Dimensionality of the Perceptual Space of Achromatic Colors Nora Umbach Research Methods and Mathematical Psychology February 2011 Color perception Stimuli Fechnerian Scaling Analysis Outline Achromatic color perception Stimulus configurations Fechnerian Scaling Analysis of data 2 | Nora Umbach Color perception Stimuli Fechnerian Scaling Analysis Achromatic color perception Stimulus configurations Fechnerian Scaling Analysis of data 3 | Nora Umbach Color perception Stimuli Fechnerian Scaling Analysis Color perception • We have a tendency to treat color as a property of objects • Experienced color is neither a property of objects, nor a property of light • The physical or physiological quantifications of color do not fully explain the psychophysical perception of color appearance 4 | Nora Umbach
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Dimensionality of the Perceptual Space ofAchromatic Colors
Nora UmbachResearch Methods and Mathematical Psychology
February 2011
Color perception Stimuli Fechnerian Scaling Analysis
Outline
Achromatic color perception
Stimulus configurations
Fechnerian Scaling
Analysis of data
2 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Achromatic color perception
Stimulus configurations
Fechnerian Scaling
Analysis of data
3 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Color perception
• We have a tendency to treat color as a property of objects
• Experienced color is neither a property of objects, nor aproperty of light
• The physical or physiological quantifications of color do notfully explain the psychophysical perception of color appearance
4 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Color perception
• We have a tendency to treat color as a property of objects
• Experienced color is neither a property of objects, nor aproperty of light
• The physical or physiological quantifications of color do notfully explain the psychophysical perception of color appearance
• In this talk we will only focus on achromatic colors
4 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Dimensionality of the perceptual space of achromatic colors
• Traditional view assumes that achromatic color perceptionmay be represented by a unidimensional achromatic colorspace (ranging from white to black)
• Logvinenko & Maloney (2006) and Niederee (2010) presentrecent evidence that this representation is at leasttwo-dimensional
• Up to now there is no systematic investigation of the structureof the perceptual space of achromatic colors
• Our experiments aim at a characterization of the perceptualspace of achromatic colors for individual observers
5 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Demonstration
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Color perception Stimuli Fechnerian Scaling Analysis
Demonstration
6 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Ratio principle
ratio principle: Lightness is determined merely by the luminanceratio between a given surface and its surround, without referenceto the level of illumination. (Gilchrist, 2006, p. 82)
• Prominent explanation of experimental results where subjectshad to match two centers presented in different surrounds(postulated by Wallach, 1948)
• Ratio principle postulates that centers will be adjusted untilratio between center and surround is (nearly) identical forboth configurations
• Infields will then be perceived as metameric (being of thesame color)
7 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Achromatic color perception
Stimulus configurations
Fechnerian Scaling
Analysis of data
8 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Stimulus presentation
(a,s) (b,t)
x
9 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Stimulus configurations
36.57 41.02 45.92 51.29 57.07
17.22
19.76
22.48
25.70
28.88
Infield ( cd
m2 )
Surround ( cd
m2 )
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Color perception Stimuli Fechnerian Scaling Analysis
Stimuli
gray1a gray2a gray3a gray4a gray5a
gray1b gray2b gray3b gray4b gray5b
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Color perception Stimuli Fechnerian Scaling Analysis
Stimuli
gray1a gray2a gray3a gray4a gray5a
gray1b gray2b gray3b gray4b gray5b
11 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Achromatic color perception
Stimulus configurations
Fechnerian Scaling
Analysis of data
12 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Probability Distance Hypothesis
The probability-distance hypothesis states that the probabilitywith which one stimulus is discriminated from another is a functionof some subjective distance between these stimuli. (Dzhafarov,2002, p. 352)
ψ(x , y) = f [D(x , y)]
13 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Discrimination Probabilities
• Most basic cognitive ability: to tell two stimuli apart fromeach other
• Fechnerian Scaling computes ‘subjective’ distances amongstimuli from their pairwise discrimination probabilities
• Subjects are required to give one of two answers: ‘x and y are
the same’ or ‘x and y are different’
ψ(x , y) = P(x and y are different)
• FS is suitable to describe spaces of arbitrary dimensionality
14 | Nora Umbach
Color perception Stimuli Fechnerian Scaling Analysis
Subjective Distances
• Subjective distances between stimuli are defined here,measuring the degree of similarity (or dissimilarity) betweenthe underlying representations
• Fechnerian distances satisfy all properties of a metric:
D(x , y) ≥ 0 non-negativity (1)
D(x , y) = 0 iff x = y identity of indiscernibles (2)
Thank you References Additional slides Fechnerian Scaling
References
Dzhafarov, E. N. (2002). Multidimensional Fechnerian Scaling:Probability-Distance Hypothesis. Journal of Mathematical Psychology,46, 352–374.
Gilchrist, A. (2006). Seeing Black and White. Oxford: University Press.
Logvinenko, A. D. & Maloney, L. T. (2006). The proximity structure ofachromatic surface colors and the impossibility of asymmetric lightnessmatching. Perception and Psychophysics, 68(1), 76–83.
Niederee, R. (2010). More than three dimensions: What continuityconsiderations can tell us about perceived color. In J. Cohen &M. Matthen (Eds.), Color Ontology and Color Science (pp. 91–122).MIT Press.
Wallach, H. (1948). Brightness Constancy and the Nature of AchromaticColors. Journal of Experimental Psychology, 38(3), 310.
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Thank you References Additional slides Fechnerian Scaling
Additional slides
Fechnerian Scaling
27 | Nora Umbach
Thank you References Additional slides Fechnerian Scaling
Thank you References Additional slides Fechnerian Scaling
Psychometric Increments
• We define psychometric increments for each observation area
φ(1) = ψ(x , y)− ψ(x , x)
φ(2) = ψ(y , x)− ψ(x , x)
• Due to regular minimality all psychometric increments arepositive
• Minima ψ(x , x) can have different values (nonconstantself-dissimilarity)
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Thank you References Additional slides Fechnerian Scaling
Discrete Object Space
In a discrete space Fechnerian computations are performed bytaking sums of psychometric increments for all possible chainsleading from one point to another (3 examples shown here).
36 | Nora Umbach
Thank you References Additional slides Fechnerian Scaling
Oriented Fechnerian Distance
• Consider a chain from si to sj , with k ≥ 2
• Psychometric length of the first kind
L(1)(x1, x2, ..., xk) =k∑
m=1
φ(1)(xm, xm+1)
• Finite number of psychometric lengths across all possiblechains connecting si and sj
• Oriented Fechnerian distance:
G1(si , sj) = L(1)min(si , sj)
• Satisfies all properties of a metric except symmetry
• Oriented distances are not computed across observation areasbut rather within observation areas
37 | Nora Umbach
Thank you References Additional slides Fechnerian Scaling
Fechnerian Distance
• For better interpretation we add up all oriented Fechneriandistances from si to sj and from sj to si
• Overall Fechnerian distance
G (si , sj) = G1(si , sj) + G1(sj , si ) = G2(si , sj) + G2(sj , si )
• Satisfies all properties of a metric
• Does not depend on observation area
• Gives us a readily interpretable measure of the ‘subjective’distance between si and sj