A modified Helmholtz line-element in brightness-colour space

The non-Riemannian nature of perceptual color space

For over 100 y, the scientific community has adhered to a paradigm, introduced by Riemann and furthered by Helmholtz and Schrodinger, where perceptual color space is a three-dimensional Riemannian space. This implies that the distance between two colors is the length of the shortest path that connects them. We show that a Riemannian metric overestimates the perception of large color differences because large color differences are perceived as less than the sum of small differences. This effect, called diminishing returns, cannot exist in a Riemannian geometry. Consequently, we need to adapt how we model color differences, as the current standard, ΔE, recognized by the International Commission for Weights and Measures, does not account for diminishing returns in color difference perception.

The scientific community generally agrees on the theory, introduced by Riemann and furthered by Helmholtz and Schrödinger, that perceived color space is not Euclidean but rather, a three-dimensional Riemannian space. We show that the principle of diminishing returns applies to human color perception. This means that large color differences cannot be derived by adding a series of small steps, and therefore, perceptual color space cannot be described by a Riemannian geometry. This finding is inconsistent with the current approaches to modeling perceptual color space. Therefore, the assumed shape of color space requires a paradigm shift. Consequences of this apply to color metrics that are currently used in image and video processing, color mapping, and the paint and textile industries. These metrics are valid only for small differences. Rethinking them outside of a Riemannian setting could provide a path to extending them to large differences. This finding further hints at the existence of a second-order Weber–Fechner law describing perceived differences.

Angular dependence of polarisation contrast sensitivity in octopus

Cephalopod photoreceptors are polarisation-sensitive, giving them an ability to discriminate between lights of different angle and degree of polarisation. While colour vision is achieved by comparison of signals of photoreceptors tuned to different parts of light spectra, polarisation vision is achieved by comparison of signals of photoreceptors tuned to different orientations of e-vector. Therefore, from a theoretical point of view, polarisation vision is similar to colour vision. In particular, detection of polarised light against an unpolarised background is analogous to detection of chromatic light against grey. The dependence of polarisation contrast sensitivity on the angle of polarisation can be theoretically predicted using a receptor noise limited model in much the same way as it has been done for predicting the shape of the increment threshold spectral sensitivity in animals with colour vision. Here we report angular dependence of polarisation contrast sensitivity in octopus (O. tetricus, Gould 1852) and compare the theoretical predictions of polarisation contrast with the experimental results. Polarisation gratings were generated using LCD screens with removed polarisers and the orientation of polarisation was changed by rotating the screen. Reaction to the stimulus was recorded using a fixation reflex. We show that, in agreement with the theoretical predictions, the maximum contrast sensitivity is achieved at horizontal and vertical orientations of polarisation. Our results demonstrate that the dependence of polarisation contrast sensitivity on the angle of polarisation can be analysed in the same way as the dependence of colour thresholds on wavelength of monochromatic light added to a grey background.

Disease recognition in philodendron leaf using image processing technique

Numerous disease recognition techniques are available to identify diseases in plant leaves. Assignment of spherical polar coordinate treated equivalent to hue, saturation, and intensity helps for disease recognition in Philodendron leaf which was identified as specks. Black vision, white vision, and color vision for the eye are possible with photopigments present in rods and cones in the retina. The highlight of this paper is converting the Philodendron leaf in natural color to grayscale and applying the technique of hue, saturation, and value to the gray image. Then running iteration for the double-sized image by allowing for the simultaneous recognition of the diseased part helps for the identification of the spots present in the leaf. This focuses specks on a brighter scale.

From lower to higher colour metrics: a historical account

BACKGROUND

'Lower colour metrics' describes the laws of colour mixture as manifest in trichromatic colour space and best known in its two-dimensional projection, the chromaticity diagram. 'Higher colour metrics' describes how distance in this colour space translates into perceptual difference. It is higher in the sense that it builds on the fundamentals of lower colour metrics.

METHODS

A historical account is given of the development of higher colour metrics, with many ups and downs, since Helmholtz started it at the end of the 19th Century.

