Reply to letter to editor by M. J. Sankeralli and K. T. Mullen published in Vision Research, 41, 53–55: Lights and neural responses do not depend on choice of color space

Sankeralli and Mullen (2001) argue that the relationship between lights and neural mechanisms depends on which color space is used. In particular, they dispute what they term the ‘orthogonality property’, according to which modulation of lights orthogonal to the direction of a linear mechanism is invisible to the mechanism. They maintain that the orthogonality property is not valid for any arbitrarily chosen color space. They claim, ‘that this orthogonality property was originally applied to a cone contrast space ... and ... is valid only for this space or any orthonormal transformation of this space.’ We disagree with their claim. Lights and neural responses do not depend on the color space used to represent the data. The orthogonality property is not a relationship between lights, but a relationship between lights and visual mechanisms. That a modulation of lights is invisible to a particular detection mechanism does not depend on the color space used to represent this relationship. A mathematical formulation can help to clarify this relationship. Color spaces used to represent human vision are constructed to give the same coordinates to lights that have the same appearance. A (linear) color mechanism is a (linear) functional defined on the coordinates of this space. The space of linear mechanisms is not the color space, itself, but its dual (Krantz, 1975; Knoblauch, 1995). A linear transformation of the color space in which lights are described induces a related but different transformation of the dual space of chromatic detection mechanisms (Kay, 1998; Lipschutz, 1968). The transformation of the dual space is represented by a matrix that is the transposed inverse of the matrix that represents the transformation of the space of coordinates of lights. The orthogonality between a chromatic stimulus and a detection mechanism is maintained when one takes care to transform both the color space and its dual space properly. Let us examine this in more detail. Suppose that a particular chromatic modulation is represented by a column vector = [ 1, 2, 3] and that a linear detection mechanism is represented by a row vector m= [m1, m2, m3]. Suppose further that the chromatic modulation is orthogonal to the mechanism (and so is invisible). This can be expressed by setting the dot product of the two vectors equal to zero.