Whoo boy, where to start? You have at least three separate issues here:

1. The *human* perception of brightness is a nonlinear function of measured luminous intensity. It is generally described as a power function with an exponent of 0.5 ("Stevens' Power Law), but this is applicable to point light sources only. The paper by Stevens and Stevens considered both point sources (for which perceived brightness and luminous intensity are the appropriate metrics) and area sources (for which perceived *lightness* and luminance are the appropriate metrics). The paper listed 0.33 as the exponent for area sources, but the actual value likely depends on the  subtended angle of the source in the visual field and the contrast ratio between the source and the background luminance.

2. The *human* perception of the brightness of colors is even more complex, as it depends on the chromaticity (perceived color) of the light source, its angular subtense within the field of view, and the background luminance and color. Saturated purple colors, for example, can appear to be as much as ten times brighter than their measured luminance compared to D65 white.

3. Sheep. Trichromatic human color vision is dependent on three specific opsins (photosensitive proteins) in the retinal cones. There are genetic variations in these opsins result in slightly different spectral responses -- some people (mostly women) have two variants of the green-sensitive opsins that results in them having tetrachromatic vision, with the resultant ability of having extraordinary capabilities of distinguishing green hues. Colorimetry is a mathematically exact science, but it is based on a "standard observer" that is only approximately true for individual observers.

There is no guarantee that sheep or their ungulate cousins have the same opsins in their retinal cones, or even trichromatic vision. Wild mice, for example, have two opsins that are sensitive to green and ultraviolet light. -- they cannot see red colors at all. In fact, this paper:

"Jacobs. G. H. et al., "1998. Photopigment Basis for Dichromatic Color Vision in Cows, Goats, and Sheep," Vis. Neuroscience 15(3):581-584.

indicates that sheep (Ovis aries) have blue and green opsins that are very similar to human opsins, but lack the red opsin.

This is actually good news for your research goal in that it implies that the color vision of sheep is similar to that of color-blind humans with protanopia. You should be able to find sufficient research on this condition to understand how such people perceive color. You might even find a test subject (one percent of males are protanopes) to validate your choice of colors.

But then sheep are sheep. Unlike humans, they have rectangular pupils that likely result in their being able to resolve vertical movement much more clearly than horizontal movement. How they will react to different colors and in general perceive the world compared to humans is an open question.

I agree with Ian's detailed comments above and want to add one more complicating factor, sheep do not have a uniform distribution of photoceptors across the retina specifically they have specialization in the horizontal and anteroinferior visual fields.

Shinozaki, Aya, et al. "Topography of ganglion cells and photoreceptors in the sheep retina." Journal of Comparative Neurology 518.12 (2010): 2305-2315.

This would necessitate a controlled presentation of the stimulus to a specific retinal area.

Dear Luis,

I have only done research on humans. In vision research on human color perception, perception of gray shades is a separate field.

First step (for us) is to distinguish between above mentioned

(1) brightness, for example shades coming from your computer screen (i.e. emitted light) and

(2) lightness, the shades you see on paper (i.e. reflected light, a composite of light source intensity and the surface reflectance).

(http://www.cs.tau.ac.il/~hezy/Vision%20Seminar/Lightness%20Perception%20and%20Lightness%20Illusions.htm)

Visual system of humans provides you with different results when asked for lightness and brightness judgment. However, the differences does not come from retinal responses but from visual cortex, such as Brodmann area 17 (V1) (http://www.pnas.org/content/98/15/8827.full)   in charge of lightness constancy – the fact that you see the same shade despite the change in illumination (http://www.sciencedirect.com/science/article/pii/S0960982207013826). Obviously there is no brightness constancy as that is the percept of illumination itself (i.e. emitted light).

The reason for my long intro is because Stevens reports his exponents from research much like mine (12 students, behavioural exp, perception judgments).

I am not sure how generalizable is this result to sheep retina, given the differences mentioned in this fantastic response by Ian Ashdown.

Normally, it is nonlinear. It depends on your research side about white/black and color/white.

For example, for human eyes, it is complicated, because your eyes/brain and my eyes/brain are not similar. So if human is the center of your research, it is naturally nonlinear.

Thank you for your contributions. It was as complicated as I expected, and more. I think I'll use other kind of ambiguous stimuli, like the position of a light (or a shadow, maybe?) in an horizontal bar, for example.

Perhaps the more important question is, what is the goal of your research? If you only need an ambiguous but quantifiable stimulus that your test subjects will respond to, two distinctive sounds played at the same time with differing relative volumes may suffice. (See https://en.wikipedia.org/wiki/Stevens'_power_law for an overview of perceived versus measured stimuli -- I would expect that sheep and primates have approximately the same response to audio stimuli.)

Dear Luis,

as we can read above things are not so easy from a scientific point of view. I will nevertheless try a practical answer. First I need to make some assumptions. When you mention grey values of 0.25, 0.5 and 0.75, I assume this is the amount of reflection of a surface you want to use as the distal stimulus. Otherwise, if it is the luma value noted in a graphics or video file, you need to consider the luminance transfer function of the electronic display you want to employ, where the gamma parameter has a role. Furthermore you start talking about grey, but later use the word color. I assume you are refering to achromatic colors.

Now for the practical side. If you want greys for sheep without deeper knowledge about their visual system your best bet is to present them with cyan colored stimuli, which - given what has been stated by others above - should look fairly desaturated (almost grey) to them. For producing the shades we can resort to one characteristic of visual pathways, i.e. compression, due to which the felt brightness of a color is roughly proportional to a power function of the stimulus intensity. The exact form of the function might be different, for example s-shaped, and its parameters might be dependent on the viewing situation and on whether the animal is viewing direct or reflected light, but what you can exploit is the fact that you are always dealing with a negatively accelerated function, which can be approximated by some power function, if you are well above the perception shreshold.

This said, your hands-on solution could be to produce a sequence of stimuli in logarithmically equal steps. You may need a light meter for that. It does not matter in this case that such a meter has been made to measure what humans perceive. 

You want to make (say) cardboards with sheep brightness resonses R1, R2, R3, R4, R5, where R3 is halfway between R5 and R1, R2 is halfway between R3 and R1, and R4 is halfway between R5 and R3. Start with producing the stimuli with light intensities S1 for R1 and S5 for R5 with cyan colors in the desired intensities. Make sure the hues don't change too much. It is best too make S5 first and to add black for producing S1.

Your intermediate shades then need to have the following light intensities:

Si = exp [log S1 + (i-1)*0.25*(log S5 -log S1)] for i=2,3,4

My guess is such stimuli would be good enough for a couple of experiments with sheep like testing which of two conditioned responses to S1 and S5 would occur with which frequency given S2,S3, S4.

Hope that helped and that I am not too much mistaken.

Cheers

Detlef