RESULTS

Despite long periods of silence, Helmholtz's basic ideas have survived by successfully extended modelling, which could also account for seemingly paradoxical effects of luminance and saturation on colour discrimination.

CONCLUSION

The subject theme, which presently is at a low tide of interest, deserves the renewed interest of colour vision researchers.

Physiological modeling for detecting degree of perception of a color-deficient person

Physiological modeling of retina plays a vital role in the development of high-performance image processing methods to produce better visual perception. People with normal vision have an ability to discern different colors. The situation is different in the case of people with color blindness. The aim of this work is to develop a human visual system model for detecting the level of perception of people with red, green and blue deficiency by considering properties like luminance, spatial and temporal frequencies. Simulation results show that in the photoreceptor, outer plexiform and inner plexiform layers, the energy and intensity level of the red, green and blue component for a normal person is proved to be significantly higher than for dichromats. The proposed method explains with appropriate results that red and blue color blindness people could not perceive red and blue color completely.

Wavelength Discrimination in Drosophila Suggests a Role of Rhodopsin 1 in Color Vision

Among the five photoreceptor opsins in the eye of Drosophila, Rhodopsin 1 (Rh1) is expressed in the six outer photoreceptors. In a previous study that combined behavioral genetics with computational modeling, we demonstrated that flies can use the signals from Rh1 for color vision. Here, we provide an in-depth computational analysis of wildtype Drosophila wavelength discrimination specifically considering the consequences of different choices of computations in the preprocessing of the behavioral data. The results support the conclusion that Drosophila wavelength discrimination behavior can best be explained by a contribution of Rh1. These findings are corroborated by results of an information-theoretical analysis that shows that Rh1 provides information for discrimination of natural reflectance spectra.

Is discrimination enhanced at a category boundary? The case of unique red

Is chromatic discrimination enhanced at the boundary between different hues? In previous studies, we gave a positive answer for the case of the locus of unique blues and yellows, the boundary that divides color space into reddish and greenish hues. But we did not find enhancement at the locus of unique green, the boundary between yellowish and bluish hues. In the present study, we examined discrimination near the locus of unique red. In interleaved experimental runs, we obtained (1) discrimination thresholds using a four-alternative spatial forced choice and (2) phenomenological judgments of the locus of unique red. When measurements were made along lines parallel to the locus of unique blues and yellows in a MacLeod-Boynton diagram, the locus of minimal thresholds coincided approximately with the locus of unique red; however, this was not the case when measurements were made along lines orthogonal to the locus of unique blues and yellows. To account for these and earlier results, we suppose that the neural channel that determines the discrimination threshold will sometimes coincide with the channel that determines the perceptual hue equilibrium and sometimes will not. If a given point in chromaticity space is a unique hue, then it is expected to remain a unique hue independently of the direction in which measurements are made; however, discrimination thresholds almost certainly will depend on different underlying channels when measurements are made in different directions through the same point in chromaticity space.

Is discrimination enhanced at the boundaries of perceptual categories? A negative case

The human visual system imposes discrete perceptual categories on the continuous input space that is represented by the ratios of excitations of the cones in the retina. Is discrimination enhanced at the boundaries between perceptual hues, in the way that discrimination may be enhanced at the boundaries between speech sounds in hearing? In the chromaticity diagram, the locus of unique green separates colours that appear yellowish from those that appear bluish. Using a two-alternative spatial forced choice and an adapting field equivalent to the Daylight Illuminant D65, we measured chromatic discrimination along lines orthogonal to the locus of unique green. In experimental runs interleaved with these performance measurements, we obtained estimates of the phenomenological boundary from the same observers. No enhancement of objectively measured discrimination was observed at the category boundary between yellowish and bluish hues. Instead, thresholds were minimal at chromaticities where the ratio of long-wave to middle-wave cone excitation was the same as that for the background adapting field.

