencies ranging from 0.63 to 13.33 Hz (and thus second-harmonic frequencies from 1.34 to 26.67 Hz). Subjectively, and in terms of the matching techniques that could be used, this frequency range split into two distinct regions: the low-frequency region from 0.63 to 2 Hz in which the hue variations produced by the second harmonic can be followed by the observer, and the region from 4 to 13.33 Hz in which observers report the hue variations as chromatic flicker superimposed on a steady light, the hue of which varies with second-harmonic phase.