Hyperbolic geometry for colour metrics

It is well established from both colour difference and colour order perpectives that the colour space cannot be Euclidean. In spite of this, most colour spaces still in use today are Euclidean, and the best Euclidean colour metrics are performing comparably to state-of-the-art non-Euclidean metrics. In this paper, it is shown that a transformation from Euclidean to hyperbolic geometry (i.e., constant negative curvature) for the chromatic plane can significantly improve the performance of Euclidean colour metrics to the point where they are statistically significantly better than state-of-the-art non-Euclidean metrics on standard data sets. The resulting hyperbolic geometry nicely models both qualitatively and quantitatively the hue super-importance phenomenon observed in colour order systems.

Neurogeometry of color vision

In neurogeometry, principles of differential geometry and neuron dynamics are used to model the representation of forms in the primary visual cortex, V1. This approach is well-suited for explaining the perception of illusory contours such as Kanizsa's figure (see Petitot (2008) for a review). In its current version, neurogeometry uses achromatic inputs to the visual system as the starting-point for form estimation. Here we ask how neurogeometry operates when the input is chromatic as in color vision. We propose that even when considering only the perception of form, the random nature of the cone mosaic must be taken into account. The main challenge for neurogeometry is to explain how achromatic information could be estimated from the sparse chromatic sampling provided by the cone mosaic. This article also discusses the non-linearity involved in a neural geometry for chromatic processing. We present empirical results on color discrimination to illustrate the geometric complexity for the discrimination contour when the adaptation state of the observer is not conditioned. The underlying non-linear geometry must conciliate both mosaic sampling and regulation of visual information in the visual system.

A spectral theory of color perception

The paper adopts the philosophical stance that colors are real and can be identified with spectral models based on the photoreceptor signals. A statistical setting represents spectral profiles as probability density functions. This permits the use of analytic tools from the field of information geometry to determine a new kind of color space and structure deriving therefrom. In particular, the metric of the color space is shown to be the Fisher information matrix. A maximum entropy technique for spectral modeling is proposed that takes into account measurement noise. Theoretical predictions provided by our approach are compared with empirical colorfulness and color similarity data.

Sensory recoding via neural synchronization: integrating hue and luminance into chromatic brightness and saturation

If neural spike trains carry information in the frequency and timing of the spikes, then neural interactions--such as oscillatory synchronization--that alter spike frequency and timing can alter the encoded information. Using coupled oscillator theory, we show that synchronization-based processing can be used to integrate sensory information, resulting in new second-order sensory percepts signaled by the compromise frequency of the coupled system. If the signals to be coupled are nonlinearly compressed, the coupled system behaves as if it signals the product or ratio of the uncoupled signals, e.g., chromatic brightness can be signaled by the compromise frequency of coupled neurons responding to hue and luminance, and chromatic saturation can be signaled by the coupled frequency of neurons responding to hue and brightness, with a power- (Stevens's) law scaling like that observed psychophysically. These emergent properties of coupled sensory systems are intriguing because multiplicative processing and power-law scaling are fundamental aspects of sensory processing.

Colour thresholds and receptor noise: behaviour and physiology compared

Photoreceptor noise sets an absolute limit for the accuracy of colour discrimination. We compared colour thresholds in the honeybee (Apis mellifera) with this limit. Bees were trained to discriminate an achromatic stimulus from monochromatic lights of various wavelengths as a function of their intensity. Signal-to-noise ratios were measured by intracellular recordings in the three spectral types of photoreceptor cells. To model thresholds we assumed that discrimination was mediated by opponent mechanisms whose performance was limited by receptor noise. Most of the behavioural thresholds were close to those predicted from receptor signal-to-noise ratios, suggesting that colour discrimination in honeybees is affected by photoreceptor noise. Some of the thresholds were lower than this theoretical limit, which indicates summation of photoreceptor cell signals.

An n-dimensional Weber Law and the Corresponding Fechner Law

Weber's law of 1834, DeltaS/S=c for the just noticeable difference (jnd), can be written as S+DeltaS=kS, k=1+c. It follows that the stimulus decrement required to elicit one jnd of sensation is S-DeltaS*=k(-1)S. If generalized for two stimulus dimensions and two corresponding response dimensions, Weber's law would have to state such equations for all directions of change in the plane. A two-dimensional Weber law with exactly these properties is realized by [S(x)+DeltaS(x)(straight theta), S(y)+DeltaS(y)(straight theta)]=[k(sin(straight theta))S(x), k(cos(straight theta))S(y)] which determines the stimulus coordinates for all stimuli just noticeably different from the stimulus (S(x), S(y)) in all directions 0</=straight theta</=2pi. Fechner's problem now is understood as finding a transformation of the plane which maps the set of stimuli one jnd apart from the standard stimulus onto a unit circle around the standard stimulus' image. This transformation (R(2)(+)-->R(2)) is [x, y]mapsto[log(k)(x), log(k)(y)]. The solution is generalized to arbitrarily many dimensions by substituting the sin and cos in the generalized Weber law by the standard coordinates of a unit vector. Copyright 2000 Academic Press.

Receptor noise as a determinant of colour thresholds

Inferences about mechanisms at one particular stage of a visual pathway may be made from psychophysical thresholds only if the noise at the stage in question dominates that in the others. Spectral sensitivities, measured under bright conditions, for di-, tri-, and tetrachromatic eyes from a range of animals can be modelled by assuming that thresholds are set by colour opponency mechanisms whose performance is limited by photoreceptor noise, the achromatic signal being disregarded. Noise in the opponency channels themselves is therefore not statistically independent, and it is not possible to infer anything more about the channels from psychophysical thresholds. As well as giving insight into mechanisms of vision, the model predicts the performance of colour vision in animals where physiological and anatomical data on the eye are available, but there are no direct measurements of perceptual thresholds. The model, therefore, is widely applicable to comparative studies of eye design and visual ecology.

Optimum probabilistic processing in colour perception. II. Colour vision as template matching

A statistical approach to account for psychophysical phenomena in human colour vision is presented. The central visual processor is viewed as an optimum recognizer of stochastic patterns supplied by the periphery. The processor makes an optimum estimate of the spectral parameters of the stimulus, given the wavelength filter characteristics of the periphery, the stochastic nature of the information and an internal template to which the external stimulus is matched. The estimate is constrained in ways inferred from empirical phenomena. Subjective brightness of mono­chromatic stimuli and related constant brightness manifolds in the colour space constitute the constraint for brightness estimation. Results analogous and in accord with those of earlier line element theories are obtained. The Bezold-Brücke hue shift constitutes the basic constraint for hue estimation. The hue estimate involves interrelation between the fields in the experiment. Similarities and differences both in basic conceptions and results introduced by the template matching notions are discussed.

Spectral-luminosity functions, scalar linearity, and chromatic adaptation

We report data for three experiments that assess the effect on the luminosity function of chromatic adaptation arising from the measurement stimuli. First, we report spectral-sensitivity functions (wavelength range, 510-640 nm) measured by heterochromatic flicker photometry for a luminance range of 25-5000 Td. The data were fitted to a linear combination of cone fundamentals. The data narrowed and the fits deteriorated with an increase in luminance level, which indicates that at high luminances chromatic adaptation that is dependent on the spectral composition of the standard and test lights is a factor in spectral-luminosity determination. Second, we report heterochromatic modulation photometry as measured with two spectral lights at constant time-averaged chromaticity and luminance for luminances from 1.6 to 1300 Td. For a time-averaged chromaticity of 570 nm, the red-green ratio of the photometric match was invariant with luminance. For a time-averaged chromaticity of 605 nm, the red-green ratio increased by almost 0.3 log unit for a 2-log-unit increase in luminance, which is indicative of chromatic adaptation to the 605-nm chromaticity. Third, we measured flicker increment detection (wavelength range, 510-640 nm) on 570- and 605-nm backgrounds of 25-5000 Td. The data were fitted to a linear combination of cone fundamentals and showed good fits at all luminances. Fits to the 570-nm-background data set showed little variation in the proportions of the cone fundamentals with luminance. Fits to the 605-nm-background data set required an increased weighting of the middle-wavelength-sensitive cone with luminance. These three experiments indicate that luminance-dependent variation in the spectral-luminosity function as assessed by flicker techniques is caused primarily by chromatic adaptation to the measurement stimuli.

Theory of wavelength discrimination in tritanopia

Many theories of color discrimination predict a discontinuity in the wavelength-discrimination function of a tritanope at the point in the spectrum at which the rate of change of the visual signal constrained to an equiluminant plane passes through zero (near 460 nm). The predicted discontinuity follows from the use of a first-order approximation for which the reciprocal of the slope of the response function that generates the visual signal is proportional to the discrimination limen. In view of the good discrimination shown by such observers elsewhere in the spectrum, however, such a singularity is impossible. I show that the inclusion of the higher-order terms produces a finite value in the 460-nm region that falls in the range of values from the literature that have been obtained experimentally.

Detection mechanisms in L-, M-, and S-cone contrast space

Detection thresholds were obtained for a 2 degrees Gaussian-blurred spot flashed for 200 ms on an 8.9 degrees white adapting field of 1070 trolands. The spot's contrast was represented in an L-, M-, and S-cone contrast space. Detection thresholds were obtained for many vectors close to specific but theoretically important planes within this space. A three-dimensional surface was fitted to the data generated by the probability summation of three mechanisms, each a weighted sum of cone contrasts. The fit revealed a red-green chromatic mechanism driven by delta L/L--delta M/M with no S-cone input that was 1 order of magnitude more sensitive than the two other mechanisms. The latter consisted of a luminance mechanism with little S-cone input and a blue-yellow chromatic mechanism with the S cone opposed to L and M cones.

Similarity of normalized discrimination ellipses in the constant-luminance chromaticity plane

Discrimination steps were measured for three subjects, along oblique axes passing through nine points in a 25 td constant-luminance chromaticity plane. When plotted in a normalized cone-excitation chromaticity diagram, the best-fitting discrimination ellipses for a given subject have approximately the same shape and orientation regardless of the reference chromaticity. Their orientation is consistent with the hypothesis that excitation of B-cones affects the red-green opponent balance, otherwise determined by R- and G-cone excitations, in a manner independent of initial cone-excitation levels. The CIELAB formula predicts an orientation for normalized ellipses in agreement with the data, but it also predicts systematic changes in the ratio of minor to major axes which are not observed experimentally.

The pi mechanisms of W S Stiles: an historical review

The origins, development, and status of the pi mechanism theory are reviewed. The paper is divided into four sections. In the first section Stiles's general ideas about 'color mechanisms' are examined, and it is concluded that foremost amongst these is a mathematical theory that specifies certain formal rules or laws that should govern a certain class of observations. In the case of pi mechanisms, the class of observations is that of two-color thresholds, and the defining laws are the two well-known displacement laws. Five other laws that two-color increment-threshold observations should obey, if the latter are governed by ideal pi mechanisms, are abstracted from Stiles's writings. In the second section literature pertinent to the testing of the seven Stilesian laws is reviewed, and it is asked whether or not the seven pi mechanisms of Stiles do in fact obey the laws. In the third section the relation of the pi mechanism concept to physiological concepts is examined, and its relation to the 'cone fundamental' is discussed; the evidence pertinent to the question: "Are any of the pi mechanisms of the single-fundamental type?" is then reviewed. The last section is devoted to the evolution of Stiles's ideas in the period after 1959 when Stiles's own investigations and those of others propelled him to reject the initial (1953) pi mechanism theory as an adequate characterization of the data of the two-color threshold.

Modified line-element theory for spatial-frequency and width discrimination

Recent data from several laboratories have shown that spatial-frequency discrimination is not a smooth function of frequency but rather exhibits alternate peaks and troughs. A model for spatial-frequency discrimination analogous to line-element models for color discrimination is presented here and shown to provide a reasonable fit to the available data. This model is based on the predicted responses of six spatial-frequency-tuned mechanisms, whose sensitivity curves have been estimated in previously published masking experiments. In order to fit the data it is necessary to pool responses from units centered under the stimulus as well as from spatially neighboring units. Thus it appears that the visual system utilizes both spatial and spatial-frequency information in discrimination tasks.

Spatial and temporal discrimination ellipsoids in color space

Three-dimensional discrimination ellipsoids are presented for a number of representative points in color space. These ellipsoids have been obtained not with the conventional split field but with flickering grating patterns. Thus our study extends the well-known results of Brown and MacAdam [J. Opt. Soc. Am. 39, 808-813 (1949)] to cases in which the image is structured in space and time. As expected, we find that the discrimination ellipsoids depend on the spatiotemporal structure of the stimulus. This has potential consequences for color-difference formulas as used in industry and commerce: no single formula will do when it is important to treat patterns with different structure. We present analytical descriptions, based on the Vos-Walraven [Vision Res. 12, 1327-1365 (1972)] line element augmented with spatiotemporal frequency-dependent coefficients that fit our results reasonably well. For coarse gratings (approximately 1 cycle per degree) or slowly modulated fields (approximately 1 Hz) our results prove to be compatible with the results of Brown and MacAdam obtained with a bipartite 2 degree field.

Sensitivity to spatiotemporal combined luminance and chromaticity contrast

Contrast-detection thresholds for various combinations of chromaticity and luminance differences were obtained for spatiotemporal square-wave modulation of a yellow field. The results are expressed in terms of excitation of the Vos-Walraven R,G primaries. For every spatiotemporal frequency the thresholds can be approximated by an ellipse in the red-green plane. Large variations were found in the orientation, magnitude, and eccentricity of the discrimination ellipses. It seems that a simple threshold function appears to be sufficient to describe the experimental data. Although the eye does not perceive hue contrast for high spatial frequencies, its sensitivity is not governed mainly by summation of the red and green channels.

A general zone theory of color and brightness vision. I. Basic formulation

A general theory of color and brightness vision, developed from basic principles of the Helmholtz and Hering points-of-view on color vision is presented in a general mathematical form suitable for quantitative analysis. Visual sensation is described by a vector expressed in terms of Hering-like elements for color and brightness which underlie in their spatial-temporal variations the perceptions of form and change. The photic stimulus of vision is recognized to act first and only through photoabsorption producing a Helmholtz-like vector of quantum absorptions. The physiological transformation of the Helmholtz photochemical excitations into the Hering sensation responses is represented as a vector of general operators. The result is a mathematical framework encompassing traditional psychophysical and sensory scaling experiments. The theory is utilized to demonstrate that for many traditional (Class A) psychophysical observations, the physiological operator reduces to a linear (matrix) transformation. For static, uniform, focal stimulation, this reduction is seen to be the basis for earlier specific linear models of color vision. We also illustrate that static intensity-level effects (Bezold-Brücke hue shifts, unique hue invariance) can be modeled from the theory by power, but not logarithmic, intensity-level dependence for the sensation elements.

Measurement Structures and Psychological Laws

Empirical laws psychology may be based on physical measurements (for example, voltages, times), counting, ordering, or just classifying. It is a pointless, though widespread practice to use a physical measure or a count as a "definition" of a psychological variable; this practice obscures the fact that all one has done is measured a physical variable, or counted. What is important are the empirical laws that are established by use of such quantitative or qualitative observations. Some kinds of empirical relations and laws yield measurement structures, akin to the qualitative structures underlying fundamental measurement in physics. Measurement structures are empirical structures that can be described most simply by introduction of a new numerical function; such a function is a new measure, and is typically interpreted as measuring some particular psychological variable Measurement structures, formulated abstractly, sometimes provide valuable tools for formulating new empirical hypotheses to be tested; but in many instances, other kinds of theory may be more appropriate. The main focus of research ought always to be the discovery of simple laws; these may or may not lead to new measures.